# Algebraic K-theory

@inproceedings{Ktheory2005AlgebraicK, title={Algebraic K-theory}, author={Algebraic K-theory and B. Guillou}, year={2005} }

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

#### 375 Citations

Commutative algebraic groups up to isogeny. II

- 2017

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 is… Expand

Algebraic K-Theory and Quadratic Forms

- 2005

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 an… Expand

K-Theory of Azumaya Algebras over Schemes

- Mathematics
- 2009

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) are… Expand

Unstable operations on K-theory for singular schemes

- Mathematics
- 2021

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 simplicial… Expand

K0 and the dimension filtration for p-torsion Iwasawa modules

- Mathematics
- 2006

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 of… Expand

Mixed Weil cohomologies

- Mathematics
- 2012

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

- Mathematics
- 2012

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 long… Expand

Non-commutative Fitting invariants and annihilation of class groups

- Mathematics
- 2010

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 important… Expand

OVERGROUPS OF ELEMENTARY SYMPLECTIC GROUPS

- Mathematics
- 2004

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 the… Expand

Topological K-theory of complex noncommutative spaces

- 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 cyclic… Expand

#### References

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Algebraic K-Theory and Its Applications

- Mathematics
- 1995

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 broad… Expand

Presentations of groups

- Mathematics
- 1976

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 than… Expand

AN INTRODUCTION TO ALGEBRAIC K-THEORY

- 2005

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