Algebraic K-theory

  title={Algebraic K-theory},
  author={Algebraic K-theory and B. Guillou},
The idea will be to associate to a ring R a set of algebraic invariants, Ki(R), called the K-groups of R. We can even do a little better than that: we will associated an (infinite loop) space K(R) to R and the K-groups will be the homotopy groups of this space. In fact, the first example of interest was not the K-theory of a ring but rather of a category of coherent sheaves on a scheme. The K-theory of a ring R is defined to be the K-theory of the category of finitely generated projective… Expand
Commutative algebraic groups up to isogeny. II
This paper develops a representation-theoretic approach to the isogeny category C of commutative group schemes of finite type over a field k, studied in [Br16]. We construct a ring R such that C isExpand
Algebraic K-Theory and Quadratic Forms
The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to anExpand
K-Theory of Azumaya Algebras over Schemes
Let X be a connected, noetherian scheme and 𝒜 be a sheaf of Azumaya algebras on X, which is a locally free 𝒪 X -module of rank a. We show that the kernel and cokernel of K i (X) → K i (𝒜) areExpand
Unstable operations on K-theory for singular schemes
Abstract We study the algebraic structures, such as the lambda ring structure, that arise on K-theory seen as an object of some homotopy categories coming from model categories of simplicialExpand
K0 and the dimension filtration for p-torsion Iwasawa modules
Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k ofExpand
Mixed Weil cohomologies
Abstract We define, for a regular scheme S and a given field of characteristic zero K, the notion of K-linear mixed Weil cohomology on smooth S-schemes by a simple set of properties, mainly:Expand
Recollements of derived categories II: Algebraic K-theory
For a recollement of derived module categories of rings, we provide sufficient conditions to guarantee the additivity formula of higher algebraic K-groups of the rings involved, and establish a longExpand
Non-commutative Fitting invariants and annihilation of class groups
Abstract One can associate to each finitely presented module M over a commutative ring R an R-ideal Fitt R ( M ) which is called the (zeroth) Fitting ideal of M over R and which is an importantExpand
Let R be a commutative ring, and let l ‚ 2; for l = 2 it is assumed additionally that R has no residue fields of two elements. The subgroups of the general linear group GL(n;R) that contain theExpand
Topological K-theory of complex noncommutative spaces
  • A. Blanc
  • Mathematics
  • Compositio Mathematica
  • 2015
The purpose of this work is to give a definition of a topological K-theory for dg-categories over $\mathbb{C}$ and to prove that the Chern character map from algebraic K-theory to periodic cyclicExpand


Algebraic K-Theory and Its Applications
Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broadExpand
Presentations of groups
Primarily an introduction to combinatorial group theory, this book has the secondary aim of introducing a wide variety of examples of groups and types of groups. The emphasis is algebraic rather thanExpand
These are the notes of an introductory lecture given at The 20th Winter School for Geometry and Physics, at Srni. It was meant as a leisurely exposition of classical aspects of algebraic K-theory,Expand
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Algebraic K-Theory and Its Applications. Graduate texts in mathematics
  • Algebraic K-Theory and Its Applications. Graduate texts in mathematics
  • 1994
Presentations of Groups. London Mathematical Society Student Texts n @BULLET 15
  • Presentations of Groups. London Mathematical Society Student Texts n @BULLET 15
  • 1990
Let S be a symmetric monoidal groupoid. The K-theory space K(S) of S is then defined to be B(S −1 S). For a general symmetric monoidal category S, we define the K-theory space of S
    Of course this induces an equivalence on
    • Of course this induces an equivalence on
    Theorem 12. If C is split exact
    • Theorem 12. If C is split exact