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

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