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Waldhausen K-theory of a dg-category

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable Homotopy theory

Contents

Idea

The Waldhausen K-theory of a dg-category is defined as the Waldhausen K-theory of a Waldhausen category associated to it. It is an additive invariant in the sense of noncommutative motives. There is a nonconnective variant studied by Marco Schlichting, which is a localizing invariant (again in the sense of noncommutative motives).

On the other hand, one could define the K-theory of a pretriangulated dg-category as the algebraic K-theory of its dg-nerve, which is a stable infinity-category. This should coincide with the Waldhausen K-theory (presumably).

Definition

Let AA be a dg-category. Consider the triangulated category D(A)D(A) of dg-presheaves on AA, and let perf(A)D(A)perf(A) \subset D(A) denote its full subcategory of compact objects. There is the structure of a Waldhausen category on perf(A)perf(A) where the weak equivalences are objectwise quasi-isomorphisms and the cofibrations are degreewise split monomorphisms. The Waldhausen K-theory of AA is the Waldhausen K-theory of perf(A)perf(A) with this Waldhausen structure.

References

  • Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006, pdf.

  • Marco Schlichting, Negative K-theory of derived categories, Math. Z. 253 (2006), no. 1, 97 - 134, pdf.

  • Goncalo Tabuada, Higher K-theory via universal invariants, Duke Mathematical Journal, 145 (2008), no. 1, 121-206, arXiv.

Last revised on January 5, 2015 at 16:42:21. See the history of this page for a list of all contributions to it.