representation of groups in group theory

Group Representation

Two representations are considered equivalent if they are similar . For example, performing similarity transformations of the above matrices by

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Subject classifications

The Structure of Groups

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.

The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

• 1.1 Permutation groups
• 1.2 Matrix groups
• 1.3 Transformation groups
• 1.4 Abstract groups
• 1.5 Groups with additional structure
• 2.1 Finite group theory
• 2.2 Representation of groups
• 2.3 Lie theory
• 2.4 Combinatorial and geometric group theory
• 3 Connection of groups and symmetry
• 4.1 Galois theory
• 4.2 Algebraic topology
• 4.3 Algebraic geometry
• 4.4 Algebraic number theory
• 4.5 Harmonic analysis
• 4.6 Combinatorics
• 4.8 Physics
• 4.9 Chemistry and materials science
• 4.10 Cryptography
• 6 Licensing

Main classes of groups

The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.

Permutation groups

The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations ) that is closed under compositions and inverses, G is a group acting on X . If X consists of n elements and G consists of all permutations, G is the symmetric group S n ; in general, any permutation group G is a subgroup of the symmetric group of X . An early construction due to Cayley exhibited any group as a permutation group, acting on itself ( X = G ) by means of the left regular representation.

In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5, the alternating group A n is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals.

Matrix groups

The next important class of groups is given by matrix groups , or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the n -dimensional vector space K n by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G .

Transformation groups

Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space X preserving its inherent structure. In the case of permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure.

The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.

Abstract groups

Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations ,

A significant source of abstract groups is given by the construction of a factor group , or quotient group, G / H , of a group G by a normal subgroup H . Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X , the factor group G / H is no longer acting on X ; but the idea of an abstract group permits one not to worry about this discrepancy.

The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.

Groups with additional structure

An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion),

are compatible with this structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then G is a topological group, a Lie group, or an algebraic group.

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G , the geometry and analysis pertaining to G yield important results about Γ . A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p -adic analytic group G has a family of quotients which are finite p -groups of various orders, and properties of G translate into the properties of its finite quotients.

Branches of group theory

Finite group theory.

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.

During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

Representation of groups

Saying that a group G acts on a set X means that every element of G defines a bijective map on the set X in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism:

where GL( V ) consists of the invertible linear transformations of V . In other words, to every group element g is assigned an automorphism ρ ( g ) such that ρ ( g ) ∘ ρ ( h ) = ρ ( gh ) for any h in G .

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group G : often, the group operation in G is abstractly given, but via ρ , it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V (via Schur's lemma).

Given a group G , representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1 , acting on the L 2 -space of periodic functions.

A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

Combinatorial and geometric group theory

Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free.

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X , for example a compact manifold, then G is quasi-isometric (i.e. looks similar from a distance) to the space X .

Connection of groups and symmetry

Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example

• If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups.
• If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X .
• If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example.

The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.

Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.

The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

Applications of group theory

Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.

Galois theory

Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S 5 , the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.

Algebraic topology

Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.

Algebraic geometry

Algebraic geometry likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.

Algebraic number theory

Algebraic number theory makes uses of groups for some important applications. For example, Euler's product formula,

captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem.

Harmonic analysis

Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.

Combinatorics

In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. Transformational theory models musical transformations as elements of a mathematical group.

In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group.

Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution.

Chemistry and materials science

In chemistry and materials science, point groups are used to classify regular polyhedra, and the symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals.

Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule.

In chemistry, there are five important symmetry operations. They are identity operation ( E) , rotation operation or proper rotation ( C n ), reflection operation ( σ ), inversion ( i ) and rotation reflection operation or improper rotation ( S n ). The identity operation ( E ) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis ( C n ) consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/ n , where n is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, n = 2, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the C n axis having the largest value of n is the highest order rotation axis or principal axis. For example in boron trifluoride (BF 3 ), the highest order of rotation axis is C 3 , so the principal axis of rotation is C 3 .

In the reflection operation ( σ ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σ h (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical ( σ v ) or dihedral ( σ d ).

Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, methane and other tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation ( S n ) requires rotation of  360°/ n , followed by reflection through a plane perpendicular to the axis of rotation.

Cryptography

Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.

Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry.

Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory.

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

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Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $\mathrm{Mod}_{g,n}$ of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if \[\rho: \pi_1(\Sigma_{g,n})\to \mathrm{GL}_r(\mathbb{C})\]is a representation whose conjugacy class has finite orbit under $\mathrm{Mod}_{g,n}$, and $r \lt\sqrt{g+1}$, then $\rho$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.

Aaron Landesman. Daniel Litt. "Canonical representations of surface groups." Ann. of Math. (2) 199 (2) 823 - 897, March 2024. https://doi.org/10.4007/annals.2024.199.2.6

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Digital Object Identifier: 10.4007/annals.2024.199.2.6

Subjects: Primary: 11G99 , 14C30 , 14H10 , 34M56 , 57K20

Keywords: Character varieties , Hodge theory , mapping class groups , non-Abelian Hodge theory , surface groups

Rights: Copyright © 2024 Department of Mathematics, Princeton University

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Relative assembly maps and the K -theory of Hecke algebras in prime characteristic

• Open access
• Published: 23 August 2024

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• W. Lück   ORCID: orcid.org/0000-0003-1141-1422 1  

We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic K -theory of twisted group rings of a group G with coefficients in a regular ring R or, more generally, with coefficients in a regular additive category. They are known to be isomorphisms rationally. We show that it suffices to invert only those primes p for which G contains a non-trivial finite p -group and p is not invertible in R . The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group F after localizing at p in terms of the p -subgroups of F using Verschiebungs and Frobenius operators. We construct and exploit the structure of a module over the ring of big Witt vectors on the Nil-terms. We analyze the algebraic K -theory of the Hecke algebras of subgroups of reductive p -adic groups in prime characteristic.

Avoid common mistakes on your manuscript.

1 Introduction

We first state and discuss the main results of this paper. In this introduction we only consider rings as coefficients for simplicity. Many of the results will extend to additive categories as coefficients. Moreover, in all cases one can allow a twisting by a G -action on R or a G -action on the additive category. Groups are understood to be discrete, unless explicitly stated otherwise.

1.1 On the K -theoretic relative assembly map from \({{{\mathcal {F}}}\textrm{in}}\) to \({{{\mathcal {V}}}\textrm{cyc}}\) for regular coefficient rings

If \(\mathcal {P}\) is a set of primes and \(f :A \rightarrow B\) is a homomorphisms of abelian groups, we call f a \(\mathcal {P}\) - isomorphism if the map \({{\,\textrm{id}\,}}_{\mathbb {Z}[\mathcal {P}^{-1}]} \otimes _ {\mathbb {Z}} f :\mathbb {Z}[\mathcal {P}^{-1}] \otimes _{\mathbb {Z}} A \rightarrow \mathbb {Z}[\mathcal {P}^{-1}] \otimes _{\mathbb {Z}} B\) is bijective, where the ring \(\mathbb {Z}[\mathcal {P}^{-1}]\) satisfies \(\mathbb {Z}\subseteq \mathbb {Z}[\mathcal {P}^{-1}] \subseteq \mathbb {Q}\) and is obtained from \(\mathbb {Z}\) by inverting all primes in \(\mathcal {P}\) .

Notation 1.1

For a group G and a ring R , let \(\mathcal {P}(G,R)\) be the set of primes, which are not invertible in R and for which G contains a non-trivial finite p -subgroup.

If G is torsionfree or, more generally, the order of any finite subgroup of G is invertible in R , then \(\mathcal {P}(G,R)\) is empty and \(\mathcal {P}\) -isomorphism means just isomorphism. If \(R = \mathbb {Z}\) , then \(\mathcal {P}(G,\mathbb {Z})\) is the set of primes p for which G contains an element of order p .

Theorem 1.2

Let R be a regular ring coming with a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(R)\) to the group of ring automorphisms of R . Then the relative assembly map

is a \(\mathcal {P}(G,R)\) -isomorphism for all \(n \in \mathbb {Z}\) .

Theorem  1.2 improves [ 28 , Theorem 0.3], where \(\mathcal {P}(G,R)\) is required to be the set of all primes. See also [ 20 , Theorem 5.11] for \(R = \mathbb {Z}\) . If the order of any finite subgroup of G is invertible in R and the action \(\rho \) is trivial, then Theorem  1.2 has already been proved in [ 26 , Proposition 2.6 on page 686]. Theorem  1.2 is a special case of Theorem  9.1 . Note that the relative assembly map appearing in Theorem  1.2 is always split injective, see [ 10 , Theorem 1.3], [ 28 , Theorem 0.1].

For more information about the relative assembly map appearing in Theorem  1.2 and the K -theoretic Farrell–Jones Conjecture, we refer to Remark  6.4 and [ 25 ].

( The Full Farrell–Jones Conjecture ) The Full Farrell–Jones Conjecture is stated in [ 25 , Conjecture 13.27]. Here we only need to know that it implies that the assembly map

is bijective for all \(n \in \mathbb {Z}\) and any ring R coming with a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(R)\) , see [ 25 , Theorem 13.61 (i)].

Note that the Full Farrell–Jones Conjecture is known to be true for a large class of groups including hyperbolic groups, CAT(0)-groups, lattices in locally compact second countable Hausdorff groups, and fundamental groups of manifolds of dimension \(\le 3\) and has useful inheritance properties, e.g., passing to subgroups and overgroups of finite index, see for instance [ 25 , Chapter 15].

Theorem 1.5

Suppose G satisfies the Full Farrell–Jones Conjecture, Let R be a regular ring coming with a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(R)\) such that the order of any finite subgroup of G is invertible in R .

Then the canonical map

is an isomorphism and

where \(R_{\rho }[G]\) denotes the \(\rho \) -twisted group ring.

1.2 On the K -theoretic relative assembly map from \({{{\mathcal {F}}}\textrm{in}}\) to \({{{\mathcal {V}}}\textrm{cyc}}\) for Artinian coefficient rings

In the case that R is an Artinian ring, we get even an integral result in degree \(n \le 0\) without the assumption that the order of any finite subgroup of G is invertible in R .

Theorem 1.6

Let G be a discrete group which satisfies the Full Farrell–Jones Conjecture. Let R be an Artinian ring coming with a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(R)\) .

Then the canoncial map

This is proved in [ 25 , Theorem 13.61 (v)] for trivial \(\rho \) . It directly extends to the case where \(\rho \) is non-trivial. \(\square \)

1.3 On the K -theoretic relative assembly map from \({{{\mathcal {F}}}\textrm{in}}\) to \({{{\mathcal {V}}}\textrm{cyc}}\) for fields as coefficients

For the reader’s convenience we summarize what happens in the special case, where R is a skew-field F . Note that a skew-field of characteristic zero is regular and satisfies \(\mathcal {P}(G,F) = \emptyset \) and any skew-field is an Artinian ring. Hence we conclude from Theorems  1.2 , 1.5 , and 1.6

Theorem 1.7

Let G be a group and F be a skew-field coming with a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(F)\) into the group of field automorphism of F . Then:

The relative assembly map

is bijective for every \(n \in \mathbb {Z}\) if one of the following condition holds:

F has characteristic zero;

There exists a prime p such that F has characteristic p and G contains no non-trivial finite p -group;

If p is a prime and F is a skew-field of characteristic p , then the map

is bijective for every \(n \in \mathbb {Z}\) ;

The canonical map

is bijective if G satisfies the Full Farrell–Jones Conjecture;

We have \(K_n(F_{\rho }[G])\) for \(n \le -1\) , if G satisfies the Full Farrell–Jones Conjecture.

1.4 On the K -theoretic relative assembly map from \({{{\mathcal {F}}}\textrm{in}}\) to \({{{\mathcal {V}}}\textrm{cyc}}\) for \(\mathbb {Z}\) as coefficients

For the reader’s convenience we summarize what is known in the special case where R is the ring \(\mathbb {Z}\) of integers.

Theorem 1.8

Let G be a group. Then

is a \(\mathcal {P}(G;\mathbb {Z})\) -isomorphism for every \(n \in \mathbb {Z}\) , where \(\mathcal {P}(G,\mathbb {Z})\) is the set of primes p , for which G contains a non-trivial finite p -group;

is bijective and \(K_n(\mathbb {Z}[G]) = 0\) for \(n \le -2\) if G satisfies the Full Farrell–Jones Conjecture;

Suppose that G is torsionfree and satisfies the Full Farrell–Jones Conjecture. Then \(K_n(\mathbb {Z}G)\) for \(n \le -1\) , the reduced projective class group \(\widetilde{K}_0(\mathbb {Z}G)\) , and the Whitehead group \({{\,\textrm{Wh}\,}}(G)\) are trivial.

(i) This follows directly from Theorem  1.2 .

(ii) See [ 25 , Theorem 13.61 (vi)].

(iii) See [ 25 , Theorem 13.61 (iii) and (iv)]. \(\square \)

1.5 Totally disconnected groups

So far G has been a discrete group. Now we want to deal with td-groups, i.e., locally compact second countable totally disconnected topological Hausdorff groups, and the algebraic K -theory of their Hecke algebras. In some special cases or in a weaker form, we extend the main results of [ 5 ] from characteristic zero to prime characteristic.

Let R be a (not necessarily commutative) ring. We will need the following assumption to make sense of the notion of a Hecke algebra. It is taken from [ 14 , page 9].

Assumption 1.9

There exists a compact open subgroup \(\widetilde{U}\) of G such that for any compact open subgroup \(\widetilde{U}'\subseteq \widetilde{U}\) of \(\widetilde{U}\) the index \([\widetilde{U}: \widetilde{U}']\) is invertible in R .

This assumption is automatically satisfied if G is discrete, since then we can take \(\widetilde{U} = \{1\}\) , or if \(\mathbb {Q}\subseteq R\) . If p is a prime number which is invertible in R , then Assumption  1.9 is satisfied for any subgroup of a reductive p -adic group G by [ 30 , Lemma 1.1] and Lemma  11.2 .

Suppose that Assumption  1.9 is satisfied. Then the Hecke algebra \(\mathcal {H}(G;R)\) is defined as the algebra of locally constant functions \(G \rightarrow R\) with compact support and multiplication given by convolution, see for instance [ 7 , Section 11].

One may consider the K -groups \(K_n(\mathcal {H}(G;R))\) of \(\mathcal {H}(G;R)\) for \(n \in \mathbb {Z}\) . There is an assembly map

Next we want to explain what we can say in the case, where the condition that R is uniformly regular and satisfies \(\mathbb {Q}\subseteq R\) is weakened to the condition that R is uniformly regular and only certain primes have to be invertible in R or to the condition that \(N \cdot 1_R = 0\) holds in R for some natural number N .

Theorem 1.12

Let p be a prime. Assume that G is modulo a compact subgroup isomorphic to a closed subgroup of a reductive p -adic group. Let N be a natural number and let R a be ring with unit \(1_R\) .

Suppose that \(N \cdot 1_R = 0\) and that Assumption  1.9 is satisfied. Then the assembly map ( 1.10 ) induces an isomorphism

for every \(n \in \mathbb {Z}\) ;

Suppose that \(N \cdot 1_R = 0\) , and that R is Artinian, e.g., R is a field of prime characteristic q for \(p \not = q\) and we take \(N = q\) . Suppose that Assumption  1.9 is satisfied. Then

and the map induced by ( 1.11 )

is bijective;

Suppose that R is uniformly regular and for any two compact open subgroups \(U_0\) and \(U_1\) of G with \(U_0 \subseteq U_1\) the index \([U_1: U_0]\) is invertible in R . Then:

Assumption  1.9 is satisfied;

The map ( 1.10 ) is bijective for all \(n \in \mathbb {Z}\) ;

We have \(K_n (\mathcal {H}(G;R))= 0\) for \(n \le 1\) ;

The canoncial map

is bijective.

One can define more general Hecke algebras \(\mathcal {H}(G,R,\rho ,\omega )\) allowing a G -action \(\rho \) on R and central character \(\omega \) and everything carries over to this more general setting, see Remark  11.6 .

The proof of Theorem  1.12 will be given in Sect.  11.4 and is based on a version of the Farrell–Jones Conjecture for totally disconnected groups with categories with G -support as coefficients.

1.6 On the twisted Nil-terms of finite groups

The proof of some of the results above relies on the following theorem.

Let F be a finite group and \(\alpha :F \rightarrow F\) be a group automorphism. Let \(F \rtimes _{\alpha } \mathbb {Z}\) be the semidirect product associated to \(\alpha \) , where for the standard generator \(t \in \mathbb {Z}\) we have \(tft^{-1} = \alpha (f)\) for \(f \in F\) . Let R be a ring coming with a group homomorphism \(\mu :F \rtimes _{\alpha } \mathbb {Z}\rightarrow {{\,\textrm{aut}\,}}(R)\) . Let \(\rho :F \rightarrow {{\,\textrm{aut}\,}}(R)\) be the restriction of \(\mu \) to F . Our goal is to get information about the structure of the Nil-groups

with respect to the ring automorphism \(\psi :R_{\rho }[F] \xrightarrow {\cong } R_{\rho }[F]\) sending \(r\cdot f\) for \(r \in R\) and \(f \in F\) to \(\mu (t)(r) \cdot \alpha (f)\) , where \(\overline{K}_{n-1}({{\,\textrm{Nil}\,}}(R_{\rho }[F],\Psi ))\) is defined in Notation  4.6 taking \(\mathcal {A}= \underline{R_{\rho }[F]}\) for the \(\mathbb {Z}\) -category \( \underline{R_{\rho }[F]}\) , which has precisely one object and whose \(\mathbb {Z}\) -module of endomorphisms is \(R_{\rho }[F]\) . These Nil-groups appear in the twisted Bass-Heller-Swan decomposition for \(R_{\mu }[F \rtimes _{\alpha }\mathbb {Z}] = (R_{\rho }[F])_{\psi }[\mathbb {Z}]\) , see ( 4.5 ).

Fix a prime number p . Let \(T_p\) be the set of triples ( P ,  k ,  y ) consisting of a p -subgroup P of F , an integer k with \(k \ge 1\) , and an element \(y \in F\) such that \(c_y \circ \alpha ^k(P) = P\) holds for the automorphism \(c_y :F \rightarrow F\) sending z to \(yzy^{-1}\) . Let \(\psi _{(P,k,y)} :R_{\rho |_P}[P] \xrightarrow {\cong } R_{\rho |_P}[P]\) be the ring automorphism sending \(r \cdot p\) for \(r \in R\) and \(p \in P\) to \(\mu (yt^k)(r) \cdot c_y \circ \alpha ^k(p)\) .

Given a triple \((P,k,y) \in T_p\) , define a functor of Nil-categories

by sending an object in \({{\,\textrm{Nil}\,}}(R[P],\psi |_P)\) given by a nilpotent R [ P ]-endomorphism \(\varphi :(c_y \circ \alpha ^k)_* Q = RP \otimes _{c_y \circ \alpha ^k} Q \rightarrow Q\) for a finitely generated projective R [ P ]-module Q to the object in \({{\,\textrm{Nil}\,}}(R[F],\alpha ^k)\) given by the nilpotent R [ F ]-endomorphism \((\alpha ^k)_* \bigl (R[F] \otimes _{R[P]} Q\bigr ) = RF \otimes _{\alpha ^k} \bigl (R[F] \otimes _{R[P]} Q\bigr ) \rightarrow R[F] \otimes _{R[P]} Q\) sending \(f_0 \otimes (f_1 \otimes q)\) to \((\alpha ^{-k}(f_0 ) f_1t^{-k}y^{-1}) \otimes \varphi (1 \otimes q)\) . It induces for every \(n \in \mathbb {Z}\) a homomorphism

be the homomorphism induced by the Verschiebungs operator \(V_k\) , see ( 3.2 ).

Theorem 1.13

The homomorphism

is surjective for every \(n \in \mathbb {Z}\) , where the subscript ( p ) stands for localization at the prime p .

The untwisted version of Theorem  1.13 , i.e., \(\alpha = {{\,\textrm{id}\,}}_F\) and trivial \(\mu \) , appears already in [ 21 , Theorem A].

Note that this does not mean that the Nil-groups are computable after localizing at p by p -subgroups groups, since the maps \(\gamma (P,k,y)_n\) are not given just by induction with the inclusion \(P \rightarrow F\) . One can check by inspecting Lemmas  7.6 and 8.11 that the Nil-groups are computable by p -elementary groups.

Corollary 1.14

Let R be a regular ring. Then \(\mathbb {Z}[\mathcal {P}(F,R)^{-1}] \otimes _{\mathbb {Z}} N\!K_n(R[F],\alpha )\) vanishes for \(\mathcal {P}(F,R)\) defined in Notation  1.1 .

We mention that the second Nil-group of \(F_2[\mathbb {Z}/2]\) is non-trivial, see [ 40 ]. So one needs to invert certain primes in Corollary  1.14 .

On the other hand, given a prime p , we get \(N\!K_n(\mathbb {Z}[\mathbb {Z}/p])=0\) for \(n \le 1\) , see [ 11 , Theorem 10.6 on page 695], [ 12 , 25 , Theorem 6.21]. So Theorem  1.13 implies that for a finite group G , for which \(p^2\) does not divide the order of G , we have \(N\!K_n(\mathbb {Z}G)_{(p)} = 0\) for \(n \le 1\) . As an application we get a new proof of the result of Harmon [ 22 ] that \(N\!K _n(\mathbb {Z}G) = 0\) for \(n \le 1\) if the order of G is square-free.

Remark 1.15

We mention without giving the details that the proof appearing in [ 25 , Theorem 6.21] can be generalized to the twisted setting showing that Harmon’s result extends to twisted group rings and that in Theorem  1.13 the terms \(N\!K_n(R[P],c_y \circ \alpha ^k)_{(p)}\) vanish if \(|P| \le p\) , \(R= \mathbb {Z}\) , and \(n \le 1\) hold. This implies that for a group G for which the order of any finite subgroup is squarefree the relative assembly map

is an isomorphism for every \(n \in \mathbb {Z}\) with \(n \le 1\) . If G satisfies the Full Farrell–Jones Conjecture and the order of any finite subgroup is squarefree, then the assembly map

is an isomorphism for every \(n \in \mathbb {Z}\) with \(n \le 1\) . Examples for such G are given by extensions \(1 \rightarrow \mathbb {Z}^n \rightarrow G \rightarrow \mathbb {Z}/m \rightarrow 1\) for a squarefree natural number m .

Remark 1.16

( K-theory of stable \(\infty \) -categories ) One may ask whether the results of this paper can be extended from the K -theory of additive categories to the K -theory of stable \(\infty \) -categories. For the extension of the statement and proofs in some cases of the Full Farrell–Jones Conjecture, we refer to Bunke–Kasprowski–Winges [ 15 ]. Dominik Kirstein and Christian Kremer are working on a twisted Bass–Heller–Swan decomposition in this setting generalizing [ 29 , 33 ]. Efimov has announced that the Nil-terms are modules over TR on the spectrum level, which would yield a module structure of Nil-groups over the ring of big Witt vectors also for stable \(\infty \) -categories. However, algebraic K -theory for additive categories can be viewed as a Mackey functor over the Green functor given by the Swan ring, see Sect.  5.4 . This is very unlikely to be the case for the algebraic K -theory of stable \(\infty \) -categories, where an A -theoretic version of the Swan group is needed, see [ 39 ]. Therefore the induction theorems, which we use here for instance to prove Theorem  1.13 , are not available in the setting of stable \(\infty \) -categories. It is completely unclear in the setting of stable \(\infty \) -categories whether Theorems  1.2 or 1.13 are still true and how one can formulate the statement of Theorem  1.12 .

2 Basics about (additive) \(\Lambda \) -categories

If a \(\Lambda \) -category comes with an appropriate notion of a finite direct sums, it is called an additive \(\Lambda \) -category . An additive G - \(\Lambda \) -category is an additive \(\Lambda \) -category, which comes with a G -action by automorphisms of additive \(\Lambda \) -categories. If \(\Lambda \) is \(\mathbb {Z}\) , we often omit \(\Lambda \) and talk just about an additive category or additive G -category.

Note that a \(\Lambda \) -category and an additive \(\Lambda \) -category respectively is in particular a \(\mathbb {Z}\) -category and an additive category respectively thanks to the canonical ring homomorphism \(\mathbb {Z}\rightarrow \Lambda \) .

One can assign to an additive category \(\mathcal {A}\) its non-connective K -theory spectrum \(\textbf{K}(\mathcal {A})\) . We denote \(K_n(\mathcal {A}) = \pi _n(\textbf{K}(\mathcal {A}))\) for \(n \in \mathbb {Z}\) .

Given a ring R , define \(\underline{R}\) to be the \(\mathbb {Z}\) -category, which has precisely one object \(*_R\) and whose \(\mathbb {Z}\) -module of endomorphisms is R . Composition is given by the multiplication in R .

For a ring R let \(K_n(R)\) for \(n \in \mathbb {Z}\) be its n -algebraic K -group, which can be defined for instance as the (non-connective) K -theory of the exact category of finitely generated projective R -module. It can be identified with \(K_n(\underline{R}_{\oplus })\) . Note that we do not have to pass to the idempotent completion of \(\underline{R}_{\oplus }\) , as we are working with non-connective K -theory.

All these classical notions are summarized with references to the relevant papers in [ 4 , Section 2 and 3].

We fix some conventions concerning matrices of morphisms. For an object A in \(\mathcal {A}\) we denote by \(A^m\) the direct sum \(\bigoplus _{i=1}^m A\) . For two finite direct sums \(\bigoplus _{i= 1}^m A_i\) and \(\bigoplus _{j= 1}^n B_j\) a morphism \(U :\bigoplus _{i= 1}^m A_i \rightarrow \bigoplus _{j= 1}^n B_j\) is the same as a ( m ,  n )-matrix

of morphisms \(u_{i,j} :A_i \rightarrow B_j\) . One may think of an element in \(\bigoplus _{i= 1}^m A_i\) as a (1,  m )-matrix

and thinks of the image of this element under U as the (1,  n )-matrix

Note that m is the number of columns and n is the number of rows with these conventions. If \(V :\bigoplus _{j= 1}^n B_j \rightarrow \bigoplus _{k= 1}^o C_k\) is another morphism given by the ( n ,  o )-matrix V , then the composite \(V \circ U :\bigoplus _{i= 1}^m A_i \rightarrow \bigoplus _{k= 1}^o C_k\) is given by the ( m ,  o ) matrix W whose ( i ,  k )-entry is

So one could think of W as a kind of product of matrices \(V \cdot U\) .

3 Frobenius and Verschiebungs operators

Let \(\mathcal {A}\) be an additive category and \(\Phi \) be an automorphism of \(\mathcal {A}\) .

Definition 3.1

( Nilpotent morphisms and Nil-categories )

A morphism \(\varphi :\Phi (A)\rightarrow A\) of \(\mathcal {A}\) is called \(\Phi \) - nilpotent , if for some \(n \ge 1 \) the n -fold composite

is trivial;

The category \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) has as objects pairs \((A, \varphi )\) , where \(\varphi :\Phi (A)\rightarrow A\) is a \(\Phi \) -nilpotent morphism in \(\mathcal {A}\) . A morphism from \((A, \varphi )\) to \((A', \varphi ')\) is a morphism \(u:A\rightarrow A'\) in \(\mathcal {A}\) such that the diagram

is commutative.

The category \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) inherits the structure of an exact category from \(\mathcal {A}\) , a sequence in \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) is declared to be exact if the underlying sequence in \(\mathcal {A}\) is (split) exact.

Next we define for \(k \in \{1,2, \ldots \}\) the Verschiebung operator \(V_k\) and the Frobenius operator \(F_k\)

Given an object \((A, \varphi )\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi ^k)\) , define \(V_k(A, \varphi )\) to be the object in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) that is given by the object \(\bigoplus _{i =0}^{k-1} \Phi ^i(A)\) in \(\mathcal {A}\) together with the \(\Phi \) -nilpotent morphism

A morphisms \(u :(A, \varphi ) \rightarrow (A', \varphi ')\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi ^k)\) given by a morphism \(u :A \rightarrow A'\) in \(\mathcal {A}\) is sent to the morphism \(V_k(A,\varphi ) \rightarrow V_k(A',\varphi ')\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) given by the morphism \(\bigoplus _{i =0}^{k-1} \Phi ^i(u) :\bigoplus _{i =0}^{k-1} \Phi ^i(A) \rightarrow \bigoplus _{i =0}^{k-1} \Phi ^i(A')\) .

Given an object \((A, \varphi )\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) , define \(F_k(A, \varphi )\) to be \((A, \varphi ^{(k)})\) , see Definition  3.1 . A morphism \(u :(A,\varphi ) \rightarrow (A',\varphi ')\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi )\) given by a morphism \(u :A \rightarrow A'\) in \(\mathcal {A}\) is sent to the morphism \(u :(A,\varphi ^{(k)}) \rightarrow (A',\varphi '^{(k)})\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi ^k)\) given by u again.

The elementary proof of the next lemma is left to the reader.

The composite

sends an object \((A,\varphi )\) to the object given by

It sends a morphism \(u :(A,\varphi ) \rightarrow (A',\varphi ')\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi ^k)\) given by a morphism \(u :A \rightarrow A'\) in \(\mathcal {A}\) to the morphism \(F_k \circ V_k(A,\varphi ) \rightarrow F_k \circ V_k(A',\varphi ')\) in \({{\,\textrm{Nil}\,}}(\mathcal {A}, \Phi ^k)\) given by the morphism \(\bigoplus _{i= 0}^{k-1} \Phi ^i(u) :\bigoplus _{i= 0}^{k-1} \Phi ^i(A) \rightarrow \bigoplus _{i= 0}^{k-1} \Phi ^i(A')\) in \(\mathcal {A}\) .

4 Frobenius and Verschiebungs operators and induction and restriction

Let \(\mathcal {A}\) be an additive category with an action \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(\mathcal {A})\) of the (discrete) group G by automorphisms of additive categories. Then we obtain a new additive category

as follows. The set of objects of \(\mathcal {A}_{\rho }[G]\) is the set of objects of \(\mathcal {A}\) . A morphism \(f :A \rightarrow A'\) in \(\mathcal {A}_{\rho }[G]\) is a finite formal \(\sum _{g \in G} (f_g :gA \rightarrow A') \cdot g\) , where \(f_g :gA \rightarrow A'\) is a morphism in \(\mathcal {A}\) from gA to \(A'\) and finite means that for only finitely many elements g in G the morphism \(f_g\) is different from the zero-homomorphism. If \(f' :A' \rightarrow A''\) is a morphism in \(\mathcal {A}_{\rho }[G]\) given by the finite formal sum \(\sum _{g' \in G} (f'_{g'} :g'A' \rightarrow A'') \cdot g'\) , then define their composite \(f' \circ f :A \rightarrow A''\) by the finite formal sum

If R is a unital ring coming with a G -action \(\rho _R :G \rightarrow {{\,\textrm{aut}\,}}(R)\) and we take \(\mathcal {A}\) to be the category of finitely generated free R -modules with the obvious G -action \(\rho :G \rightarrow {{\,\textrm{aut}\,}}(\mathcal {A})\) coming from \(\rho _R\) by induction, then \(\mathcal {A}_{\rho }[G]\) is equivalent to the additive category of finitely generated free modules over the twisted group ring \(R_{\rho }[G]\) .

If \(\Phi :\mathcal {A}\xrightarrow {\cong } \mathcal {A}\) is an automorphism of an additive category \(\mathcal {A}\) , then we define

for the \(\mathbb {Z}\) -action \(\rho _{\Phi } :\mathbb {Z}\rightarrow {{\,\textrm{aut}\,}}(\mathcal {A}), \; n \mapsto \Phi ^n\) .

Let \(i_k :\mathbb {Z}\rightarrow \mathbb {Z}\) be the group homomorphism given by multiplication with k . Next we define functors of additive categories

The functor \((i_k)_*\) sends an object A in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) , which is given by an object A in \(\mathcal {A}\) , to the object in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) given by A again. Consider a morphism

in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) . It is sent by \(( i_k)_*\) to the morphism

in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) .

The functor \(i_k^*\) sends a object A in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) , which is given by an object A in \(\mathcal {A}\) , to the object \(i_k^*(A)\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) given by the object \(\bigoplus _{i = 0}^{k-1} \Phi ^i(A)\) in \(\mathcal {A}\) . Consider a morphism \( f = \sum _{l \in \mathbb {Z}} \bigl (f_l :\Phi ^{l}(A) \rightarrow A'\bigr ) \cdot t^l :A \rightarrow A'\) in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) . It is sent by \(i_k^*\) to the morphism

in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) defined as follows. By additivity we have only to specify \(i_k^*(f_l \cdot t^l)\) . For this purpose we have to define for every \(i,j \in \{0,1,\ldots , (k-1)\}\) a morphisms \(i_k^*(f_l \cdot t^l)_{i,j} :\Phi ^i(A) \rightarrow \Phi ^j(A')\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) . It is given by \(\bigl (\Phi ^j(f_l) :\Phi ^{i+mk}(A) \rightarrow \Phi ^j(A')\bigr ) \cdot t^m\) if there exists an integer m with \(i+mk +l =j\) , and by zero otherwise.

If \(\mathcal {A}\) is given by \(\underline{R}\) for a ring R coming with a ring automorphism \(\Phi :R \xrightarrow {\cong } R\) , \((i_k)_*\) and \(i_k^*\) corresponds to induction and restriction with respect to the change of ring homomorphism of twisted group rings \(R_{\Phi ^k}[\mathbb {Z}] \rightarrow R_{\Phi }[\mathbb {Z}]\) associated to \(i_k\) .

In the sequel we use the notation of [ 29 ]. We get by taking homotopy groups from [ 29 , Theorem 0.1] for \(n \in \mathbb {Z}\) an isomorphism

Here \(\textbf{T}_{\textbf{K}(\Phi ^{-1})}\) is the mapping torus of the map induced on non-connective K -theory spectra \(\textbf{K}(\Phi ^{-1}) :\textbf{K}(\mathcal {A}) \rightarrow \textbf{K}(\mathcal {A})\) . There is a long exact Wang sequence

where \(\textbf{j}:\textbf{K}(\mathcal {A}) \rightarrow \textbf{T}_{\textbf{K}(\Phi ^{-1})} \) is the inclusion. If \(\Phi = {{\,\textrm{id}\,}}_{\mathcal {A}}\) , this boils down to an isomorphism

We define \(K_{n}({{\,\textrm{Nil}\,}}(A,\Phi ^k)) = \pi _n(\textbf{K}_{{{\,\textrm{Nil}\,}}}^{\infty }(\mathcal {A},\Phi ^k))\) for \(n \in \mathbb {Z}\) , where the non-connective K -theory spectrum \( \textbf{K}_{{{\,\textrm{Nil}\,}}}^{\infty }(\mathcal {A},\Phi )\) is constructed in [ 27 , Remark 6.3 and Lemma 6.5]. It is likely but we have no detailed proof that the group \(K_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) can be identified for \(n \le -1\) with the K -groups associated to the exact category \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) in the sense of Schlichting [ 34 ] see [ 27 , Remark 6.11]. Fortunately, we do not need this identification for our purposes. There is the inclusion functor \(I :\mathcal {A}\rightarrow {{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) that sends an object A to the object ( A , 0) and the projection functor \(P :{{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ) \rightarrow \mathcal {A}\) that sends an object \((A,\varphi )\) to A . Obviously \(P \circ I = {{\,\textrm{id}\,}}_{\mathcal {A}}\) .

Notation 4.6

We define \(\overline{K}_n({{\,\textrm{Nil}\,}}(A,\Phi ))\) to be the cokernel of the split injective homomorphism \(K_n(I) :K_n(\mathcal {A}) \rightarrow K_n({{\,\textrm{Nil}\,}}(A,\Phi ))\) for \(n \in \mathbb {Z}\) .

Let \(T_{\textbf{K}(\Phi ^{-1}),k}\) be the k -fold mapping torus of \(\textbf{K}(\Phi ) :\textbf{K}(\mathcal {A}) \rightarrow \textbf{K}(\mathcal {A})\) , which is a \(\mathbb {Z}/k\) -spectrum. There is a k -fold covering \(\textbf{p}_k :T_{\textbf{K}(\Phi ^{-1}),k} \rightarrow T_{\textbf{K}(\Phi ^{-1})}\) and a homotopy equivalence \(\textbf{f}:T_{\textbf{K}(\Phi ^{-1}),k} \rightarrow \textbf{T}_{\textbf{K}(\Phi ^{-k})}\) . These correspond to the following construction on the level of spaces for a map \(\Phi :X \rightarrow X\) . Namely, \(p_k :T_{\Phi ,k} \rightarrow T_{\Phi }\) is the k -sheeted covering obtained by the pull back of the k -sheeted covering \(S^1 \rightarrow S^1\) sending z to \(z^k\) with the canonical map \(T_{\Phi } \rightarrow S^1\) . Explicitly \(T_{\Phi ,k}\) is obtained from \(\coprod _{i = 1}^{k} X \times [i-1,i]\) by identifying \((x,i) \in X \times [i-1,i]\) with \((\Phi (x),i)\) in \(X \times [i,i+1]\) for \(i = 1,2 \ldots k-1\) and \((x,k) \in X \times [k-1,k]\) with \((\Phi (x),0)\) in \(X \times [0,1]\) . Obviously \(T_{\Phi ,1} = T_{\Phi }\) . The map \(p_k :T_{\Phi ,k} \rightarrow T_{\Phi }\) sends the class of \((x, j) \in X \times [i-1,i]\) to the class of \((x,j-i-1) \in X \times [0,1]\) for \(i = 1,2, \ldots , k\) . The homotopy equivalence \(f :T_{\Phi ,k} \rightarrow T_{\Phi ^k}\) sends the class of \((x, j) \in X \times [i-1,i]\) to the class of \((\Phi ^{k-i}(x),\frac{j}{k}) \in X \times [0,1]\) for \(i = 1,2, \ldots , k\) . On homotopy groups we obtain an isomorphism

and a homomorphism induced by \(\textbf{p}_k\)

Since \(\textbf{p}_k\) is a k -sheeted covering, there is a transfer homomorphism

Note that the Frobenius and the Verschiebungs operator are functors of exact categories and hence induces homomorphism

Since \(K_n(F_k) \circ K_n(I) = K_n(I)\) and \(K_n(V_k) \circ K_n(I) = K_n(I)\) holds, they induce homomorphisms

The functors \((i_k)_*\) and \(i_k^*\) are functors of additive categories and induce homomorphisms

The main result of this section is

Theorem 4.16

Let \(k \ge 1\) be a natural number and \(\Phi :\mathcal {A}\xrightarrow {\cong } \mathcal {A}\) be an automorphism of an additive category \(\mathcal {A}\) . Then:

The following diagram commutes for \(n \in \mathbb {Z}\)

where the upper horizontal isomorphisms is the one defined in ( 4.5 ) for \(\Phi ^k\) , the lower horizontal isomorphisms is the one defined in ( 4.5 ) for \(\Phi \) , and the vertical arrows have been defined in ( 4.7 ), ( 4.8 ), ( 4.12 ), and ( 4.14 );

where the upper isomorphisms is the one defined in ( 4.5 ) for \(\Phi \) , the lower isomorphisms is the one defined in ( 4.5 ) for \(\Phi ^k\) , and the horizontal arrows have been defined in ( 4.7 ), ( 4.9 ), ( 4.13 ), and ( 4.15 ).

(i) We give only an outline of the proof and leave some details to the reader. In the sequel we use the notation of [ 29 ].

We obtain from [ 29 , Theorem 0.1 (i)] for \(n \in \mathbb {Z}\) an isomorphism

where \(N\!K_n(\mathcal {A}_{\Phi }[t^{\pm }])\) is the kernel of the map \(K_n(\mathcal {A}_{\Phi }[t^{\pm }]) \rightarrow K_n(\mathcal {A})\) coming from the functor \(\mathcal {A}_{\Phi }[t^{\pm }] \rightarrow \mathcal {A}\) given taking the coefficient of \(t^0\) . One easily checks that the functors \((i_k)_*\) and \(i_k^*\) of ( 4.3 ) and ( 4.4 ) induce homomorphisms

Then following diagram commutes for \(n \in \mathbb {Z}\)

where the upper horizontal arrow is the isomorphism ( 4.17 ) for \(\Phi ^k\) , the lower horizontal arrow is the isomorphism ( 4.17 ) for \(\Phi \) and the vertical homomorphism have been defined in ( 4.7 ), ( 4.8 ), ( 4.14 ) and ( 4.18 ). The proof of commutativity for the terms \(N\!K_n(\mathcal {A}_{\Phi ^k}[t])\) and \(N\!K_n(\mathcal {A}_{\Phi ^k}[t^{-1}])\) is obvious, the one for the term \(\pi _n(\textbf{T}_{\textbf{K}(\Phi ^{-k})})\) is left to the reader.

We obtain from [ 29 , Theorem 0.1 (ii)] for \(n \in \mathbb {Z}\) an isomorphism

and hence an isomorphism

Therefore it remains to show that the diagram

commutes for \(n \in \mathbb {Z}\) . Fix a natural number m . Then the diagram ( 4.22 ) is a retract of the corresponding diagram

where we have replaced \(\mathcal {A}\) by \(\mathcal {A}[\mathbb {Z}^m]\) and \(\Phi \) by \(\Phi [\mathbb {Z}^m]\) , see [ 27 , Lemma 2.2, Theorem 6.2, Remark 6.3]. Hence we can assume without loss of generality that \(n \ge 1 \) when proving the commutativity of ( 4.22 ), since our result shall be true for any additive category \(\mathcal {A}\) . So we can use the connective K -theory spectrum when dealing with the commutativity of ( 4.22 ). By inspecting [ 29 ] one sees that this boils down to show that the following diagram commutes for \(n \ge 1\) ,

where the upper and the lower homomorphism are induced by the functors

Next we explain the key ingredients in the proof of the commutativity of ( 4.23 ) and leave it to the reader to figure out the routine to fill in the details based on standard fact about connective K -theory such as the Additivity Theorem for Waldhausen categories.

Consider an object \(\varphi :\Phi ^k(A) \rightarrow A\) in \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ^k)\) given by a nilpotent endomorphism \(\varphi :\Phi ^k(A) \rightarrow A\) in \(\mathcal {A}\) . We have defined its Verschiebung \(V_k(\varphi )\) as an object in \({{\,\textrm{Nil}\,}}(A,\Phi )\) given by a specific nilpotent endomorphism

in \(\mathcal {A}\) , see ( 3.2 ). To it we can assign the morphism

in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) . It is given by the following ( k ,  k )-matrix

of morphisms in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) .

On the other hand we can assign to \(\varphi :\Phi ^k(A) \rightarrow A\) in \(\mathcal {A}\) the morphism in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) given

Its image under the induction functor \((i_k)_*\) is the morphism in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) given by

Consider the morphism \(w :\bigoplus _{i = 1}^k \Phi ^i(A) \xrightarrow {\cong } \bigoplus _{i = 1}^k \Phi ^i(A)\) in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) given by the ( k ,  k )-matrix

of morphisms in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) . Note that w is the same as the identity ( k ,  k )-matrix from the K -theoretic point of view by its block structure. Then the composite

is given by the ( k ,  k )-matrix

of morphisms in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) . Note that this composite looks like \({{\,\textrm{id}\,}}_A \cdot t^{-k} - \varphi \cdot t^0\) from the K -theoretic point of view by its block structure. This finishes the proof of assertion (i).

(ii) The following diagram commutes for \(n \in \mathbb {Z}\)

where the upper horizontal arrow is the isomorphism ( 4.17 ) for \(\Phi \) , the lower horizontal arrow is the isomorphism ( 4.17 ) for \(\Phi ^k\) and the vertical homomorphism have been defined in ( 4.7 ), ( 4.9 ), ( 4.15 ) and ( 4.19 ). The proof of commutativity for the terms \(N\!K_n(\mathcal {A}_{\Phi ^k}[t])\) and \(N\!K_n(\mathcal {A}_{\Phi ^k}[t^{-1}])\) is obvious, the one for the term \(\pi _n(\textbf{T}_{\textbf{K}(\Phi ^{-k})})\) is left to the reader.

commutes for \(n \in \mathbb {Z}\) . By the same argument as it appears in the commutativity of the diagram ( 4.22 ), one can show that it suffices to prove the commutativity of ( 4.26 ) for \(n \ge 1\) , or, in other words for connective K -theory. By inspecting [ 29 ] one sees that this boils down to show that the following diagram

commutes for \(n \ge 1\) .

Next we explain the key ingredients in the proof of the commutativity of ( 4.27 ) for connective K -theory and leave it to the reader to figure out the routine to fill in the details based on standard fact about K -theory such as the Additivity Theorem for Waldhausen categories.

Let \(\Phi :\mathcal {A}\xrightarrow {\cong } \mathcal {A}\) be an automorphism of an additive category. Consider a nilpotent endomorphism \(\varphi :\Phi (A) \rightarrow A\) representing an element in \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) . Consider the morphism \({{\,\textrm{id}\,}}_A \cdot t^{-1} - \varphi \cdot t^0 :\Phi (A) \rightarrow A\) in \(\mathcal {A}_{\Phi }[\mathbb {Z}]\) . The functor \(i_k^* :\mathcal {A}_{\Phi }[\mathbb {Z}] \rightarrow \mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) sends it to the morphism \(\bigoplus _{i = 1}^k \Phi ^i(A) \rightarrow \bigoplus _{j = 0}^{k-1} \Phi ^j(A)\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) that is given by the ( k ,  k )-matrix

If we apply the Frobenius \(F_k\) operator to the object \(\varphi :\Phi (A) \rightarrow A\) of \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) , we obtain the object \(\varphi ^{(k)} :\Phi ^k(A) \xrightarrow {\Phi ^{k-1}(\varphi )} \Phi ^{k-1}(A) \xrightarrow {\Phi ^{k-2}(\varphi )} \cdots \xrightarrow {\varphi } A\) of \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ^k)\) . To it we can assign the morphism \({{\,\textrm{id}\,}}_A \cdot t^{-1} - \varphi ^{(k)} \cdot t^0 :\Phi ^k(A) \rightarrow A\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) .

Consider the morphism

in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) that is given by the ( k ,  k )-matrix

of morphisms in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) . Note that from a K -theoretic point of view this morphism should give the same element in K -theory as the morphisms \({{\,\textrm{id}\,}}_A \cdot t^{-1} - \varphi ^{(k)} \cdot t^0 :\Phi ^k(A) \rightarrow A\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) , just view its special block structure.

We also have the automorphism

in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) given by the ( k ,  k )-matrix

of morphisms in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\) . Note that v is the same as the identity ( k ,  k )-matrix from the K -theoretic point of view by its block structure.

Now one easily check the equality of morphisms \(\bigoplus _{i = 1}^k \Phi ^i(A) \rightarrow \bigoplus _{i = 0}^{k} \Phi ^j(A)\) in \(\mathcal {A}_{\Phi ^k}[\mathbb {Z}]\)

Now Theorem  4.16 follows. \(\square \)

be the homomorphism coming from the Bass-Heller-Swan homomorphism, see ( 4.5 ), and the inclusion of the first Nil-term. Let

be the homomorphisms coming from the Bass–Heller Swan homomorphism, see ( 4.5 ), and the projection onto the first Nil-term. Obviously we have \(\overline{r}_n \circ \overline{s}_n = {{\,\textrm{id}\,}}\) .

Theorem  4.16 implies

Corollary 4.30

We have for every \(n \in \mathbb {Z}\) the commutative diagram

where the vertical arrows have been defined n ( 4.28 ) and the horizontal arrows have been defined in ( 4.12 ) and ( 4.14 );

where the vertical arrows have been defined in ( 4.29 ) and the horizontal arrows have been defined in ( 4.13 ) and ( 4.15 ).

Corollary  4.30 has already been proved for commutative rings as coefficients in the untwisted case by Stienstra [ 35 , Theorem 4.7].

For a natural number D , let \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )_D\) be the full subcategory of \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )\) consisting of objects \((P,\varphi )\) satisfying \(\varphi ^{(D)} = 0\) . Obviously \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ) = \bigcup _{d \ge 0 } {{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )_d\) . Hence the canonical maps

are bijective. Given an element \(z \in \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) and a natural number D , we say that z is of nilpotence degree \(\le D\) if z lies in the image of \(\overline{K_n}({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )_D)\rightarrow \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) .

The next result is known for rings as coefficient in the untwisted case, see Farrell [ 19 , Lemma 3] for \(n = 1\) , Grunewald [ 20 , Prop. 4.6], Stienstra [ 35 , p. 90], and Weibel [ 41 , p. 479].

Fix an integer \(n \in \mathbb {Z}\) Consider an element \(z \in \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) of nilpotence degree \(\le D\) .

Then the composite

sends z for every \(k \ge D\) to zero, where \(\overline{s}_n\) and \(K_n(i_k^*)\) have been defined in ( 4.28 ) and ( 4.15 ).

Since the composite \({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )_D\rightarrow {{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ) \xrightarrow {F_k} {{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ^k)\) sends an object \((P,\phi )\) to ( P , 0) for \(k \ge D\) , the composite \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )) \xrightarrow {\overline{K}_n(F_k)} \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ^k))\) sends z to zero for \(k \ge D\) . Now Lemma  4.33 follows from Corollary  4.30  (i). \(\square \)

5 Mackey and Green functors for finite groups

For the reader’s convenience we recall some basics about Mackey and Green functors and Dress induction following [ 3 , Section 2 and 3]. Throughout this section F is a finite group.

5.1 Mackey functors

and a contravariant functor

Definition 5.1

Double Coset formula

the following diagram of functors of abelian groups commutes

One easily checks that the condition Additivity implies \(M(\emptyset ) = 0\) and is equivalent to the requirement that \(M_*(i) \oplus M_*(j) :M(X) \oplus M(Y) \rightarrow M(X \amalg Y)\) is bijective, since the double coset formula implies that \({{\,\textrm{id}\,}}\times {{\,\textrm{id}\,}}:M(\emptyset ) \rightarrow M(\emptyset ) \times M(\emptyset )\) is an isomorphism and that \((M^*(i) \times M^*(j )) \circ (M_*(i) \oplus M_*(j))\) is the identity.

is a family of \(\Lambda \) -bilinear maps

5.2 Green functors

Definition 5.5.

( Green functor ) A Green functor for the finite group F with values in \(\Lambda \) -modules is a Mackey functor U together with a pairing

A (left) U -module M is a Mackey functor for the group F with values in \(\Lambda \) -modules together with a pairing

5.3 Dress induction

is surjective, where \({{\,\textrm{pr}\,}}_H :F/H \rightarrow F/F\) is the projection. The next result is a mild generalization of the fundamental work of Dress on induction, see [ 17 , 18 ].

where \({{\,\textrm{pr}\,}}_H :F/H \rightarrow F/F\) is the projection.

By assumption we can write the unit \(1_{F/F} \in \mathcal {U}(F/F)\) as the sum

for appropriate elements \(u_H \in M(F/H)\) . Now we compute using the various axioms for the structure of a Green functor and a module over a Green functor

\(\square \)

5.4 The Green functor \({{\,\textrm{Sw}\,}}_F\)

Let G be a group. Next we define the Swan ring \({{\,\textrm{Sw}\,}}(G)\) associated to G . As an abelian group \({{\,\textrm{Sw}\,}}(G)\) is defined in terms of generators and relations as follows. Generators [ M ] are given by \(\mathbb {Z}G\) -isomorphism classes of \(\mathbb {Z}G\) -modules M , which are finitely generated free as \(\mathbb {Z}\) -modules. If we have an exact sequence \(0 \rightarrow M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow 0\) of such modules, then we require the relation \([M_1] = [M_0] + [M_2]\) . The multiplication in the Swan ring is given by the tensor product over \(\mathbb {Z}\) with the diagonal G -action. Note that \({{\,\textrm{Sw}\,}}(G)\) is a commutative ring, whose unit is given by \(\mathbb {Z}\) equipped with the trivial G -action.

In [ 3 , Section 3] the definition of \({{\,\textrm{Sw}\,}}(G)\) is extended from groups to small groupoids. For a small groupoid \(\mathcal {G}\) an element in \({{\,\textrm{Sw}\,}}(\mathcal {G})\) is given by a covariant functor from \(\mathcal {G}\) to the category of finitely generated free abelian groups. Actually, \({{\,\textrm{Sw}\,}}(\mathcal {G})\) is the Grothendieck group of such functors under the obvious objectwise notion of a short exact sequence of such functors. The structure of a commutative ring comes from the tensor product of abelian groups applied objectwise. The unit is represented by the constant functor, which sends every object in \(\mathcal {G}\) to \(\mathbb {Z}\) and every morphism in \(\mathcal {G}\) to \({{\,\textrm{id}\,}}_{\mathbb {Z}}\) .

Given a G -set X , denote by \(\mathcal {T}^G(X)\) its transport groupoid . Its set of objects is X , its set of morphism from \(x_0 \rightarrow x_1\) is \(G_{x_0,x_1} = \{g \in G \mid x_1 = gx_0\}\) , and composition comes from the multiplication in G . Now we can consider \({{\,\textrm{Sw}\,}}(\mathcal {T}^G(X))\) for a G -set X . For every subgroup H of G there is an obvious isomorphism

coming from the facts that \(\mathcal {T}^G(G/H)\) is a connected groupoid and the automorphism group of eH is H .

The elementary proof that we get for a finite group F a Green functor \({{\,\textrm{Sw}\,}}^F\) , which is given for a finite F -set X by \({{\,\textrm{Sw}\,}}^F(X) = {{\,\textrm{Sw}\,}}(\mathcal {T}^F(X))\) , can be found in [ 3 , Section 3].

Let F be a finite group. Then the Green functor with values in \(\mathbb {Z}_{(p)}\) -modules \(\mathbb {Z}_{(p)} \otimes _{\mathbb {Z}} {{\,\textrm{Sw}\,}}_F\) is computable for \(\mathcal {H}_p\) .

See for instance [ 3 , Proof of Lemma 4.1 (c)] which is based on [ 38 , 6.3.3], [ 36 , Lemma 4.1] and [ 37 , Section 12]. \(\square \)

Let p be a prime and denote by \(\mathcal {H}_p\) be the family of p -hyperelementary subgroups of the finite group F . We get from Lemmas  5.6 and 5.8

holds in \(\mathcal {M}(F/F)_{(p)} = \mathbb {Z}_{(p)} \otimes _{\mathbb {Z}} \mathcal {M}(F/F)\) , where \({{\,\textrm{pr}\,}}_H :F/H \rightarrow F/F\) is the projection.

Remark 5.10

There is also a version of \({{\,\textrm{Sw}\,}}(H)\) for groups H and \({{\,\textrm{Sw}\,}}(\mathcal {G})\) for groupoids \(\mathcal {G}\) , where one replaces finitely generated free abelian group by finitely generated abelian group. These two versions are isomorphic, see [ 32 , page 890] and [ 3 , Section 3] for groupoids.

It is more convenient to work with the version for finitely generated free abelian groups, since for finitely generated free abelian groups M the functor \(M \otimes _{\mathbb {Z}} -\) is exact.

6 K -theoretic functors associated to G - \(\mathbb {Z}\) -categories

Let G be a group and let \(\Lambda \) be a commutative ring.

6.1 The K -theoretic covariant -spectrum associated to a G - \(\mathbb {Z}\) -category

Let \(\mathcal {A}\) be a G - \(\Lambda \) -category. For a G -set X we define a \(\Lambda \) -category \(\mathcal {A}(X)\) as follows. Objects are pairs ( A ,  x ) with \(A \in {{\,\textrm{ob}\,}}(\mathcal {A})\) and \(x \in X\) . A morphism \(\varphi :(A,x) \rightarrow (A',x')\) is a formal finite sum \(\varphi = \sum _{g \in G_{x,x'}} \varphi _g \cdot g\) , where \(G_{x,x'} = \{g \in G \mid x' = gx\}\) and \(\varphi _g\) is a morphism in \(\mathcal {A}\) from gA to \(A'\) . For a morphism \(\varphi ' :(A',x) \rightarrow (A'',x'')\) given by the formal finite sum \(\varphi ' = \sum _{g' \in G_{x',x''}} \varphi '_g \cdot g'\) , we define the composite \(\varphi ' \circ \varphi :(A,x) \rightarrow (A'',x'')\) by the finite formal sum \(\varphi ' \circ \varphi = \sum _{g'' \in G_{x,x''}} (\varphi ' \circ \varphi )_{g''} \cdot g''\) , where for \(g'' \in G_{x,x''}\) we put

Given a G -map \(f :X \rightarrow Y\) , define a functor of \(\Lambda \) -categories \(\mathcal {A}(f) :\mathcal {A}(X) \rightarrow \mathcal {A}(Y)\) by sending an object ( A ,  x ) to the object ( A ,  f ( x )) and a morphism \(\varphi :(A,x) \rightarrow (A',x')\) given by the formal sum \(\varphi = \sum _{g \in G_{x,x'}} \varphi _g \cdot g\) to the morphism given the same finite formal sum. This makes sense because of \(G_{x,x'} \subseteq G_{f(x),f(x')}\) . Hence we get a covariant functor

It induces the covariant functor

Let \(H_*^G(-;\textbf{K}_{\mathcal {A}})\) be the G -homology theory associated to \(\textbf{K}_{\mathcal {A}}\) . It has the property that \(H_n^G(G/H;\textbf{K}_{\mathcal {A}}) = \pi _n(\textbf{K}_{\mathcal {A}}(G/H))\) holds for any subgroup H of G . Denote by \(E_{{{{\mathcal {F}}}\textrm{in}}}(G)\) and \(E_{{{{\mathcal {V}}}\textrm{cyc}}}(G)\) respectively the classifying space for the family \({{{\mathcal {F}}}\textrm{in}}\) of finite subgroups and of the family \({{{\mathcal {V}}}\textrm{cyc}}\) of virtually cyclic subgroups of G , see for instance [ 23 ]. The projection \(E_{{{{\mathcal {V}}}\textrm{cyc}}}(G) \rightarrow G/G\) induces the assembly map for \(n \in \mathbb {Z}\)

The Farrell–Jones Conjecture predicts that it is bijective for all \(n \in \mathbb {Z}\) . If R is a ring and we take for \(\mathcal {A}\) the additive category \(\underline{R}_{\oplus }\) , then this assembly map can be identified with the assembly map  1.4 .

The canoncial map \(E_{{{{\mathcal {F}}}\textrm{in}}}(G) \rightarrow E_{{{{\mathcal {V}}}\textrm{cyc}}}(G)\) induces the relative assembly map

If R is a ring and we take for \(\mathcal {A}\) the additive category \(\underline{R}_{\oplus }\) , then this assembly map can be identified with the relative assembly map appearing in Theorem  1.2 .

For more information about assembly maps and the Farrell–Jones Conjecture we refer for instance to [ 24 , 25 , 26 ].

6.2 Restriction

For a G -map \(f :X \rightarrow Y\) , we think of \(K_n(\mathcal {A}(f)_{\oplus }) :K_n(\mathcal {A}(X)_{\oplus }) \rightarrow K_n(\mathcal {A}(Y)_{\oplus })\) as induction. We can define a kind of restriction, if we assume that \(f^{-1}(y)\) is finite for every \(y\in Y\) , as follows. Next we define a functor of \(\Lambda \) -categories

It sends an object ( A ,  y ) to \(\bigoplus _{x \in f^{-1}(y)} (A,x)\) . Consider a morphism \(\varphi :(A,y) \rightarrow (A',y')\) given by a formal sum \(\sum _{g \in G_{y,y'}} \varphi _g \cdot g\) . Then \({{\,\textrm{res}\,}}(f)(\varphi ) :\bigoplus _{x \in f^{-1}(y)} (A,x) \rightarrow \bigoplus _{x' \in f^{-1}(y')} (A',x')\) is given by the collection of morphisms \({{\,\textrm{res}\,}}(f)(\varphi )_{x,x'} :(A,x) \rightarrow (A',x')\) for \(x \in f^{-1}(y)\) and \(x' \in f^{-1}(y')\) if we put \({{\,\textrm{res}\,}}(f)(\varphi )_{x,x'} = \sum _{g \in G_{x,x'}} \varphi _g \cdot g\) . This makes sense because of \(G_{x,x'} \subseteq G_{y,y'}\) . The functor \({{\,\textrm{res}\,}}(f)\) induces a functor of additive \(\Lambda \) -categories \({{\,\textrm{res}\,}}(f)_{\oplus } :\mathcal {A}(Y)_{\oplus } \rightarrow (\mathcal {A}(X)_{\oplus })_{\oplus }\) . Composing it with the obvious equivalence of additive \(\Lambda \) -categories \((\mathcal {A}(X)_{\oplus })_{\oplus } \xrightarrow {\simeq } \mathcal {A}(X)_{\oplus }\) yields a functor of additive \(\Lambda \) -categories, denoted in the same way

6.3 The pairing with the Swan group

In this section we construct a bilinear pairing for a G -set X and \(n \in \mathbb {Z}\)

We have to construct for any covariant functor M from \(\mathcal {T}^G(X)\) to the category of finitely generated free abelian groups a functor of \(\mathbb {Z}\) -categories \(F(M) :\mathcal {A}(X) \rightarrow \mathcal {A}(X)_{\oplus }\) , since then we can define for the element \([M] \in {{\,\textrm{Sw}\,}}(\mathcal {T}^G(X))\) represented by M the homomorphism \(P([M],-) :K_n(\mathcal {A}(X)_{\oplus }) \rightarrow K_n(\mathcal {A}(X)_{\oplus })\) to be \(K_n(F(M)_{\oplus })\) . For \(x \in X\) let r ( x ) be the rank of the finitely generated free abelian group M ( x ) and choose an ordered basis \(\{b_1, \ldots , b_{r(x)}\}\) of M ( x ). Consider \(x,x' \in X\) with \(G_{x,x'} \not = 0\) . Then \(r(x) = r(x')\) and for \(g \in G_{x,x'}\) we get an invertible ( r ( x ),  r ( x ))-matrix with entry \(\rho (g)_{i,i'} \in \mathbb {Z}\) for \(i \in \{1, \ldots , r(x)\}\) and \(i' \in \{1, \ldots , r(x')\}\) , which describes the homomorphism of finitely generated free abelian groups \(M(g) :M(x) \rightarrow M(x')\) for the morphism \(g :x \rightarrow x'\) in \(\tau ^G(X)\) with respect to the chosen ordered bases. Now F ( M ) sends an object ( A ,  x ) to \(\bigoplus _{i =1 }^{r(x)} A(x)\) . A morphism \(\sum _{g \in G} f_g \cdot g :(A,x) \rightarrow (A',x')\) is sent to the morphism \(\bigoplus _{i =1 }^{r(x)} A(x) \rightarrow \bigoplus _{i' =1 }^{r(x')} A'(x')\) , whose component for \(i \in \{1, \ldots , r(x)\}\) and \(i' \in \{1, \ldots , r(x')\}\) is given by \(\sum _{g \in G} \rho (g)_{i,i'} \cdot f_g \cdot g :(A,x) \rightarrow (A',x')\) . Note that the choice of basis does not matter, since it does not change the equivalence class of F ( M ). We leave it to the reader to check that we get a well-defined bilinear pairing ( 6.6 ).

6.4 The special case of a ring as coefficients

The following example is illuminating and the most important one.

Let \(\Lambda \) be a commutative ring and let R be a \(\Lambda \) -algebra, where we tacitly always require that \(\lambda r = r \lambda \) holds for every \(\lambda \in \Lambda \) and \(r \in R\) . Consider a group homomorphism \(\rho :G \rightarrow {{\,\textrm{aut}\,}}_{\Lambda }(R)\) to the group of automorphisms of the \(\Lambda \) -algebra R . The twisted group ring with R -coefficients \(R_{\rho }[G]\) is defined as follows. Elements are finite formal sums \(\sum _{g \in G} r_g \cdot g\) for \(r_g \in R\) . The multiplication in \(R_{\rho }[G]\) is given by

Note that \(R_{\rho }[G]\) inherits the structure of \(\Lambda \) -algebra from R .

Define \(\underline{R}\) to be the \(\Lambda \) -category, which has precisely one object \(*_R\) and whose \(\Lambda \) -module of endomorphisms is R . Composition is given by the multiplication in R and the \(\Lambda \) -structure on the endomorphisms comes from the structure of a \(\Lambda \) -algebra. From \(\rho \) we obtain the structure of G - \(\Lambda \) -category on \(\underline{R}\) and we can consider the \(\Lambda \) -category \(\underline{R}(X)\) for any G -set X .

Consider a subgroup \(H \subseteq G\) . There is an equivalence of \(\Lambda \) -categories

that sends the object \(*_{R_{\rho |_H}[H]}\) to the object \((*_R, eH)\) and a morphism in \(\underline{R_{\rho |_H}[H]}\) , which is given by the element \(\sum _{h \in H} r_h \cdot h\) in \(R_{\rho |_H}[H]\) , to the morphism \((*_R, eH) \rightarrow (*_R, eH)\) in \(\underline{R}(G/H)\) given by \(\sum _{h \in H} r_h \cdot h\) again. Obviously T ( H ) is fully faithful. Every object in \(\underline{R}(G/H)\) is isomorphic to an object in the image of T ( H ), namely, for any object \((*_R, gH)\) we get an isomorphism \((*_R, eH) \xrightarrow {\cong } (*_R, gH)\) by \(1_R \cdot g\) . Hence \(T_H\) is an equivalence of \(\Lambda \) -categories. From ( 6.7 ) we obtain an equivalence of additive \(\Lambda \) -categories

and hence for every \(n \in \mathbb {Z}\) an isomorphism

Consider subgroups \(H \subseteq K \subseteq G\) with \([K:H] < \infty \) . Let \({{\,\textrm{pr}\,}}:G/H \rightarrow G/K\) be the projection and \(i :H \rightarrow K\) be the inclusion. Then \(K_n(\underline{R}({{\,\textrm{pr}\,}})) :K_n(\underline{R}(G/H)) \rightarrow K_n(\underline{R}(G/K))\) corresponds under the isomorphism ( 6.9 ) to the induction homomorphism \(i_* :K_n(R_{\rho |_H}[H]) \rightarrow K_n(R_{\rho |_K}[K])\) , whereas \(K_n({{\,\textrm{res}\,}}({{\,\textrm{pr}\,}})_{\oplus }) :K_n(\underline{R}(G/K)_{\oplus }) \rightarrow K_n(\underline{R}(G/H)_{\oplus })\) corresponds under the isomorphism ( 6.9 ) to the restriction homomorphism \(i^* :K_n(R_{\rho |_K}[K]) \rightarrow K_n(R_{\rho |_H}[H])\) . The pairing P of ( 6.6 ) corresponds under the isomorphisms ( 5.7 ) and ( 6.9 ) to the pairing \( {{\,\textrm{Sw}\,}}(H) \times K_n(R_{\rho |_H}[H]) \rightarrow K_n(R_{\rho |_H}[H])\) coming from the fact that, for a \(\mathbb {Z}H\) -module M , whose underlying abelian group is finitely generated free, and a finitely generated projective \(R_{\rho |_H}[H]\) -module P , we can equip \(M \otimes _{\mathbb {Z}} P\) with the \(R_{\rho |_H}[H]\) -module structure given by

for \(\sum _{h \in H} r_h \cdot h \in R_{\rho |_H}[H]\) , \(m \in M\) , and \(p \in P\) and thus obtain a finitely generated projective \(R_{\rho |_H}[H]\) -module.

6.5 The Green and Mackey structure of a finite quotient group

Fix a (not necessarily finite) group G and a surjective group homomorphism \(\nu :G \rightarrow F\) onto a finite group F . Let \(\mathcal {A}\) be a G - \(\mathbb {Z}\) -category. Fix \(n \in \mathbb {Z}\) . Then we can define a module over the Green functor \({{\,\textrm{Sw}\,}}_F\) for F as follows. The underlying Mackey functor \(M(Z,n) = (M(Z,n)_*,M(Z,n)^*)\) is on an object X which is a finite F -set X , given by

where \(\nu ^*X\) is the G -set obtained from the F -set X by restriction with \(\nu \) . The covariant functor \(M(Z,n)_*\) sends an F -map \(f :X \rightarrow Y\) to the homomorphism \(K_n(\mathcal {A}(\nu ^*f)_{\oplus }) :K_n(\mathcal {A}(\nu ^*X)_{\oplus }) \rightarrow K_n(\mathcal {A}[\nu ^*Y]_{\oplus })\) defined in ( 6.2 ). The contravariant functor \(M(Z,n)_*\) sends an F -map \(f :X \rightarrow Y\) to \(K_n({{\,\textrm{res}\,}}(\nu ^*f)_{\oplus }) :K_n(\mathcal {A}[\nu ^*Y]_{\oplus }) \rightarrow K_n(\mathcal {A}(\nu ^*X)_{\oplus })\) for the functor of additive categories \({{\,\textrm{res}\,}}(\nu ^*f)_{\oplus }\) defined in ( 6.5 ). For a finite F -set X we get a map of abelian groups

by precomposing a covariant functor M ( Z ,  n ) from \(\tau ^F(X)\) to the category of finitely generated free abelian groups with the obvious functor \(\mathcal {T}^G(\nu ^*X) \rightarrow \mathcal {T}^F(X)\) induced by \(\nu \) . Now we obtain from ( 6.6 ) and ( 6.10 ) for all finite F -sets X a pairing

We leave the lengthy but straightforward proof to the reader that these data define a Mackey functor \(\mathcal {M}\) for F and the structure of a Green-module over \({{\,\textrm{Sw}\,}}_F\) on it. This fact is not surprising in view of Sect. 6.4 , since the proof in this special case is well-known.

7 Infinite covirtually cyclic groups and their finite quotients

7.1 basics about infinite covirtually cyclic subgroups.

A group V is called covirtually cyclic , if it contains a normal finite subgroup \(K \subseteq V\) such that V / K is cyclic. Equivalently, V is either a finite group or there is an extension \(1 \rightarrow K \rightarrow V \xrightarrow {\pi _V} C\rightarrow 1\) of groups such that K is finite and C is infinite cyclic. An infinite covirtually cyclic group is the same as a infinite virtually cyclic group of type I.

Let V be a covirtually cyclic subgroup of G which is infinite. Then it contains a finite subgroup \(K = K_V\) with the property such that any finite subgroup of G is contained in K . Note that K is uniquely determined by this property, is a characteristic and in particular normal subgroup of V , and the quotient \(C_V:= V/K_V\) is infinite cyclic.

Definition 7.1

( Polarization ) A polarization of V is a choice of an element \(t \in V\) whose image under the projection \(\pi = \pi _V :V \rightarrow C_V\) is a generator.

7.2 Basic definitions and notation

Fix an integer \(M \ge 1\) . Let m be the order of the automorphism \(c_t :K \xrightarrow {\cong } K\) given by conjugation with t . Then \(t^mkt^{-m} = k\) holds for all \(k \in K_V\) . Hence the infinite cyclic subgroups \(\langle t^{Mm}\rangle \) and \(\langle t^{m}\rangle \) of V are normal and their intersection with \(K_V\) is trivial. Define finite groups

Denote by \(\nu :V \rightarrow F\) and \(\beta :F \rightarrow \widehat{F}\) the canonical projections. We have the exact sequence \(1 \rightarrow \ker (\beta ) \xrightarrow {j} F\xrightarrow {\beta } \widehat{F} \rightarrow 1\) .

Note that the following square commutes

where the horizontal maps are the obvious isomorphism given by the elements \(t \in V\) and \(\nu (t) \in F\) , and \(\beta \circ \nu (t) \in \widehat{F}\) , and \(p_{Mm} :\mathbb {Z}\rightarrow \mathbb {Z}/Mm\) and \(\overline{p}_M :\mathbb {Z}/Mm \rightarrow \mathbb {Z}/m\) are the obvious projections.

7.3 Some properties of subgroups

Consider a subgroup H of F and a prime q . We compute

This implies

Given a prime p , a finite group H is called p - hyperelementary , if it can be written as an extension \(1 \rightarrow C \rightarrow H \rightarrow P \rightarrow 1\) for a cyclic group C and a p -group P , and and is called p - elementary if it is isomorphic to \(C \times P\) or a cyclic group C and a p -group P . One can always arrange that the order of C is prime to p .

Let p be a prime with \(p \not = q\) . Let \(i :K \rightarrow F\) be the injective group homomorphism given by restricting \(\nu :V \rightarrow F\) to K . Suppose that q divides \(|i^{-1}(H)|\) and that H is a p -hyperelementary group. Then

Note that for \(k \in K_V\) and \(y \in \ker (\beta )\) we have \(\nu (k)\cdot j(y) = j(y) \cdot \nu (k)\) . This implies that we get a well-defined group homomorphism

It is injective by the following calculation for \((k,y) \in (\nu ^{-1}(H) \cap K) \times j^{-1}(H)\) using the fact that \(\beta \circ i\) is injective

Since H is p -hyperelementary and \(p \not = q\) , the q -Sylow subgroup of H is cyclic. Hence the q -Sylow subgroup of \(i^{-1}(H) \times j^{-1}(H)\) is cyclic. Since q divides \(|i^{-1}(H)|\) by assumption, q does not divide \(|j^{-1}(H)|\) and we get \(\log _q(|j^{-1}(H)|) = 0\) . Now apply ( 7.3 ). \(\square \)

Suppose that H is p -hyperelementary and that \(i^{-1}(H)\) is a p -group. Then H is p -elementary.

By [ 21 , Lemma 3.1] it suffices to show that for every prime q with \(q\mid |H|\) and \(q \not = p\) there exists an epimorphism from H onto a non-trivial finite cyclic group of q -power order. Let \(\alpha :F \rightarrow C_V/(Mm\cdot C_V)\) be the canonical projection, whose kernel is K . Since \(\alpha \) induces an epimorphism \(H \rightarrow \alpha (H)\) onto a finite cyclic group \(\alpha (H)\) , it suffices to show that the q -Sylow subgroup of \(\alpha (H)\) is non-trivial for any prime q with \(q\mid |H|\) and \(q \not = p\) . Choose an element \(h \in H\) and an integer \(a \ge 1\) with \(h \not = e_H\) and \(h^{q^a} = e_H\) . It suffices to show that \(\alpha (h)\) is not the the unit element. Suppose the contrary. Then we can find \(x \in i^{-1}(H)\) with \(i(x) = h\) . Since \(i^{-1}(H)\) is a finite p -group, we can choose an integer \(b \ge 1\) satisfying \(x^{p^b} = e_K\) . As \(p \not = q\) , we can find integers \(\lambda , \mu \) with \(\lambda q^a + \mu \cdot p^b = 1\) . We compute

Hence \(x = e_K\) which implies \(h = e_H\) , a contradiction. \(\square \)

8 On Nil-terms for infinite covirtually cyclic subgroups

For the remainder of this section we fix an infinite covirtually cyclic group V with a polarization \(t \in V\) and a V - \(\mathbb {Z}\) -category \(\mathcal {A}\) . Let the covariant functor

be the functor coming from the functor in ( 6.1 ) for \(G = V\) and \(\nu :V \rightarrow F\) taken from Sect.  7.2

8.1 The basic diagram  8.7

In the sequel we denote for two subgroups \(H_0\) and \(H_1\) of V with \(H_0 \subseteq H_1\) by \({{\,\textrm{pr}\,}}:V/H_0 \rightarrow V/H_1\) the V -map given by the canonical projection. Given subgroups \(H_0\) and \(H_1\) of V and \(v \in V\) in with \(v^{-1}H_0v \subseteq H_1\) , we denote by \(R_{v} :V/H_0 \rightarrow V/H_1\) the V -map sending \(v'H_0\) to \(v'vH_1\) . Note that for \(v_0,v_1 \in V\) with \(v_i^{-1}H_0v_i \subseteq H_1\) for \(i = 0,1\) we have \(R_{v_0} = R_{v_1}\) if and only if \(v_1^{-1}v_0 \in H_1\) holds.

Given \(d \in \mathbb {Z}^{\ge 1} = \{ n \in \mathbb {Z}\mid n \ge 1\}\) , let \(C[d] = d \cdot C \subseteq C\) be the subgroup of index d in C and V [ d ] be the preimage of C [ d ] under the projection \(\pi :V\rightarrow C\) . In particular we get \(V[1] = V\) .

Let \(W\subseteq V\) be a subgroup of V of finite index. Then W itself is an infinite covirtually cyclic group. Its maximal finite subgroup is \(K_W = K_V \cap W\) . Let \(d = d(W) \in \mathbb {Z}^{\ge 1}\) be the natural number given by the index d of \(\pi _V(W) \) in the infinite cyclic group C . Then \(W \subseteq V[d]\) and we have \(W = V[d]\) if and only if \(K_V \subseteq W\) , or, equivalently, \(K_W = K_V\) holds.

Fix a polarization \(t_W \in W\) . Since the following diagram of V -spaces commutes

we get from \(Z({{\,\textrm{pr}\,}}) :Z(V/K_W) \rightarrow Z(V/W)\) a functor of additive categories

The functor \(U(W,t_W)\) is for any choice of polarization \(t_W \in W\) an equivalence of additive categories and induces in particular for every \(n \in \mathbb {Z}\) an isomorphism

This follows from the isomorphism ( 6.9 ) and the obvious fact that the canonical map \(K_W \rtimes _{c_{t_W}}\mathbb {Z}\rightarrow W\) is an isomorphism. \(\square \)

We get for \(n \in \mathbb {Z}\) from the isomorphism appearing in Lemma  8.3 , the twisted Bass-Heller Swan decomposition applied to \(\mathcal {A}= Z(V/K_W)\) and \(\Phi = Z(R_{t_W})\) , see Theorem  4.16 , and the projection onto and the inclusion of the first Nil-term maps

satisfying \(r(W,t_W)_n \circ s(W,t_W)_n = {{\,\textrm{id}\,}}_{\overline{K}_{n-1}({{\,\textrm{Nil}\,}}(Z(V/K_W),Z(R_{t_W})))}\) .

Fix a polarization \(t \in V\) of V . The following diagram commutes by Theorem  4.16

where \(V_d\) is the Verschiebungs operator, see ( 3.2 ).

Let \(W\subseteq V\) be a subgroup of V of finite index. Fix a polarization t of V . Then there is \(y \in K = K_V\) with \(yt^{d} \in W\) . Fix \(y = y(W)\in K\) with \(yt^{d} \in W\) . Then the element \(yt^{d}\) is a polarization of W . As the composites \(V/K_W\xrightarrow {{{\,\textrm{pr}\,}}} V/K \xrightarrow {R_{t^{d}}} V/K\) and \(V/K_W \xrightarrow {R_{yt^{d}}} V/K_W \xrightarrow {{{\,\textrm{pr}\,}}} V/K\) agree, \(Z({{\,\textrm{pr}\,}}) :Z(V/K_W) \rightarrow Z(V/K)\) induces a map

for every \(n \in \mathbb {Z}\) . One easily easily checks that the following diagram commutes

to be the composite

Then we obtain by concatenating ( 8.4 ) and ( 8.5 ) the commutative diagram

8.2 Improving the induction results

For the remainder of this section consider the setup of Sect. 7.2 . In particular we have fixed a natural number M and a surjective group homomorphism \(\nu :V \rightarrow F\) onto a finite group F . Moreover, we fix \(n \in \mathbb {Z}\) and let M ( Z ,  n ) be the module over the Green functor \({{\,\textrm{Sw}\,}}_F\) associated to Z in Sect.  6.1 with respect to the epimorphism \(\nu :V \rightarrow F\) .

Consider any element z in \(K_n(Z(V/V)) = \mathcal {M}(Z,n)(F/F)\) . Let p be a prime number. Let \(\mathcal {H}_{p} \) be the family of p -hyperelementary subgroups of F .

Then there are elements \(a_H \in \mathbb {Z}_{(p)}\) and \(u_H \in {{\,\textrm{Sw}\,}}(H)\) for each \(H \in \mathcal {H}_{p}\) such that

holds in \(K_n(Z(V/V))_{(p)}\) , where \({{\,\textrm{pr}\,}}_H :F/H \rightarrow F/F\) is the projection.

This follows from Lemma  5.9 . \(\square \)

Consider an element \(z \in \overline{K}_{n-1}({{\,\textrm{Nil}\,}}(Z(V/K_V),Z(R_t)))\) of nilpotence degree \(\le D\) .

Then for every \(H \subseteq F\) satisfying \([F:H] \ge D\) the composite

sends z to zero.

Put \(d = [F: H] = [V: \nu ^{-1}(H)]\) . Then we have \(\nu ^{-1}(H) \subseteq V[d]\) and the homomorphism \(\mathcal {M}(Z,n)^*({{\,\textrm{pr}\,}}) :K_n(Z(V/V)) \rightarrow K_n(Z(V/\nu ^{-1}(H)))\) is the composite

The following diagram commutes

where the bijective horizontal arrows have been defined in ( 8.2 ). Now the claim follows from Lemma  4.33 , since the map \(s_n(V,t)\) is by definition the composite

Lemmas  8.8 and 8.9 imply

Consider an element z in \(z \in \overline{K}_{n-1}({{\,\textrm{Nil}\,}}(Z(V/K_V),Z(R_t)))\) of nilpotence degree \( \le D\) . Let p be a prime number. Let \(\mathcal {H}_{p} \) be the family of p -hyperelementary subgroups of F .

Then there are elements \(a_H \in \mathbb {Z}_{(p)}\) and \(u_H\) for each \(H \in \mathcal {H}_{p}\) such that

holds in \(K_n(Z(V/V))_{(p)}\) .

Define \(\overline{\mathcal {H}_{p}}\) to be the family p -hyperelementary subgroups H of F , for which \(i^{-1}(H) = H \cap F \subseteq F\) is a p -group.

Consider an element z in \(\overline{K}_{n-1}({{\,\textrm{Nil}\,}}(Z(V/K_V),Z(R_t)))\) of nilpotence degree \(\le D\) . Let p be a prime number. Assume that \(\log _q(l) < \log _q(M)\) holds for every natural number l with \(l < D\) and every prime q that satisfies \(q \le \max \{D,|\widehat{F}|\}\) and is different from p .

Then there are elements \(a_H \in \mathbb {Z}_{(p)}\) and \(u_H\) for each \(H \in \overline{\mathcal {H}_{p}}\) such that

In view of Lemma  8.10 it suffices to show under the assumptions above that \(i^{-1}(H) = H \cap F\) is a p -group for \(H \in \mathcal {H}_{p}\) , provided that \([F:H] < D\) holds. Because of Lemma  7.5 it remains to show \(\log _q([F:H]) < \log _q(M)\) for every every \(H \in \mathcal {H}_{p}\) and every prime q that divides \(|i^{-1}(H)| = |H \cap F|\) and is different from p , provided that \([F:H] < D\) holds. If q satisfies \(q \le \max \{D,|\widehat{F}|\}\) , then \(\log _q([F:H]) < \log _q(M)\) follows from the assumptions. Suppose q that satisfies \(q > \max \{D,|\widehat{F}|\}\) . Then q does divide neither [ F  :  H ] nor \(\widehat{F}\) . Since q divides \( |H \cap F|\) and hence | F | and \(|F| = M \cdot |\widehat{F}|\) holds, q divides | M | and hence \(\log _q(M) \ge 1\) . As q does not divide [ F  :  H ], we have \(\log _q([F:H])= 0\) and hence \(\log _q([F:H]) < \log _q(M)\) . \(\square \)

Notation 8.12

The isomorphism type of the abelian group \(\overline{K}_{n-1}\bigl ({{\,\textrm{Nil}\,}}(Z(V/K_{W}),Z(R_{y(W)t^{d(W)}}))\bigr )\) and the image of \(\Xi _{n-1}(W,y(W)t^{d(W)})\) are independent of the choice of y ( W ), since for any other choice \(y(W)'\) we have the commutative diagram

with an isomorphisms as vertical arrow.

Theorem 8.13

is surjective.

Consider \(z \in \overline{K}_{n-1}({{\,\textrm{Nil}\,}}(Z(V/K),Z(R_t)))\) . Choose D such that the nilpotence degree of z is \(\le D\) . Now choose a natural number M such that \(\log _q(l) < \log _q(M)\) holds for every natural number l with \(l < D\) and every prime q that satisfies \(q \le \max \{D,|\widehat{F}|\}\) and is different from p . From Lemma  8.11 we get elements \(a_H \in \mathbb {Z}_{(p)}\) and \(u_H\) for each \(H \in \overline{\mathcal {H}_{p}}\) such that

holds in \(K_n(Z(V/V))_{(p)}\) . Define for \(H \in \overline{\mathcal {H}_p}\) the subgroup \(W_H\) of V by \(W_H:= \nu ^{-1}(H)\) . For \(H \in \overline{\mathcal {H}_p}\) the following diagram

commutes because of the commutative diagram ( 8.7 ). Define the element in \(z_H \in \overline{K}_{n-1}\bigl ({{\,\textrm{Nil}\,}}(Z(V/K_{W}),Z(R_{y_{W}t^{d(W)}}))\bigr )_{(p)}\) by

Recall that we have \(r(V,t)_n \circ s(V,T)_n = {{\,\textrm{id}\,}}\) . Now we compute

Corollary 8.17

9 proof of theorems  1.2 and 1.5 for additive categories.

Let G be a group and \(\Lambda \) be a commutative ring. Let \(\mathcal {A}\) be an additive G - \(\Lambda \) -category. We have defined \(\mathcal {P}(G,\Lambda )\) in Notation  1.1 .

Theorem 9.1

Suppose that the additive category \(\mathcal {A}\) is regular in the sense of [ 4 , Definition 6.2]. Then the canonical map

is a \(\mathcal {P}(G,\Lambda )\) -isomorphism for all \(n \in \mathbb {Z}\) .

is an isomorphism for all \(n \in \mathbb {Z}\) by [ 16 , Remark 1.6], see also [ 25 , Theorem 13.44]. Hence it suffices to show that the relative assembly map

is a \(\mathcal {P}(G,\Lambda )\) -isomorphism for all \(n \in \mathbb {Z}\) . By the Transitivity Principle appearing in [ 2 , Theorem 3.3], see also [ 25 , Theorem 15.12], we can assume without loss of generality that G itself is an infinite covirtually cyclic group V . Hence we have to show for any covirtually cyclic group V that the assembly map

is a \(\mathcal {P}(G,\Lambda )\) -isomorphism, where we view \(\mathcal {A}\) as a V - \(\Lambda \) -category by restriction from G to V . Write \(V = K \rtimes _{c_t} \mathbb {Z}\) . The twisted Bass-Heller-Swan isomorphism of ( 4.5 ) applied to \(V = K \rtimes _{c_t} \mathbb {Z}\) yields an isomorphism

There is an identification

coming from the fact that a model for \(E_{{{{\mathcal {F}}}\textrm{in}}}(V)\) is \(\mathbb {R}\) with respect to the V -action given by \(v \cdot r = {{\,\textrm{pr}\,}}(v) + r\) for \(r \in \mathbb {R}\) and the canonical projection \({{\,\textrm{pr}\,}}:V \rightarrow \mathbb {Z}\) . Under these identifications the map ( 9.2 ) becomes the obvious inclusion. Hence map ( 9.2 ) is a \(\mathcal {P}(G,\Lambda )\) -isomorphism if we can show

Because of [ 4 , Lemma 7.7 and Theorem 8.1] it suffices to show that the additive category \(Z_{\mathcal {A}}(G/K_{W})\) is regular. Since it is equivalent to \(\mathcal {A}|_{K_W}[K_W]\) and \(K_W\) is a p -group, it suffices to show for any p -subgroup P of V that \(\mathcal {A}|_P[P]\) is regular. If the p -subgroup P is trivial, then \(\mathcal {A}|_P[P] = \mathcal {A}\) and hence by assumption regular. If P is non-trivial, then by definition of \(\mathcal {P}(V,\Lambda )\) the prime p is invertible in \(\Lambda \) . We leave it to the reader to check that then \(\mathcal {A}|_P[P]\) is regular, as \(\mathcal {A}\) is regular. It is not hard to extend the proof for rings in [ 7 , Lemma 7.4 (2)] to additive categories. This finishes the proof of Theorem  9.1 . \(\square \)

Corollary 9.3

Suppose G satisfies the Full Farrell–Jones Conjecture. Assume that the additive \(\Lambda \) -category \(\mathcal {A}\) is regular in the sense of [ 4 , Definition 6.2] and that the order of any finite subgroup of G is invertible in \(\Lambda \) .

where \(\mathcal {A}[G/G]\) has been defined in Sect.  6.1 .

This follows from Theorem  9.1 using the equivariant Atiyah–Hirzebruch spectra sequence and the fact that \(\mathcal {A}(G/H)\) is equivalent to the regular additive category \(\mathcal {A}|_H[H]\) for finite \(H \subseteq G\) , which implies \(K_i(\mathcal {A}(G/H)) = 0\) for \(i \le -1\) and \(|H| < \infty \) , see for instance [ 25 , Proof of Proposition 13.48 (iv)], [ 8 , Section 4]. \(\square \)

10 Nil-terms as modules over the ring of big Witt vectors

We extend the result of Weibel [ 41 , Corollary 3.2], which is based on work by Stienstra [ 35 ], that for a ring R of characteristic N for some natural number N we have \(N\!K_n(RG)[1/N] = \{0\}\) for every group G and every \(n \in \mathbb {Z}\) , to additive categories allowing twisting by an automorphism.

10.1 Review of the ring of big Witt vectors

Let \(\Lambda \) be a commutative ring. Let \(W(\Lambda )\) be the commutative ring of big Witt vectors. The underlying abelian group is the multiplicative group \(1 + t\Lambda [[t]]\) of formal power series \(1 + \lambda _1t + \lambda _2t^2 + \cdots \) in t with coefficients in \(\Lambda \) and leading term 1. We do not give the details of the multiplicative structure \(*\) but at least mention that it is the unique continuous functional with the property that \((1 -\lambda t) *(1-\mu t) = (1- \lambda \mu t)\) holds for \(\lambda , \mu \in \Lambda \) . We mention that it satisfies for \(m,n \in \mathbb {Z}^{\ge 1}\) and \(\lambda , \mu \in \Lambda \)

where d is greatest common divisor of m and n . The unit for the addition is 1, whereas the unit for the multiplication is \((1-t)\) .

For \(N \in \mathbb {Z}^{\ge 1}\) the subgroup \(I_N = 1 + t^N\Lambda [[t]]\) is actually an ideal in \(W(\Lambda )\) . We have \(W(\Lambda ) = I_1 \supset I_2 \supset I_3 \supset \cdots \) and \(\bigcap _{N \ge 1} I_N = \{1\}\) . Thus we obtain the so called t -adic topology on \(W(\Lambda )\) . Then \(W(\Lambda )\) is separated and complete in this topology. The quotient rings \(W_N(\Lambda ) = W(\Lambda )/I_{N+1}\) are called the rings of N -truncated Witt vectors. For more information we refer to [ 13 , Section I.1].

10.2 Endomorphisms rings

Note that then we get from i and r a natural isomorphism

Note that the tensor product over \(\Lambda \) induces the structure of a commutative ring on \(K_0({{\,\textrm{End}\,}}(\Lambda ))\) . Since the image of \(K_0(i)\) is an ideal, the quotient \(\overline{K}_0({{\,\textrm{End}\,}}(\Lambda ))\) inherits the structure of a commutative ring.

There is a well-defined map \(\eta :\overline{K}_0({{\,\textrm{End}\,}}(\Lambda )) \rightarrow W(\Lambda )\) , which sends the class of ( P ,  f ) for an endomorphisms \(f :P \rightarrow P\) of a finitely generated projective \(\Lambda \) -module P to its characteristic polynomial \(\det _{\Lambda }(1 -tf)\) . The next result is due to Almkvist [ 1 ].

Theorem 10.4

We obtain a well-defined injective ring homomorphism

whose image consists of all rational functions, i.e., quotients \(x/x'\) of polynomials \(x,x'\) in \(1 + t \Lambda [t] \subseteq 1 + t \Lambda [[t]]\) .

The t -adic topology on \(\overline{K}_0({{\,\textrm{End}\,}}(\Lambda ))\) is given by the filtration \(\overline{K}_0({{\,\textrm{End}\,}}(A)) = \eta ^{-1}(I_1) \supset \eta ^{-1}(I_2) \supset \eta ^{-1}(I_3) \supset \cdots \) . Let \(\overline{K}_0({{\,\textrm{End}\,}}(A))^{\widehat{\phantom{a}}}\) be the completion of \(\overline{K}_0({{\,\textrm{End}\,}}(A))\) with respect to the t -adic topology. Since \(W(\Lambda )\) is separated and complete in the t -adic topology and the image of \(\eta \) is dense in \(W(\Lambda )\) , we conclude from Theorem  10.4 that \(\eta \) induces an isomorphism of topological rings

10.3 The action on Nil-groups

Let \(\mathcal {A}\) be an additive \(\Lambda \) -category, i.e., a small category enriched over the category of \(\Lambda \) -modules coming with a direct sum \( \oplus \) compatible with the \(\Lambda \) -module structures. Let \(\Phi :\mathcal {A}\xrightarrow {\cong } \mathcal {A}\) be an automorphism of additive \(\Lambda \) -categories of \(\mathcal {A}\) . Denote by \({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )\) the associated Nil-category, which inherits the structure of an exact category. Recall that then the K -groups \(K_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) and \(\overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) are defined for \(s \in \mathbb {Z}\) .

Let \(\underline{\Lambda }_{\oplus }\) be the additive \(\Lambda \) -category obtained from the obvious \(\Lambda \) -category having precisely one object by adding finite sums. More precisely, for every \(m \in \mathbb {Z}^{\ge }\) we have the object [ m ] in \(\underline{\Lambda }\) . A morphism \(C :[m] \rightarrow [n]\) for \(m,n \ge 1\) is a ( m ,  n )-matrix \(C = (c_{i,j})\) over \(\Lambda \) . The object [0] is declared to be the zero object. Composition is given by matrix multiplication, whereas the direct sum is given on objects by assigning to two object [ m ] and [ n ] the object \([m+n]\) and on morphisms by the block sum of matrices.

We define a bilinear functor of additive \(\Lambda \) -categories

as follows. On objects it is defined by sending ([ m ],  A ) to \(\bigoplus _{i = 1}^m A\) . Consider morphisms \(C :[m] \rightarrow [n]\) and \(f :A \rightarrow B\) in \(\underline{\Lambda }\) and \(\mathcal {A}\) . Define the morphism

by \((C \oplus f)_{i,j} = c_{i,j} \cdot f :A \rightarrow B\) for \((i,j) \in \{1, \ldots , m \} \times \{1, \ldots , n \}\) . If we fix an object ([ n ],  C ) in \({{\,\textrm{End}\,}}(\underline{\Lambda })\) , we get a functor of exact categories \(F_{([n],C)} :{{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ) \rightarrow {{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )\) by sending an object \((A,\varphi )\) that is given by the nilpotent endomorphism \(\varphi :\Phi (A) \rightarrow A\) to the object that is given by the nilpotent endomorphism defined by the composite

for the canonical isomorphism \(\sigma (n,A) :\bigoplus _{i=1}^n \Phi (A) \xrightarrow {\cong } \Phi \left( \bigoplus _{i=1}^n A\right) \) . It induces a homomorphism \(\overline{K}_n(F_{([n],C)}) :\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )) \rightarrow \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) for every \(n \in \mathbb {Z}\) . Now one easily checks that the collection of these homomorphisms defines a bilinear pairing of abelian groups for every \(s \in \mathbb {Z}\)

The pairing \(T_s\) of ( 10.7 ) extends uniquely to continuous pairing

for \(s \in \mathbb {Z}\) , where we equip \(\overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) with the discrete topology and \(W(\Lambda )\) with the t -adic topology.

Moreover, this pairing turns \(\overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) into a \(W(\Lambda )\) -module.

Because of the isomorphism of topological rings ( 10.5 ), the existence of the extension \(\widehat{T}_s\) follows if we can show for every element \(y \in \overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) that there exists an integer N such that for every \(x \in \eta ^{-1}(I_N)\) we have \(T_s(x,y) = y\) . This is done as follows.

In the first step we reduce the claim to the special case \(s \ge 1\) .

From [ 27 , Definition 2.1, Theorem 3.4 and Lemma 6.5] applied to the functor sending \((\mathcal {A}, \Phi )\) to the connective K -theory spectrum \(\textbf{K}({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) , we get a natural (untwisted) Bass–Heller–Swan isomorphisms for \(s \in \mathbb {Z}\)

Restricting this isomorphism to \(K_{s-1}({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) yields a (split) injective homomorphisms \(\sigma _s :K_{s-1}({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )) \rightarrow K_{s}({{\,\textrm{Nil}\,}}(\mathcal {A}[\mathbb {Z}],\Phi [\mathbb {Z}]))\) , natural in \((\mathcal {A}, \Phi )\) . It induces a (split) injective homomorphism \(\overline{\sigma }_s :\overline{K}_{s-1}({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi )) \rightarrow \overline{K}_{s}({{\,\textrm{Nil}\,}}(\mathcal {A}[\mathbb {Z}],\Phi [\mathbb {Z}]))\) , natural in \((\mathcal {A}, \Phi )\) , since we also have the (untwisted) Bass-Heller-Swan isomorphism

Now we obtain for every \(s \in \mathbb {Z}\) a commutative diagram

Since for every integer s we can find a natural number k with \(s + k \ge 1\) and we can iterate the construction of the diagram above k -times, it suffices to show for every element \(y \in \overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) for \(s \ge 1\) that there exists an integer N such that for every \(x \in \eta ^{-1}(I_N)\) we have \(T_s(x,y) = y\) . Moreover, we can work for the rest of the proof with the connective K -theory spectrum of Waldhausen categories.

Let \({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )_N\) be the full exact subcategory of \({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )\) consisting of those objects \((A,\varphi )\) , whose nilpotence degree is less or equal to N , i.e., \(\varphi ^{(N)} :\Phi ^N(A) \rightarrow A\) is trivial. For every natural number N the construction of the pairing \(T_s\) of ( 10.7 ) yields also a pairing

These pairings \(T[N]_s\) and \(T_s\) are compatible with the map \(\overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )_N) \rightarrow \overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) coming from the obvious inclusions of full subcategories. In view of the isomorphism ( 4.31 ), it suffices to show for every natural number N and \(s \ge 1\) that \(T[N]_s(x,y) = 0\) holds for \(x \in \eta ^{-1}(I_N)\) and \(y \in \overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )_N)\) .

For a natural number \(n \ge 1\) and \(\lambda _1, \lambda _2, \dots , \lambda _n\) in \(\Lambda \) , define the automorphism \(C_n(\lambda _1, \lambda _2, \ldots , \lambda _r;\varphi ) :[n] \rightarrow [n]\) in \({{\,\textrm{End}\,}}(\Lambda )\) by the ( n ,  n )-matrix over \(\Lambda \)

Then \(\eta \bigl ([C_n(-\lambda _1, -\lambda _2, \ldots -\lambda _n)]\bigr ) = 1 +\lambda _1 t + \lambda \cdot t^2 + \cdots + \lambda _n \cdot t^n\) . Because of Theorem  10.4 it suffices to prove \(T[N]_n([C_n(\lambda _1, \lambda _2, \ldots \lambda _n)],y) = 0\) for every \(s,N \ge 1\) , elements \(\lambda _1, \lambda _2, \ldots , \lambda _n\) in \(\Lambda \) and \(y \in \overline{K}_s({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi )_N)\) , provided that \(\lambda _i = 0\) holds for \(i = 1,2 \ldots , N-1\) .

Fix \(s,N \ge 1\) and \(\lambda _1, \lambda _2, \ldots , \lambda _n\) in \(\Lambda \) such that \(\lambda _i = 0\) holds for \(i = 1,2 \ldots , N-1\) . In the sequel we use the notation of [ 29 , Section 8]. We have defined the functor of Walhausen categories

in ( 4.24 ). It induces an isomorphism

Given a natural number \(n \ge 1\) , elements \(\lambda _1, \lambda _1, \ldots , \lambda _n \in \Lambda \) , and a nilpotent endomorphism \(\varphi :\Phi (A) \rightarrow A\) in \(\mathcal {A}\) , define the morphism in \(\mathcal {A}_{\Phi }[t^{-1}]\)

by the ( n ,  n )-matrix

Thus we obtain a functor of Waldhausen categories

which induces a homomorphism

This map can easily be identified with the map sending \(y \in K_s({{\,\textrm{Nil}\,}}(\mathcal {A},\Phi ))\) to the image of \(T_s([C_n(\lambda _1, \lambda _2, \ldots \lambda _n)],y) = 0\) under the injective homomorphism ( 10.10 ). Hence it suffices to show that the image of the composite

is contained in the image of

where J is the obvious inclusion, and I sends an object A to ( A , 0).

Given a natural number \(n \ge 1\) and a nilpotent endomorphism \(\varphi :\Phi (A) \rightarrow A\) in \(\mathcal {A}\) , define the automorphism in \(\mathcal {A}_{\Phi }[t,t^{-1}]\)

Then we get in \(\mathcal {A}_{\Phi }[t,t^{-1}]\) that the composite \(F_n(\varphi ):= E_n(\varphi ) \circ U_{\lambda _1,\lambda _2, \ldots , \lambda _n}(\varphi ) :\Phi (A)^n \rightarrow A^n\) is given by the matrix

where we define for \(k = 1,2,, \ldots n\)

Note that \(E_n(\varphi )\) does not necessarily live in \(\mathcal {A}_{\Phi }[t^{-1}]\) . However the composite

does, where for any morphisms \(\psi :A \rightarrow B\) in \(\mathcal {A}_{\Phi }[t^{-1}]\) we denote by \(\psi ^n\) the morphism \(\bigoplus _{j=1}^n \psi :A^n = \bigoplus _{j=1}^n A\rightarrow B^n = \bigoplus _{j=1}^n B\) and \({{\,\textrm{id}\,}}_{\Phi ^{1-n}(A)} \cdot t^{1-n}\) is a morphism from \(A \rightarrow \Phi ^{1-n}(A)\) in \(\mathcal {A}_{\Phi }[t^{-1}]\) . So the composite

of morphisms in \(\mathcal {A}_{\Phi }[t^{-1}]\) is given by the matrix

where we define for \(k = 1,2, \ldots n\)

for the morphism \(\varphi ^{(i)} = \varphi \circ \Phi (\varphi ) \circ \cdots \circ \Phi ^i(\varphi ) :\Phi ^{i+1}(A) \rightarrow A\) in \(\mathcal {A}\) .

Note that we get functors of Waldhausen categories

sending an object \((A,\varphi )\) to \(\widehat{E}_n(\varphi )\) and

Note that \(w_n(\varphi ) = 0\) if \(\varphi ^{(N)} = 0\) , since \(\lambda _1 = \lambda _{N-1} = 0\) is assumed. Hence \(\widehat{F}_n \circ J\) is given by a lower triangular matrix, all whose entries on the diagonal are given by \({{\,\textrm{id}\,}}_{\Phi ^{1-n}(A)} \cdot t^{-n}\) . The matrix \(\widehat{E}_n(\varphi )\) is given by a lower triangular matrix, all whose entries on the diagonal are given by \({{\,\textrm{id}\,}}_{\Phi ^{-n}(A)} \cdot t^{-n}\) . This implies that the image of both \(K_s(\widehat{E}_n) \circ K_s(J)\) and \(K_s(\widehat{F}_n) \circ K_s(J) \) lie in the image of

Hence the same statement is true for \(K_s(U_{\lambda _1,\lambda _2, \ldots , \lambda _n}) \circ K_s(J)\) . This finishes the proof that the pairing \(T_s\) of ( 10.7 ) extends to a continuous pairing

for \(s \in \mathbb {Z}\) .

Since the image of \(\eta \) is dense in W ( R ), this continuous extension is unique.

Obviously the pairing \(T_n\) turns \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) into a module over the commutative ring \(\overline{K}_0({{\,\textrm{End}\,}}(\Lambda ))\) . We conclude from the uniqueness of the extension \(\widehat{T}_n\) that it turns \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) into a module over the commutative ring \(W(\Lambda )\) .

This finishes the proof of Lemma  10.8 . \(\square \)

10.4 Consequences of the \(W(\Lambda )\) -module structure on the Nil-terms

Theorem 10.11.

Let N be a natural number. Let \(\Lambda \) be a commutative ring with unit \(1_{\Lambda }\) . Consider an additive \(\Lambda \) -category \(\mathcal {A}\) together with an automorphism \(\Phi :\mathcal {A}\xrightarrow {\cong } \mathcal {A}\) of additive \(\Lambda \) -categories. Let \(\mathcal {P}\) be a set of primes. Then we get for every \(n \in \mathbb {Z}\) :

If \(\Lambda \) is an \(\mathbb {Z}[\mathcal {P}^{-1}]\) -algebra, then \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) is an \(\mathbb {Z}[\mathcal {P}^{-1}]\) -module;

If \(\Lambda \) is a \(\mathbb {Z}_p\) -algebra, then \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))\) is a \(\mathbb {Z}_p\) -module;

If \(N \cdot 1_{\Lambda }\) is zero in \(\Lambda \) , then \(\overline{K}_n({{\,\textrm{Nil}\,}}(\mathcal {A};\Phi ))[1/N]\) vanishes.

Using Lemma  10.8 the proof in Weibel [ 41 , Corollary 3.3] extends to our setting, see also [ 9 , Lemma 3.10]. \(\square \)

11 The Farrell–Jones Conjecture for totally disconnected groups at the prime p

Throughout this section G is a td-group , i.e., a locally compact second countable totally disconnected topological Hausdorff group.

11.1 Various sets of primes

We need the following sets of primes.

Notation 11.1

( \(\mathcal {P}(G)\) ) Let G be a td-group. Define \(\mathcal {P}(G)\) to be the set of primes q , for which there exist compact open subgroups \(U'\) and U of G such that \(U' \subseteq U\) holds and q divides the index \([U:U']\) .

If G is a compact td-group, then \(\mathcal {P}(G)\) is the set of all primes q for which there exists a compact open subgroup U of G such that q divides [ G  :  U ]. We have \(\mathcal {P}(\widetilde{U}) = \{p\}\) for the compact open subgroup \(\widetilde{U} \subseteq Q\) appearing in Assumption  1.9 , provided that \(\widetilde{U}\) is non-trivial.

Let \(G'\) be a (not necessarily open or compact) closed subgroup of the td-group G . Then \(\mathcal {P}(G') \subseteq \mathcal {P}(G)\) .

Consider compact open subgroups \(L'\) and L of \(G'\) satisfying \(L' \subseteq L\) . Since L is a compact subgroup of G , we can find a compact open subgroup \(L_0 \subseteq G\) with \(L \subseteq L_0\) , see [ 5 , Lemma 2.3]. Since \(L'\) is open in L , we can find an open subset \(V \subseteq L_0\) with \(L' = V \cap L\) . Since \(L'\) is a compact subgroup of \(L_0 \) and contained in V , we can find a compact open subgroup \(L'_0\) of \(L_0\) such that \(L' \subseteq L_0' \subseteq V \) holds, see [ 5 , Lemma 2.3]. Hence we get \(L'_0\cap L = L'\) . This implies that the obvious map \(L/L' \rightarrow L_0/L_0'\) is injective and hence \([L:L']\) divides \([L_0: L_0']\) . Since \(L_0\) and \(L_0'\) are compact open subgroups of G , any prime q that divides \([L:L']\) divides \([L_0: L_0']\) and hence belongs to \(\mathcal {P}(G)\) . This implies \(\mathcal {P}(G') \subseteq \mathcal {P}(G)\) . \(\square \)

Remark 11.3

Suppose that G has up to conjugacy only finitely many maximal compact open subgroups \(K_1\) , \(K_2\) , ..., \(K_n\) . (This assumption is satisfied for every reductive p -adic group.) Suppose that Assumption  1.9 is satisfied and let \(\widetilde{U}\) be the compact open subgroup appearing in Assumption  1.9 . For \(i = 1,2, \ldots n\) the subgroup \(K_i \cap \widetilde{U}\) has finite index in \(K_i\) and we define \(\mathcal {P}_i\) to be the finite set consisting of those primes that divide the index \([K_i: (K_i \cap \widetilde{U})]\) . One easily checks that \(\mathcal {P}(K_i) \subseteq \mathcal {P}(K_i \cap \widetilde{U}) \cup \mathcal {P}_i\) holds. Since \(\mathcal {P}(\widetilde{U}) \subseteq \{p\}\) , we conclude from Lemma  11.2

In particular \(\mathcal {P}(G)\) is a finite set.

The notion of a Hecke category with G - support is introduced in [ 6 , Definition 5.1]. Given a Hecke category with G -support and a open subgroup \(U \subseteq G\) , one obtains by restriction to U the Hecke category with U -support \(\mathcal {B}|_U\) , see [ 6 , Notation 5.4]. For the notions of a uniform regular or l -uniform regular additive category we refer to [ 4 , Definition 6.2].

Notation 11.4

( \(\mathcal {P}(\mathcal {B})\) ) Let \(\mathcal {B}\) be Hecke category with G -support. Let \(\mathcal {P}(\mathcal {B})\) be the largest set of primes with the property that for any \(d \in \mathbb {Z}\) with \(d \ge 0\) there is \(l(d) \in \mathbb {Z}\) with \(l(d) \ge 0\) such that for every compact open subgroup U of G with \(\mathcal {P}(U) \subseteq \mathcal {P}(\mathcal {B})\) the category \((\mathcal {B}_U)_\oplus [\mathbb {Z}^d]\) is l ( d )-uniformly regular coherent.

Notation 11.5

( \(\mathcal {P}(R)\) ) For a ring R let \(\mathcal {P}(R)\) be the set of primes which are invertible in R .

Remark 11.6

If the ring R satisfies \(\mathbb {Q}\subseteq R\) , or, equivalently, that \(\mathcal {P}(R)\) consists of all primes, then the Hecke algebra \(\mathcal {H}(G,R,\rho ,\omega )\) and the Hecke category with Q -support \(\mathcal {B}(Q,R,\rho ,\omega )\) are introduced in [ 7 , Section 2.2] and [ 6 , Sections 6.B and 6.C] for any G , N , \({{\,\textrm{pr}\,}}:G \rightarrow Q\) , \(\rho \) , and \(\omega \) as appearing in [ 7 , Section 2.1] and in [ 6 , Section 6.A]. We will assume that \(N \subseteq G\) is locally central , i.e., its centralizer in G is an open subgroup of G . If we replace the condition \(\mathbb {Q}\subseteq R\) by Assumption  1.9 applied in the case where G is replaced by Q , then \(\mathcal {H}(G,R,\rho ,\omega )\) and \(\mathcal {B}(G,R,\rho ,\omega )\) are still defined. Recall that Assumption  1.9 is satisfied if there exists a prime p such that p is invertible in R and Q is a subgroup of a reductive p -adic group, see [ 30 , Lemma 1.1] and Lemma  11.2 .

Now suppose that R is uniformly regular and that Assumption  1.9 , is satisfied. Then \(\mathcal {P}(R)\) is contained in \(\mathcal {P}(\mathcal {B}(G,R,\rho ,\omega ))\) . The proof of this fact is the same as the one of [ 7 , Theorem 7.2], one just has to observe that [ 7 , Lemma 7.4 (3)] still holds if one replaces the condition \(\mathbb {Q}\subseteq R\) by the condition that the order of the finite group | D | is invertible in R , and that \({{\,\textrm{Idem}\,}}(\mathcal {B}(G,R,\rho ,\omega )_{\oplus })\) is equivalent to \({{\,\textrm{Idem}\,}}(\underline{\mathcal {H}(G,R,\rho ,\omega )}_{\oplus })\) , see [ 6 , Lemma 6.6].

Given a set \(\mathcal {P}\) of primes, a map of abelian groups is called a \(\mathcal {P}\) - isomorphism if it becomes an isomorphism after inverting every element in \(\mathcal {P}\) . A map of spectra \(\textbf{f}:\textbf{E}\rightarrow \textbf{F}\) is called a weak \(\mathcal {P}\) -homotopy equivalence if \(\pi _n(F) :\pi _n(\textbf{E}) \rightarrow \pi _n(\textbf{F})\) is a \(\mathcal {P}\) -isomorphism for all \(n \in \mathbb {Z}\) . If \(\mathcal {P}\) is empty, this is of course the same as a weak homotopy equivalence.

11.2 The Farrell–Jones Conjecture in prime characteristic

Definition 11.7.

Definition 11.9

This is the \({\mathcal {C}\textrm{vcy}}\) - assembly map for \(\mathcal {B}\) .

We want to prove

Theorem 11.10

Let p be a prime. Assume that G is modulo a compact subgroup isomorphic to a closed subgroup of a reductive p -adic group. Let \(\mathcal {B}\) be a category with G -support.

If \(\mathcal {P}(\mathcal {B})\) consists all primes, then Theorem  11.10  (ii) has already been proved in [ 5 , Theorem 1.11 and Theorem 1.15].

In Sect.  11.3 the proof of Theorem  11.10 will be obtained by inspecting the proof in [ 5 , Theorem 1.11 and Theorem 1.15] taking Theorem  10.11   (iii) into account.

11.3 Reduction from to \({\mathcal {C}\textrm{vcy}}\)

In this subsection we give the proof of Theorem  11.10 .

We conclude from by [ 5 , Theorem 5.15],

Theorem 11.11

The \({\mathcal {C}\textrm{vcy}}\) -assembly map ( 11.9 ) is a weak homotopy equivalence for any Hecke category with G -support \(\mathcal {B}\) if G is a reductive p -adic group.

Theorem 11.12

Let \(\mathcal {P}\) be a (possibly empty) set of primes. Suppose that the \({\mathcal {C}\textrm{vcy}}\) -assembly map ( 11.9 )

is a \(\mathcal {P}\) -equivalence.

is also \(\mathcal {P}\) -equivalence.

The arguments on the proof that [ 5 , Theorem 14.7] implies [ 5 , Theorem 14.1] presented in [ 5 , Section 14.B] carry directly over to our setting and lead directly to a proof of Theorem  11.12 . Namely, in the diagram appearing [ 5 , Section 14.B] the maps \(\alpha _1\) and \(\widehat{\alpha _1}\) are \(\mathcal {P}\) -equivalences and the other four maps labeled by \(\simeq \) are also weak homotopy equivalences in our setting. \(\square \)

Lemma 11.13

Let \(\mathcal {P}\) be a set of primes (which may be empty). Then the following assertions are equivalent

is a \(\mathcal {P}\) -equivalence;

This follows from [ 5 , Proposition 14.9] and the commutative diagram [ 5 , 14.10] since the proof of [ 5 , Proposition 14.9], the construction of the commutative diagram [ 5 , 14.10] and the fact that the map (1)–(9) appearing there are weak homotopy equivalences presented in [ 5 , Subsections 14.C, 14.E, and 14.F] carry directly over to our setting, since no assumptions about \(\mathcal {B}\) are used. \(\square \)

Lemma 11.14

Let N be a natural number such that for the category \(\mathcal {B}\) with G -support the underlying \(\mathbb {Z}\) -category \(\mathcal {B}\) is obtained by restriction with the projection \(\mathbb {Z}\rightarrow \mathbb {Z}/N\) from a \(\mathbb {Z}/N\) -category \(\mathcal {B}'\) . Then the

is a \(\mathcal {P}\) -homotopy equivalence for \(\mathcal {P}\) the set of primes dividing N .

Lemma 11.15

If \(\mathcal {P}(G) \subseteq \mathcal {P}(\mathcal {B})\) holds, then then canonical map

is a weak homotopy equivalence.

The proof that the map appearing in Lemma  11.15 is a weak homotopy equivalence is a variation of the one appearing in [ 5 , Section 14.H]. By the arguments in [ 5 , Section 14.H] it suffices to check that the map [ 5 , (14.14)] is a weak homotopy equivalence, since the map appearing in Lemma  11.15 is the map (7) appearing at the very bottom of the diagram [ 5 , (14.10)]. Hence it suffices to check that the argument appearing [ 5 , Lemma 14.16] carries over if we assume \(\mathcal {P}(G) \subseteq \mathcal {P}(B)\) and do not use not the stronger [ 5 , Assumption 3.11], which is equivalent to requiring that we can choose \(\mathcal {P}(\mathcal {B})\) to be the set of all primes, since then we can apply [ 4 , Theorem. 14.1], exactly as we did in [ 5 , Section 14.H] after the proof of [ 5 , Lemma 14.16].

Let \((U_{r,i})_{r = 1,\ldots ,n, i \in \mathbb {N}_{\ge 1}}\) be the system of compact open subgroups of G as they appear in [ 5 , Section 14.H]. In view of the proof of [ 5 , Lemma 14.16], it remains to explain why for every \(i \in \{1,2, \ldots , n \}\) and every element \(\lambda \in |Q_i|\) the category \(\bigl ((\mathcal {B}|_{G_{\lambda }})_\oplus \bigr )[\mathbb {Z}^d]\) is l -uniformly regular coherent, where \(|Q_i| = G/U_{1,i} \times \cdots \times G/U_{n,i}\) . In the proof of [ 5 , Lemma 14.16] we had used [ 5 , Assumption 3.11] precisely at this place, but nowhere else in the proof of [ 5 , Theorem 14.7], and we do not want to use this assumption here. In the situation here it suffices to show \(\mathcal {P}(G_{\lambda }) \subseteq \mathcal {P}(\mathcal {B})\) . This follows from the assumption \(\mathcal {P}(G) \subseteq \mathcal {P}(\mathcal {B})\) and Lemma  11.2 . This finishes the proof of Lemma  11.15 . \(\square \)

Proof of Theorem 11.10

Theorem  11.10 follows from Theorems  11.11 , 11.12 , Lemmas  11.13 , 11.14 , and 11.15 , provided that G is a reductive p -adic group. Now the general case, where G is modulo a compact subgroup isomorphic to a closed subgroup of a reductive p -adic group, follows from the proof of [ 6 , Theorem 1.5], which directly carries over to our setting. \(\square \)

Notation 11.16

( \(\mathcal {P}(\mathcal {B},G)\) ) Let \(\mathcal {B}\) be Hecke category with G -support. Define \(\mathcal {P}(\mathcal {B},G)\) to be the set of primes q that belong to \(\mathcal {P}(G)\) but not to \(\mathcal {P}(\mathcal {B})\) .

Conjecture 11.17

is a \(\mathcal {P}(\mathcal {B},G)\) -equivalence.

If \(\mathcal {P}(G) \subseteq \mathcal {P}(\mathcal {B})\) , then Theorem  11.10  (ii) confirms Conjecture  11.17 if G is modulo a compact subgroup isomorphic to a closed subgroup of a reductive p -adic group.

11.4 Proof of Theorem  1.12

Proof of theorem 1.12.

(i) We conclude from Lemma  11.14 that the canonical map

is a \(\mathcal {P}_N\) -homotopy equivalence for \(\mathcal {P}_N\) the set of primes dividing N provided that N is a natural number such that for the category \(\mathcal {B}\) with G -support the underlying \(\mathbb {Z}\) -category \(\mathcal {B}\) is obtained by restriction with the projection \(\mathbb {Z}\rightarrow \mathbb {Z}/N\) from a \(\mathbb {Z}/N\) -category \(\mathcal {B}'\) .

(ii) In view of the arguments appearing in the proof of [ 8 , Theorem 2.16] based on the equivariant Atyiah-Hirzebruch spectra sequence of [ 8 , Theorem 2.1], it suffices to show that \(K_n(\mathcal {H}(p^{-1}(U),R,\rho |_{p^{-1}(U)},\omega )) \cong H_n^G(G/U;\textbf{K}_{\mathcal {B}(G,R,\rho ;\omega )})\) vanishes for every \(n \le -1\) and every compact open subgroup \(U \subseteq Q\) . The arguments in the proof of [ 7 , Lemma 8.1] apply also to our setting and imply that it suffices to show for any compact normal subgroup \(K \subseteq p^{-1}(U)\) with \(K \in P\) that \(K_n(\mathcal {H}(p^{-1}(U)//K,R,\rho |_{p^{-1}(U)},\omega )) = 0\) holds for \(n \le -1\) The proof of [ 7 , Lemma 7.5] carries over to our setting and implies that \(\mathcal {H}(p^{-1}(U)//K,R,\rho |_{p^{-1}(U)},\omega )\) can be identified as a crossed product ring \(R *D\) for some group D invertible in R . As R is Artinian, \(R *D\) and hence are \(\mathcal {H}(p^{-1}(U)//K,R,\rho |_{p^{-1}(U)},\omega )\) are Artinian. Now we conclude from [ 25 , Theorem 4.15 (ii)] applied in the case of a trivial group and \(k = 0\) that \(K_n(\mathcal {H}(p^{-1}(U)//K,R,\rho |_{p^{-1}(U)},\omega )) = 0\) holds for \(n \le -1\) . This finishes the proof of assertion (ii). \(K \subseteq Q\) .

(iii) The proof that the assembly ( 1.10 ) is bijective is analogous to the proof of assertion (i) but now using Lemma  11.15 instead of Lemma  11.14 . \(\square \)

Remark 11.18

It is conceivable that assertion (ii) appearing in Theorem  1.12 is still true if we do not invert N . (We have no proof.) For instance it may be true that, for a prime p , an Artinian ring R for which p is invertible in R , and a subgroup G of a reductive p -adic group,

holds and the map

is bijective, or, equivalently, that the assembly map ( 1.10 ) is bijective for \(n \le 0\) . (The latter is not true in general under the assumptions above for \(n \ge 1\) .)

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Acknowledgements

The paper is funded by the ERC Advanced Grant “KL2MG-interactions” (no. 662400) granted by the European Research Council and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy — GZ 2047/1, Projekt-ID 390685813, Hausdorff Center for Mathematics at Bonn. The author thanks Ian Hambleton, Dominik Kirstein, and Christian Kremer for helpful conversations and the (unknown) referee for his detailed and very helpful report. On behalf of all authors, the corresponding author states that there is no conflict of interest. My manuscript has no associated data. The paper is organized as follows:

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Lück, W. Relative assembly maps and the K -theory of Hecke algebras in prime characteristic. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02966-x

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Received : 08 February 2024

Revised : 27 July 2024

Accepted : 31 July 2024

Published : 23 August 2024

DOI : https://doi.org/10.1007/s00208-024-02966-x

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