Commutative Algebra Seminar
Spring 2009

Thu, 23 Jul 2009: Nicholas Baeth (University of Central Missouri) 

TITLE: Monoids of Torsion-free Modules 

Fri, 17 Jul 2009: Mike Axtell (University of St. Thomas) 

TITLE: Irreducible Divisor Graphs 

Thu, 16 Jul 2009: Ravi Rao (Tata Institute for Fundamental Research, Mumbai, India) 

TITLE: Group structures on orbits of unimodular rows 

ABSTRACT: The study of defining groups structures on the orbit space of unimodular rows over a commutative ring modulo elementary action was initiated by L.N. Vaserstein. He worked over rings of dimension two. This work was extended to higher dimensions by W. van der Kallen. We describe these results and their motivation from topology briefly, and indicate some progress and further questions on this topic. The talk will be accessible to graduate students. 

Thu, 30 Apr 2009: Leila Khatami, Northeastern Univ. 

TITLE: Bass-Type Equalities for Gorenstein Injective Dimension 

Thu, 16 Apr 2009: Micah Leamer, UNL 

TITLE: Torsion in tensor products. II 

Wed, 15 Apr 2009: Micah Leamer, UNL 

TITLE: Using numerical semigroups to determine torsion in tensor products over analytically irreducible local domains. I 

Thu, 09 Apr 2009: Sylvia Wiegand 

TITLE: Prime ideals in noetherian rings 

Wed, 08 Apr 2009: Janet Striuli (Fairfield Univ.) 

TITLE: Totally reflexive modules over almost Gorenstein rings 

Wed, 01 Apr 2009: Lars Winther Christensen (Texas Tech) 

TITLE: Where to find totally reflexive modules 

ABSTRACT: Let R be a commutative local ring that is not Gorenstein. Given that there exists a non-free totally reflexive R-module, it is known that there have to be infinitely many of them in the non-trivial sense. I.e., there has to be infinitely many indecomposable and pair wise non-isomorphic totally reflexive R-modules. In the talk I will discuss procedures for finding them. The talk is based on ongoing work with Janet Striuli. 

Thu, 26 Mar 2009: Laura Lynch, UNL 

TITLE: What annihilates local cohomology? 

Wed, 25 Mar 2009: Lori McDonnell, UNL 

TITLE: Some open questions concerning Hilbert-Kunz multiplicities 

Thu, 12 Mar 2009: Anthony Iarrobino (Northeastern University) 

Title: Commuting nilpotent matrices and the Hilbert scheme of Artinian algebras 

ABSTRACT: Let K be a closed field. Suppose that A is a nilpotent n by n Jordan matrix whose block sizes correspond to the partition P of n. What are the Jordan block sizes Q(P) for a generic nilpotent matrix B with entries in K commuting with A? V. Baranovsky, R. Basili, A. Premet used a connection between pairs A,B of commuting nlpotent matrices and Artinian algebras K[A,B] in two variables to relate the irreducibility of families of commuting nilpotent matrices with the irreducibility of a punctual Hilbert scheme parametrizing Artinian algebras in two variables. We discuss work of R. Basili and I showing that the above problem of determining Q(P) is equivalent to that of determining the Hilbert function H of the Artinian algebra K[A,B]. P. Oblak and T. Kosir have shown that K[A,B] is Gorenstein. They conclude from F.H.S. Macaulay's characterization of H for height two complete intersections that Q(P) has parts that differ pairwise by at least two. 

Thu, 05 Mar 2009: Luchezar Avramov (UNL) 

Title: Free resolutions over Koszul algebras 

Wed, 04 Mar 2009: Luchezar Avramov (UNL) 

Title: Free resolutions over graded rings. 

Thu, 26 Feb 2009: Roger Wiegand (UNL) 

Wed, 25 Feb 2009: Roger Wiegand (UNL) 

Brauer-Thrall II in dimension one. I, II 

Thu, 12 Feb 2009: Ines Henriques (UNL) 

Wed, 11 Feb 2009: Ines Henriques (UNL) 

Cohomology over short Gorenstein rings. I, II 

Wed, 04 Feb 2009: Tom Marley (UNL)

Title: "A non-coherent Gorenstein ring" 

Thu, 29 Jan 2009: Melissa Kraus (Purdue University) 

Title: A New Look at Macaulay's Theorem 

ABSTRACT: A classical result of Macaulay describes precisely the numerical functions that are realizable as the Hilbert function of a homogeneous ideal in a polynomial ring in several variable over a field. Macaulay described his original proof as long and complicated. More recent proofs focus on binomial coefficients. We discuss a different proof that concentrates on algebraic tools and involves general change of coordinates, generic initial ideals, strongly stable ideals and Green's hyperplane restriction theorem. The goal is to gain intuition that may apply to related open problems such as the Lex-Plus-Powers Conjecture. 

Wed, 28 Jan 2009: Don Stanley (Regina)

Title: Modules, Derived Categories and some Subcategories.

Abstract: Let R be a commutative Noetherian ring.
If we take the category of chain complexes of R-modules and mod out  by weak equivalences we get
the derived category of R, D(R). A thick subcategory of D(R) is a  triangulated subcategory closed under retracts.
In R-modules the corresponding abelian subcategories are called wide  subcategories.
About 20 years ago, Hopkins and Neeman classified the thick  subcategories of $D_{parf}(R)$, in terms of supports
Using these results Hovey started the classification of wide  subcategories of finitely generated R-modules,
which was finished by Takahashi using different methods and
these also turn out to be classified by supports.

In our talk we consider subcategories which are closed under fewer  operations.  In particular we don't assume closure under
taking kernels. It turns out that in R-modules such subcategories are  also classified by supports. However in D(R)
we get a more interesting classification of so called nullity classes  in terms of increasing sequences of supports.
The t-structures introduced by Beilinson, Bernstein and Deligne are  special kinds of nullity classes and we describe conjecturally how  they might be classified using this approach.

Thu, 15 Jan 2009: Charles A. Weibel (Rutgers Univ) 

Title: "The Bloch-Kato Conjecture: the Norm Residue is an Isomorphism" 

ABSTRACT: Milnor conjectured in 1970 that the etale cohomology of a field (mod 2 coefficients) should have a presentation with units as generators and simple quadratic relations (the ring with this presentation is now called the "Milnor K-theory"). This was proven by Voevodsky, but the odd version (mod p coefficients for other primes) has been open until vary recently, and has been known as the Bloch-Kato Conjecture. Using certain norm varieties, constructed by Rost, and techniques from motivic cohomology, we now know that this conjecture is true. This talk will be a non-technical overview of the ingredients that go in to the proof, and why this conjecture matters to non-specialists.