## Commutative Algebra Seminar Fall 2008

Thu, 04 Dec 2008: Claudia Miller, Syracuse

TITLE: Limit Hilbert-Kunz multiplicities

Wed, 03 Dec 2008: Srikanth Iyengar, UNL

TITLE: Frobenius and regularity

Thu, 20 Nov 2008: Sylvia Wiegand

Wed, 19 Nov 2008: Sylvia Wiegand

TITLE: Block diagonalization of matrices and 2-units over Pruefer domains (Parts 1 and 2)

ABSTRACT: We show that matrices over a large class of integral domains are equivalent to almost diagonal" matrices, a generalization of a 1972 result of L.S. Levy for Dedekind domains. Such matrices can be decomposed into a form with all the nonzero entries congregated in blocks along the diagonal, where the sizes of the diagonal blocks are determined by the class group of the integral domain. This result implies that, for $n$ sufficiently large, every $n\times n$ matrix is a sum of two invertible units. We explore connections with combinatorics and we show a few parlor tricks regarding matrices being sums of units.

Wed, 5 Nov 2008: Osamu Iyama  (Nagoya University, Japan)

Title: Tilting and cluster tilting for quotient singularities

Abstract:
Let k be a field of characteristic zero and S=k[x_1,...,x_d] the polynomial ring of d variables. For a finite subgroup
G of ${\rm SL}_d(k)$ acting on $k^d\backslash\{0\}$ freely, let R:=S^G be the corresponding quotient singularity and
$\widehat{R}$ its completion. By a classical result due to Herzog and Auslander, $\widehat{R}$ is of finite
Cohen-Macaulay type if d=2. As a generalization, there exists a $(d-1)$-cluster tilting object in the category of
maximal Cohen-Macaulay $\widehat{R}$-modules for arbitrary d. Using this, we will construct a tilting object in the category of
graded maximal Cohen-Macaulay R-modules. Consequently its stable category is triangle equivalent to the derived category of modules
over a finite dimensional k-algebra.

Thu, 30th Oct 2008: Suanne Au
Title: The Equivariant K-theory of Toric Varieties, Part II

Wed, 29 Oct 2008: Mu-wan Huang

Title : The Equivariant K-theory of Toric Varieties, Part I

Tue, 28 Oct 2008: Tony Puthenpurakal

Mon, 27 Oct 2008: Tony Puthenpurakal

Title : Properties of Koszul homology modules

ABSTRACT: We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, S_2 property and their Bass numbers. This is joint work with Uwe Nagel.

Thu, 23 Oct 2008: Irena Swanson (Reed College)

TITLE: Goto numbers of parameter ideals

Wed, 22 Oct 2008: YiHuang Shen (Purdue University)

Title: Stanley decompositions for squarefree monomial ideals

ABSTRACT: We will introduce the Stanley decompositions for multigraded modules. Herzog, Vladoiu and Zheng's method for computing Stanley depths of monomial ideals will be presented and some of our recent progress for squarefree monomial ideals will be discussed.

Wed, 15 Oct 2008: Mark Walker

TITLE : Hochster's theta pairing and the Hodge Index Theorem

Thu, 09 Oct 2008: Srikanth Iyengar

TITLE: Homology over complete intersections via exterior algebra. Part II

Wed, 08 Oct 2008: Luchezar Avramov

TITLE: Homology over complete intersections via exterior algebra. Part I

Thu, 18 Sep 2008: Tom Marley

Wed, 17 Sep 2008: Tom Marley

TITLE : Coherent Gorenstein rings (I, II)

Thu, 11 Sep 2008: Roger Wiegand

Wed, 10 Sep 2008: Roger Wiegand

TITLE: Extended modules.

ABSTRACT: This is joint work with Wolfgang Hassler on the following general problem: If R \to S is a flat local homomorphism and N is a finitely generated S-module, when is there an R-module M such that N \cong S\otimes_R M as S-modules? The talk will be understandable, possibly even enjoyable, for students who are not yet experts on commutative algebra. Some of the talk will be a bit speculative, and some open problems will be mentioned.

3 and 4 Sep 2008: Suresh Nayak (Chennai Mathemtical Institute and UNL)

Title: Semi-invariants of matrices under simultaneous left-right action

27 and 28 Aug 2008: Srikanth Iyengar (UNL)
Title: Beyond finite representation type: Representation dimension