# Motives Seminar-WS 2012/13

## Dg categories and derived Morita theory

Thanks to our visitor for this semester, Prof. Christian Haesemeyer (UCLA), the motives seminar will present a short course in dg categories and derived Morita theory, running up to the Christmas break. After that we will revert to our normal operating procedure, a working seminar in a topic still to be determined (see some suggestions below).

**Abstract:** The goal of this course is to approach an understanding of Toen's derived Morita theory ("The homotoy theory of dg categories and derived Morita theory", Inv. Math. 167, 2007 also available on the author's webpage) and in particular some of its applications like the description of Hochschild cohomology as endomorphisms of the identity functor. However we will start by following Bernard Keller's overview of dg categories ("On differential graded categories", available on the author's webpage). Should any time remain we will try to either study the theory of higher and derived stacks (yes, higher and derived are actually standing for different things here) - if quite a lot of time remains - or Tabuada's localization of the category of dg categories in which K-theory becomes representable - if only little time remains.

### Schedule:

The motives seminar/short course will meet each week on Tuesdays, 14-16h (c.t.) in the seminar room 4N (4th floor, North side). First meeting: Tues. October 23.

Talk on Sept. 18, 2012:

Viraj Navkal (UCLA) will speak on:

Title: K'-Theory of a Complete Local CM Ring of Finite Representation Type

Abstract:Let R be a complete local Cohen-Macaulay ring of finite representation type. By a theorem of Auslander and Reiten, the Grothendieck group of the category of maximal Cohen-Macaulay R-modules depends only on its Auslander-Reiten sequences. In this talk I will discuss an ongoing project to understand the relationship between the higher K-theory and Auslander-Reiten theory of such a category.

We're taking suggestions for topics for the motives seminar this semester. If you have a suggestion, send it to me at

and I'll post it here (without your name, so feel free!).

## Suggested Topics

1. Moritz Kerz's work on Gersten's conjecture for Milnor K-theory.

The Gersten conjecture for Milnor K -theory. Invent. Math. 175 (2009), no. 1, 1-33.

2. The work of Uwe Jannsen, and Shuji Saito on Kato's conjecture.

References: The survey article: Shuji Saito, "Recent progress on the Kato conjecture". Quadratic forms, linear algebraic groups, and cohomology, 109-124, Dev. Math., 18, Springer, New York, 2010.

and preprints

"Kato conjecture and motivic cohomology over finite fields". Uwe Jannsen, Shuji Saito, arXiv:0910.2815v1

"Hasse principles for higher-dimensional fields" Uwe Jannsen, arXiv:0910.2803v1

3. Grothendieck-Witt groups and the Quillen-Lichtenbaum conjecture for Hermitian *K*-theory:

http://homepages.warwick.ac.uk/~masiap/research/QLGW.pdf

4. Mixed Tate motives from various angles. This could include the article of Bloch-Kriz, *Mixed Tate motives*. Ann. of Math. (2) **140** (1994), no. 3, 557–605, and Francis Brown's Annals paper,* Mixed Tate motives over Z*, Annals of Math. (2)

**175**(2012)no. 2, 949-976

5. Harbater-Hartman-Krashen patching. This gives methods for proving local to global principals for homogeneous spaces in certain situations.

References. Harbater, Hartmann, Krashen:

*Patching subfields of division algebras*. Trans. Amer. Math. Soc. 363 (2011), no. 6, 3335–3349.

*Patching over fields*. Israel J. Math. 176 (2010), 61–107.

* Applications of patching to quadratic forms and central simple algebras*. Invent. Math. 178 (2009), no. 2, 231–263.