Equivariant Algebraic K-theory Research Articles

Parametrised Presentability Over Orbital Categories

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.

Homotopy Mackey functors of equivariant algebraic K-theory

Given a finite group G acting on a ring R, Merling constructed an equivariant algebraic K-theory G-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide an interpretation of the homotopy Mackey functors of equivariant algebraic K-theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the nth algebraic K-groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along homomorphisms of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic K-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K-theory of endomorphism rings, the K-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings.

Spectral Mackey functors and equivariant algebraic K-theory, II

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal $\infty$-categories and a suitable generalization of the second named author's Day convolution, we endow the $\infty$-category of Mackey functors with a well-behaved symmetric monoidal structure. This makes it possible to speak of spectral Green functors for any operad $O$. We also answer a question of A. Mathew, proving that the algebraic $K$-theory of group actions is lax symmetric monoidal. We also show that the algebraic $K$-theory of derived stacks provides an example. Finally, we give a very short, new proof of the equivariant Barratt-Priddy-Quillen theorem, which states that the algebraic $K$-theory of the category of finite $G$-sets is simply the $G$-equivariant sphere spectrum.

Rigidity for equivariant pseudo pretheories

We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic K-theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology.

Quantum K-theoretic geometric Satake: the case

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group$G$and the spherical perverse sheaves on the affine Grassmannian$Gr$of its Langlands dual group. Bezrukavnikov and Finkelberg developed a derived version of this equivalence which relates the derived category of$G^{\vee }$-equivariant constructible sheaves on$Gr$with the category of$G$-equivariant${\mathcal{O}}(\mathfrak{g})$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group$U_{q}\mathfrak{g}$. We define a convolution category$K\operatorname{Conv}(Gr)$whose morphism spaces are given by the$G^{\vee }\times \mathbb{C}^{\times }$-equivariant algebraic K-theory of certain fibre products. We conjecture that$K\operatorname{Conv}(Gr)$is equivalent to a full subcategory of the category of$U_{q}\mathfrak{g}$-equivariant${\mathcal{O}}_{q}(G)$-modules. We prove this conjecture when$G=\operatorname{SL}_{n}$. A key tool in our proof is the$\operatorname{SL}_{n}$spider, which is a combinatorial description of the category of$U_{q}\mathfrak{sl}_{n}$representations. By applying horizontal trace, we show that the annular$\operatorname{SL}_{n}$spider describes the category of$U_{q}\mathfrak{sl}_{n}$-equivariant${\mathcal{O}}_{q}(\operatorname{SL}_{n})$-modules. Then we use quantum loop algebras to relate the annular$\operatorname{SL}_{n}$spider to$K\operatorname{Conv}(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.

Equivariant iterated loop space theory and permutative G–categories

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V-fold loop G-spaces to several avatars of a recognition principle for infinite loop G-spaces. We then explain what genuine permutative G-categories are and, more generally, what E_{\infty} G-categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt-Priddy-Quillen theorem as a statement about genuine G-spectra and use it to give a new, categorical, proof of the tom Dieck splitting theorem for suspension G-spectra. Other examples are geared towards equivariant algebraic K-theory.

Categorical models for equivariant classifying spaces

Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Fixed points of the equivariant algebraic 𝐾-theory of spaces

In a recent work Malkiewich and Merling proposed a definition of the equivariant K K -theory of spaces for spaces equipped with an action of a finite group. We show that the fixed points of this spectrum admit a tom Dieck-type splitting. We also show that this splitting is compatible with the splitting of the equivariant suspension spectrum. The first of these results has been obtained independently by John Rognes.

Equivariant algebraic K-theory of G-rings

A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example, that the map of spectra with G-action induced by a G-map of G-rings is not equivariant. We define a version of equivariant algebraic K-theory which encodes a group action on the input in a functorial way to produce a genuine algebraic K-theory G-spectrum for a finite group G. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological K-theory, Atiyah’s Real K-theory, and existing statements about algebraic K-theory spectra with G-action. We recover the map from the Quillen–Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory.

Spectral Mackey functors and equivariant algebraic K-theory (I)

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory.More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors.Finally, we employ this technology to introduce fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. We use this to give what we think is the first general construction of π1ét-equivariant algebraic K-theory for profinite étale fundamental groups. This is key to our approach to the “Mackey functor case” of a sequence of conjectures of Gunnar Carlsson.

The completion problem for equivariant K-theory

Abstract In this paper, we study the Atiyah–Segal completion problem for the equivariant algebraic K-theory. We show that this completion problem has a positive solution for the action of connected groups on smooth projective schemes. In contrast, we show that this problem has a negative solution for non-projective smooth schemes, even if the action has only finite stabilizers.

K-theory and mapping spaces: an appendix to “Equivariant semi-topological K-homology and a theorem of Thomason”

AbstractWe discuss a geometric model for equivariant algebraic K-theory of quasiprojective varieties. This is inspired by and extends earlier work of Grayson-Walker on algebraic K-theory and related theories. Our main result is used by Hornbostel-Heller in their work on equivariant semi-topological K-homology and a theorem of Thomason.

An exotic Deligne-Langlands correspondence for symplectic groups

Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules.

Localization theorem in equivariant algebraic K-theory

In this paper, it is shown that the localized equivariant algebraic K-group of a scheme with a smooth diagonalizable group scheme action is isomorphic to that of its fixed point scheme. By using it, the trace formula for a coherent sheaf with a smooth diagonalizable group scheme action is proved.

AN E∞ RING G-SPECTRUM REPRESENTING THE EQUIVARIANT ALGEBRAIC K-THEORY OF A BIPERMUTATIVE G-CATEGORY

Journal Article AN E∞ RING G-SPECTRUM REPRESENTING THE EQUIVARIANT ALGEBRAIC K-THEORY OF A BIPERMUTATIVE G-CATEGORY Get access KAZUHISA SHIMAKAWA KAZUHISA SHIMAKAWA Research Institute for Mathematical Sciences, Kyoto UniversitySakyo-ku,Kyoto 606,Japan Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 45, Issue 3, September 1994, Pages 367–395, https://doi.org/10.1093/qmath/45.3.367 Published: 01 September 1994 Article history Received: 14 October 1991 Published: 01 September 1994

Hermitian forms and higher algebraic 𝐾-theory

Introduction. This note promulgates certain invariants for Hermitian forms over a ring with antistructure which are defined utilizing the algebraic ^-theory of the ring in a Z2-equivariant fashion. Low degree cases of these invariants extend the classical rank, discriminant, and Hasse-Witt invariants. The principal result described here is that these invariants (using all degrees) yield the stable classification of forms over many fields and algebras for which low degree cases alone do not usually suffice (because of signature invariants that may be defined). The manner in which this is accomplished is quite amusing, in that it sheds light on why the reductions of the signatures modulo certain powers of 2 are in fact determined by the low degree invariants. More general results can be obtained, e.g. for higher unitary algebraic A'-theory. However, these are suppressed here in favor of an outline of the equivariant algebraic Ktheory upon which this work so crucially relies. This will be dealt with at length in [Gi2], and arithmetic aspects will be pursued further elsewhere. 1. Hermitian forms. If R is a ring and MR is the category of right R-modules, then PR will denote the full, exact subcategory of M# determined by the finitely generated projective modules, and KnR, n E Z, will denote the (GerstenWagoner) algebraic K-theory groups of R (cf. [Ge] ). Recall that an antistructure (a, u) on R consists of an antiautomorphism r »—* r of R and a unit uGR such that uu = 1 and r* = unT, rER. An Hermitian form over (R, a, w), or an (a, u)-form over R, is a pair (P, A) with P E ?R and X: P x P —+ R such that \(x, y 4zf) = X(x, y) + X(x, z)r, \(y, x) = \(x, y)u for xf y, z E P, r E R, and (Ad X): P —> Da? is an isomorphism where (Ad XX*)00 = X(JC, y), x, y E R. Here, Da: VR —* VR is the a-conjugate duality cofunctor Dal = Hom/2(?, R) where HomR(P, R) has right R-module structure (f° r)(x) = r /(x), x E P, r E R, ƒ E Z)aP. There are evident notions of isometry and orthogonal direct sum of (a, w)-forms over R (cf. [Wa] ), resulting in a permutative category U(R, a, u) and (Gersten-Wagoner type) unitary algebraic K-theory groups KUn(R, a, w), n E Z, of (R, a, w). Of primary interest here is KU0(R, a, u) which is just the Grothendieck group of U(R, a, u).