## Commutative Algebra SeminarFall 11 & Spring 12

Seminars will be in  Avery 351 and will be held twice a week:

Wednesdays  3:30 - 4:20 pm
Thursdays      2:30 - 3:20 pm

#### This week's seminar

Past seminars
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Algebraic Geometry Seminar

University of Kansas Algebra Seminar

### Seminars

25th and 26th April  2012 (Wednesday and Thursday)
Roger Wiegand (UNL)
Title: Vanishing of Tor over complete intersections

Abstract: Suppose  M  and  N  are non-zero finitely generated modules over a local ring  R.  If  the tensor product  M \otimes_R N  has nice depth properties, e.g, is torsion-free, reflexive, or maybe maximal Cohen-Macaulay, one expects (perhaps incorrectly) that  M  and  N  too should have such properties.  Attempts to prove such results inevitably lead to consideration of  Tor_i^R(M,N)  and, in particular, to rigidity questions (vanishing of  Tor_i  for a few  i  forces vanishing of all higher  Tor's).  I will sketch Auslander's argument (from half a century ago!) that leads to these issues, show in detail how vanishing and rigidity questions impact questions on depths of tensor products, and then give some consequences of the vanishing of a certain pairing introduced recently by Hailong Dao.  The emphasis will be on motivating and tying together these ideas, rather than on technical details.  The talks will be based on joint work with Olgur Celikbas and Greg Piepmeyer.

19th April  2012 (Thursday)
Branden Stone (University of Kansas, Lawrence)
TBA

18th April  2012 (Wednesday)
Amanda Croll (UNL)
Title: The Jordan module of a local ring

4th and 5th April  2012 (Wednesday and Thursday)
Paolo Mantero (Purdue University, West Lafayette)
Title: Associated graded rings, Hilbert functions and a conjecture of Sally

Abstract: The associated graded ring of an ideal in a local ring R tends not to inherit good properties of R such as Cohen-Macaulayness or even Gorensteiness. However, Sally conjectured that if R is Cohen-Macaulay and its multiplicity is small (almost minimal), then the associated graded ring with respect to the maximal ideal of R is either Cohen-Macaulay or almost Cohen-Macaulay. The conjecture was proved independently by Rossi-Valla and Wang. We discuss joint work with Y. Xie on generalizing these results to the case of ideals not primary to the maximal ideals.

29th March  2012 (Thursday)
Kiriko Kato (Osaka Prefecture University, Japan)
Title: Triangulated categories of extensions and the Second Isomorphism Theorem for triangulated categories

Abstract: This is joint work with Peter J\orgensen. Let $\sT$ be a triangulated category with triangulated subcategories $\sX$ and $\sY$.  We show that the subcategory of extensions $\sX *\sY$ is triangulated if and only if $\sY * \sX \subseteq \sX * \sY$. In this situation, we show the following analogue of the Second Isomorphism Theorem: $( \sX * \sY ) / \sX \simeq \sY / ( \sX \cap \sY)$ and $( \sX * \sY ) / \sY \simeq \sX / ( \sX \cap \sY )$. This follows from the existence of a stable t-structure $\big( \frac{\sX }{ \sX \cap \sY } , \frac{ \sY }{ \sX \cap \sY } \big)$ in $( \sX* \sY ) / ( \sX \cap \sY )$.  We use the machinery to give a recipe for constructing triangles of recollements and recover some triangles of recollements from the literature.

8th March  2012 (Thursday)
Sukhendu Mehrotra (University of Wisconsin, Madison)
Title: Generalized deformations of K3 surfaces

Abstract: Moduli spaces of stable sheaves on K3 surfaces are examples of a very special class of varieties called irreducible holomorphic symplectic manifolds. These moduli spaces have been extensively studied by O'Grady, Huybrechts, Yoshioka and others. While any K3 X deforms in a 20 dimensional family, the Kuranishi space of such a moduli space M is known to be 21 dimensional. This means that the general deformation of M is not a moduli space of sheaves. The aim of this talk is to provide a description of these general deformations in terms of noncommutative'' K3s. This is joint work with Eyal Markman.

7th March  2012 (Wednesday)
Jeanette Shakalli Tang (Texas A & M)
Algebraic Deformation Theory

Abstract: A deformation of an algebra is obtained by slightly modifying its multiplicative structure. Algebraic deformation theory arises in many different areas of mathematics, such as combinatorics, representation theory and orbifold theory. In this talk, we will give an introduction to the general theory of deformations and provide some basic examples. We will show that, in general, finding a deformation of an algebra is quite a challenging problem. However, for a special type of algebra, there exists a method for finding deformations, which we will apply to obtain a new class of deformations.

29th February  2012
(Wednesday)
Yi Zhang (University of Minnesota)
Title: Some results on local cohomology in positive characteristic

Abstract:  Let R be a polynomial ring, in indeterminates x_1,...,x_n, over a field k of positive characteristic p. We will give a lower bound on the dimension of associated primes of local cohomology modules of R with respected to an ideal I in R, in terms of the degrees of the generators of I. Given homogeneous ideals I_1,...,I_s of R, we will describe grading of the  H^i_{m}(H^{j_1}_{I_1}\circ \cdots \circ H^{j_s}_{I_s}(R)), where m is the ideal generated by
x_1,...,x_n, and also give two algorithms to calculate it.

22nd and 23rd February  2012
(Wednesday and Thrusday)
Jon Carlson (University of Georgia)
The stable module category of a finite group

16th February  2012 (Thursday)
Chin-Yi Jean Chan (Central Michigan University)
Relational equivalence and the stability of the Hilbert-Kunz Function

Abstract: In this talk, we discuss how the the rational equivalence that defines the Chow group contributes to the study of the stability of the Hilbert-Kunz function of a module over an integral domain under certain conditions.  We will also estimate the "additive error" of the Hilbert-Kunz function on short exact sequences. This is a joint work with Kazuhiko Kurano, inspired by the paper of Huneke, McDermott and Monsky (Math. Res. Lett. 11 (2004) 539--546).

13th February  2012 (Monday) - Joint seminar with Algebraic Geometry
Tokuji Araya (Tokuyama College of Technology, Japan)
2:30 - 3:20 PM, Avery 351
Thick subcategories over graded simple singularities

Abstract: Ryo Takahashi classified the thick subcategories of the stable category of maximal Cohen-Macaulay modules over a hypersurface local ring. By his classification, we can see that if the base ring has a simple singularity, then the thick subcategories are trivial. On the other hand, if the base ring is graded, then there exist non-trivial thick subcategories. In this talk, we will classify the thick subcategories of the stable category of graded maximal Cohen-Macaulay modules over a graded hypersurface which has a simple singularity.

9th February  2012 (Thursday)
Yuji Yoshino (Okayama University, Japan)
Deformations and degenerations of modules

Abstract: Fixing an algebra, we can consider the deformations and the degenerations of modules. In the seminar I plan to give some of my recent results on the degenerations of modules and their stable analogy. I will discuss several examples to show how we can describe the degeneration order for maximal Cohen-Macaulay modules.

2nd February  2012 (Thursday)
Tom Marley (UNL)
The Frobenius functor and injective modules II

26th January 2012 (Thursday)
Tom Marley (UNL)
The Frobenius functor and injective modules I

23rd and 24th November (Wednesday and Thursday)
Thanksgiving

17th November (Thursday)
Math Day (UNL)

16th November (Wednesday)
Piyush Shroff (Texas A&M)

Finite Generation of Cohomology

Abstract:  Cohomology is a collection of algebraic invariants used to study geometric and algebraic objects. Cohomology of algebras contains lots of information. Especially in commutative algebra, many properties of algebras in which people are interested have a homological interpretation making it easier to organize information. The property of being finitely generated is very important because it is much easier to understand a finitely generated algebra. In this talk, first I will give some basic definitions and then describe my result about finite generation of cohomology of quotients of PBW algebras.

10th November (Thursday)

Mike Hopkins (Harvard  University)
Equivariant multiplicative closure

9th November (Wednesday)
Ryo Takahashi (UNL and Shinsu University)

Finiteness of dimensions of resolving subcategories

Abstract:  Let R be a commutative Noetherian ring, and let mod R be the category of finitely generated R-modules. In this talk, we define the dimension of a resolving subcategory of mod R. Our main results are concerning its finiteness, which are also related to the celebrated theorem of Auslander-Huneke-Leuschke-Wiegand and a recent result of Oppermann-Stovicek. This talk is based on joint work with Hailong Dao.

3rd November (Thursday)
Jesse Burke (Bielefeld University, Germany)

Support and free resolutions for modules over complete intersections via non-affine matrix factorizations

Abstract:  We first describe an equivalence, due to Orlov, that allows one to study many questions about modules over a complete intersection rings via coherent sheaves on a related non-affine hypersurface. We then discuss how such coherent sheaves may be described by matrix factorizations. Finally, we use these matrix factorizations to construct a support theory and free resolutions for modules over a complete intersection. This is joint work with Mark Walker.

2nd November (Wednesday)
Parker Lowery (University of Western Ontario, Canada)

A geometric moduli stack classifying the bounded derived category

Abstract:  We discuss the geometricity of the classifying stack of pseudo-coherent objects on a projective scheme.  This classifies what is commonly called the bonded derived category.   We then use this geometric stack to give a derived motivic Hall algebra for the scheme.  This algebra extends the Hall algebra associated to coherent sheaves and is important in calculating invariants associated to the given scheme.

27th October (Thursday)
Christian Haesemeyer (University of California, Los Angeles)

Rational points, zero cycles of degree one, and A^1-homotopy theory

Abstract:  In ordinary topology, whether a principal bundle has sections can be checked using obstruction theory - by computing cohomology classes (an example would be the question whether a family of tori over a Riemann surafce has a topological section). The analogous problem in algebraic geometry is the question whether a variety has a rational point (an example is the question whether a genus one curve over the function field of an algebraic curve has a rational point). I will try to describe an approach of importing topological methods into algebra to - in principle - construct an obstruction theory for existence of a rational point. This is joint work with Aravind Asok.

26th October (Wednesday)
Marcy Robertson (University of Western Ontario, Canada)

Koszul Duality and Periodic Homology Theories

Abstract:  We will survey the topology which led to the original bar and cobar constructions, viewing this as a part of the larger theory of Koszul duality of operads. We will use this classical example to discuss operadic algebra as a form of non-commutative geometry and redefine Koszul duality via a type of derived Morita theory of operads. Time permitting, we will explain how to use derived Morita theory to construct periodic (topological) homology theories.
No previous knowledge of operads will be assumed.

19th and 20th October (Wednesday and Thursday)
Mark Walker (UNL)

On the vanishing of Hochster's theta invariant and the Herbrand difference

Abstract:  For a hypersurface R with an isolated singularity, Hochster's theta invariant, \theta(M,N), and the Herbrand difference, h(M,N), are two closely related invariants of pairs of R-modules. When R = \C[[x_0, ..., x_n]]/(f), Polishchuk-Vaintrob have recently established an intriguing formula for the Herbrand difference that involves matrix factorizations and residues. In particular, they prove the Herbrand difference and hence the theta invariant vanish for all pairs of modules when n is even. More recently, Buchweitz-van Straten have established another interpretation of the invariants \theta and h,  in terms of the link pairing on the homology of the Milnor fiber associated to f(x_0, ..., x_n), and in particular they reproduce the vanishing result of Polishchuk-Vaintrob.

In this talk, I will describe purely algebraic generalizations of the results of Buchweitz-van Straten.  In particular, I will establish the characteristic p > 0 analogue of the vanishing result mentioned above.

13th October (Thursday)
Enrico Carlini (Dipartimento di Matematica Politecnico di Torino)

The non-negative rank of a matrix

Abstract:  Given a matrix M with non-negative real entries, one can ask: Is it possible to write M as the sum of r rank 1 matrices having non-negative entries? The minimum r such that the answer is affirmative is called the non-negative rank of M. In this talk we will explore some known and some new properties of the non-negative rank. In particular, we will study how small changes in M affect the non-negative rank. This talk is base on a SIAMAX joint paper with C. Bocci and F. Rapallo.

12th October (Wednesday)
Kosmos Diveris (Syracuse University)

Finitistic Extension Degree

Abstract:  The AC condition concerning the vanishing of cohomology over a ring originates from the work of Auslander. More recently Christensen and Holm have shown that several longstanding homological conjectures hold for rings having the AC condition. In this talk we define a new condition that generalizes the uniform AC condition. We show that many of the known results for AC rings hold for rings having our condition. We will also discuss some examples of rings for which our condition holds and some where it fails to hold.

5th and 6th October (Wednesday and Thursday)
Rest cure

18th and 29th September (Wednesday and Thursday)
Rest cure

21st and 22nd September (Wednesday and Thursday)
Alexandra Seceleanu (UNL)

Syzygies and order ideals

Abstract:  I will introduce order ideals and the significance they have for analyzing ranks of syzygy modules. I will then explain some techniques of obtaining new from old information regarding order ideals, with the main focus on the case of hypersurface rings.

14th and 15th September (Wednesday and Thursday)
Rest cure

7th and 8th September (Wednesday and Thursday)
Daniel Murfet  (UCLA)

Matrix factorizations and knot invariants

1st September (Thursday)
Osamu Iyama  (Nagoya)

Auslander-Reiten duality for non-isolated singularities and maximal modification algebras

31st August (Wednesday)
Kamran Divaani-Aazar (Az-Zahra University, Tehran, Iran)

A unified approach to formal local cohomology and local Tate cohomology

Abstract: This talk is based on joint work with Mohsen Asgharzadeh. Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology,  local cohomology and local Tate cohomology through some natural isomorphisms. Also, we investigate vanishing of formal local cohomology modules.

### Visitors in Fall 2011 and Spring 2012

• Lutz Hille, University of Muenster, Muenster, Germany, April 2012 (tentative)
• Paolo Montero, Purdue University, West Lafayette, 1st - 5th April 2012
• Kiriko Kato, Osaka Prefecture University, Sakai, Japan, 25th March - 6th April 2012
• Sukhendu Mehrotra, University of Wisconsin, Madison, 7th - 9th March 2012
• Janet Shakalli-Tang, Texas A & M, College Station, 6th - 9th March 2012
• Yi Zhang, University of Minnesota, Minneapolis, 25th February - 1st March 2012
• Jon Carlson, University of Georgia, 21st - 25th February 2012
• Chin-Yi Jean Chan, Central Michigan University, Mt. Pleasant, 15th - 17th February 2012
• Tokuji Araya, Tokuyama College of Technology, Yamaguchi, Japan, 8th - 14th February 2012
• Yuji Yoshino, Okayama University, Okayama, Japan, 8th - 14th February 2012
• Piyush Shroff, Texas A & M, College Station, 15th - 19th November 2011
• Richard Green, University of Colorado, Boulder, 10th & 11th November 2011
• Mike Hopkins, Harvard University, 10th & 11th November 2011
• Eric Wofsey, Harvard University, 8th - 11th November 2011
• Jesse Burke, University of Bielefeld, Bielefeld, Germany, 1st - 15th November 2011
• Parker Lowrey, University of Western Ontario, London, Canada, 1st - 4th November 2011
• Marcy Robertson, University of Western Ontario, London, Canada, 25th - 28th October 2011
• Kosmos Diveris, Syracuse University, Syracuse, 11th - 16th October 2011
• Daniel Murfet, UCLA, Los Angeles, 6th - 14th September 2011
• Osamu Iyama, Nagoya University, Nagoya, Japan, 27th - 30th August 2011
• Kamran Divaani-Aazar, Az-Zahra University, Tehran, Iran,  9th July - 10th September 2011

### Past seminars

Fall 2010 & Spring 2011   Fall 2009 & Spring 2010 Spring 2009  Fall 2008    Spring 2008    Fall 2007    Spring 2007    Fall 2006    Spring 2006    Fall 2005    2004

Maintained by Srikanth Iyengar

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