Related papers: A linearization map for genuine equivariant algebr…

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…

We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global…

In this paper, we build on the work from our previous paper (arXiv:2002.01556) to show that periodic rational $G$-equivariant topological $K$-theory has a unique genuine-commutative ring structure for $G$ a finite abelian group. This means…

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git…

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type,…

A group equivariant $KK$-theory for rings will be defined and studied in analogy to Kasparov's $KK$-theory for $C^*$-algebras. It is a kind of linearization of the category of rings by allowing addition of homomorphisms, imposing also…

We study the $K$-theory and Swan theory of the group ring $R[G]$, when $G$ is a finite group and $R$ is any ring or ring spectrum. In this setting, the well-known assembly map for $K(R[G])$ has a companion called the coassembly map. We…

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with an anti-involution $a_G$ which is the composition of a Lie group involutive automorphism $\sigma_G$ and the group inversion. We view $(G, a_G)$ as a Real $(G,…

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied…

We give a description of the value of a finitary localizing invariant, such as algebraic $K$-theory, on the category of sheaves on a locally coherent space $X$. This in particular includes all spaces that arise as spectra of commutative…

Let X be a pointed connected simplicial set with loop group G. The linearisation map in K-theory as defined by Waldhausen uses G-equivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on…

Recent work has applied scissors congruence $K$-theory to study classical cut-and-paste ($SK$) invariants of manifolds. This paper proves the conjecture that the squares $K$-theory of equivariant $SK$-manifolds arises as the fixed points of…

In this article we describe the $\tG\times \tG$-equivariant $K$-ring of $X$, where $\tG$ is a {\it factorial} cover of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the…

We give a general formula for the equivariant complex $K$-theory $K_G^*(V)$ of a finite dimensional real linear space $V$ equipped with a linear action of a compact group $G$ in terms of the representation theory of a certain double cover…

We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the…

We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real…

A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. We answer…

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are…

In this paper, we calculate the image of the connective and periodic rational equivariant complex $K$-theory spectrum in the algebraic model for naive-commutative ring $G$-spectra given by Barnes, Greenlees and K\k{e}dziorek for finite…

Let $G$ be a connected semisimple Lie group with its maximal compact subgroup $K$ being simply-connected. We show that the twisted equivariant $KK$-theory $KK^{\bullet}_{G}(G/K, \tau_G^G)$ of $G$ has a ring structure induced from the…