相关论文: K-theory and G-theory of derived algebraic stacks

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and…

We compute the K-theory of ring C*-algebras for polynomial rings over finite fields. The key ingredient is a duality theorem which we had obtained in a previous paper. It allows us to show that the K-theory of these algebras has a ring…

We examine the theory of connective algebraic K-theory, CK, defined by taking the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded…

Computation of the K- and KO-theory for the classifying G-spaces for proper actions of certain infinite discrete groups G via a special version of the equivariant Atiyah- Hirzebruch spectral sequence.

This paper investigates the derived and spectral analogs of logarithmic geometry. We develop the deformation theory for animated log rings and $\mathbb{E}_\infty$-log rings and examine the corresponding theories of derived and spectral log…

This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see http://www.claymath.org/programs/outreach/academy/colloquium2005.php These lectures were intended to very young (and motivated) college…

Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank…

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 $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and let $\mu \in \mathfrak{g}^{*}$. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of $\mu$ and the set of…

Algebraic $K$-theory is a homology theory that behaves very well on sufficiently nice objects such as stable $C^*$-algebras or smooth algebraic varieties, and very badly in singular situations. This survey explains how to exploit this to…

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in…

The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of…

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, matrix-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, kk_*(A,B),…

Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor…

Both boundary maps in K-theory are expressed in terms of surjections from projective C*-algebras to semiprojective C*-algebras.

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric…

This paper extends the notion of geometric control in algebraic K-theory from additive categories with split exact sequences to other exact structures. In particular, we construct exact categories of modules over a Noetherian ring filtered…

We develop a theory of good moduli spaces for derived Artin stacks, which naturally generalizes the classical theory of good moduli spaces introduced by Alper. As such, many of the fundamental results and properties regarding good moduli…

We introduce bivariant K-theory for nonarchimedean bornological algebras over a complete discrete valuation ring $V$. This is the universal target for dagger homotopy invariant, matricially stable and excisive functors, similar to bivariant…

To any left system of diagram categories or to any left pointed derivateur (in the sense of Grothendieck) a K-theory space is associated. This K-theory space is shown to be canonically an infinite loop space and to have a lot of common…