Commutative Algebra and Algebraic Geometry

Saturday 10/19/2013

Sunday 10/20/2013

Talk Abstracts

Sara Faridi
The Combinatorics of Betti Numbers

This talk is about geometric interpretations of Betti numbers of monomial ideals. These numbers are invariants coming from minimal free resolutions of the ideal. We will demonstrate, using the Taylor complex, how some of these Betti numbers can be read off from the facet complex of the ideal. We will focus on the case of graphs, and show how for many graphs one can tell what the projective dimension of the edge ideal is by looking at the graph itself.

Laura Ghezzi
The first Hilbert coefficients

The set of the first (after the multiplicity) Hilbert coefficients of parameter ideals in a Noetherian local ring (R,m) codes for significant information about its structure. In joint work with S. Goto, J. Hong, K. Ozeki, T.T. Phuong, and W.V. Vasconcelos, we studied noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology.
In this talk we give an overview of our main results and we present recent work on "variation". We give estimations for the first two Hilbert coefficients e_0(I) and e_1(I) when the m-primary ideal I is enlarged (in the case of e_1 in the same integral closure class).

Alina Iacob
Gorenstein injective envelopes

We prove that the class of Gorenstein injective modules is enveloping over commutative noetherian rings with dualizing complexes. This is joint work with Edgar Enochs.

Anthony Iarrobino
Joint work with Leila Khatami (Union College), Rui Zhao (U. Missouri), Bart Van Steirteghem (Medger Evers College, CUNY)
When do two nilpotent matrices commute?

Given a nilpotent matrix B of Jordan type -- sizes of Jordan blocks -- a partition P of n, what is the Jordan type Q(P) of the generic nilpotent matrix A commuting with B? We describe results from several groups who have studied this problem recently. An almost rectangular partition is one whose largest part minus smallest part is at most one. It is known that Q(P) has r_P parts that differ pairwise by at least two, whereherer P is the minimum number of almost rectangular partitions whose union is P. We discuss a poset attached to the problem of determining Q(P), and some results and conjectures of Polona Oblak and others about the map $P\to Q(P)$ and its inverse. We conjecture that there is a bijection between the set of partitions P of n with r_P=k and the set of partitions having kxk Durfee square.

Kuei-Nuan Lin
Projective Dimension of Square-Free Monomials

This is joint work with Paolo Mantero. Given a square-free monomial ideal over a polynomial ring over a field k, we can associate a hypergraph H with I which was first introduced by Kimura-Terai-Yoshida. We find a combinatorial descriptions of projective dimension of I when H is a string or a cycle, i.e. we give an exclusive formula of projective dimension of I.

George McNinch
Levi decompositions of linear algebraic groups

A linear algebraic group G over a field K is a closed subgroup of GL(V) for a finite dimensional K-vector space V. A closed subgroup H of G is split unipotent if H has a filtration by closed subgroups for which the successive quotients are isomorphic to the additive group of K.
If K is a perfect field, G has a *unipotent radical* R which is connected, normal, and split unipotent subgroup. The quotient G/R is a *reductive* algebraic group. If K is a field of characteristic 0, the group G is always isomorphic to the semidirect product of G/R and R; equivalently, there is a closed subgroup M of G mapped isomorphically to G/R by the quotient homomorphism. When K has positive characteristic, we say that G has a Levi decomposition if such a closed subgroup M can be found; the group M is thus a Levi factor of G.
The talk will discuss conditions for the existence and conjugacy of Levi factors of G for (perfect) fields of positive characteristic.

Saeed Nasseh
DG homological algebra: Application to a question in commutative algebra
Abstract

Hamid Rahmati
Duality for Koszul homology over Gorenstein rings

Over a Gorenstein ring if an ideal satisfies the property that all its Koszul homologies are Cohen-Macaulay then by a classical result, due to Herzog, the Koszul homology algebra satisfies Poincar\'e duality. We will discuss a slightly weaker version of this duality that holds for all ideals in a Gorenstein ring. We will also give some applications of this duality. This is a joint work with Claudia Miller and Janet Striuli.

Jenna Rajchgot
Type A quiver loci and Schubert varieties

I'll describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, I'll discuss a similar result up to some factors of general linear groups.
These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.
This work is joint with Ryan Kinser.

Branden Stone
Non-simplicial decompositions of Betti diagrams of complete intersections

The framework of Boij-oderberg theory allows us to decompose Betti diagrams over a polynomial ring into pure diagrams. In this talk, we relax the requirement that the degree sequences in such pure diagrams be totally ordered. As a result, we were able to dene a multiplication law for Betti diagrams that respects the decomposition. Given the Betti diagram of any complete intersection, this new law allows us to write a simple pure diagram decomposition in terms of the degrees of the minimal generators for the complete intersection. This work was done as part of a Mathematical Sciences Research Institute summer graduate workshop in 2011.

Janet Striuli
Construction of totally reflexive modules

In this talk we give a generalization of a theorem of Celikbas, Gheibi, Takahashi. In particular we are able to describe an infinite family of totally reflexive modules in the presence of an exact pair of zero divisors.

Alexandre Tchernev
Symmetric powers of modules of low projective dimension

Symmetric and exterior powers of modules arise in many areas of commutative algebra and algebraic geometry, and their torsion properties are key to understanding the properties of related geometric objects. We will discuss how one can obtain explicit and easy to verify necessary and sufficient conditions that characterize torsion freeness for symmetric powers of finitely generated modules of projective dimension 1 over a commutative Noetherian ring.

Gordana Todorov
(Joint work with Osamu Iyama, Idun Reiten and Hugh Thomas)
Lattice structure of torsion classes for hereditary algebras

In this work we consider certain subalgebras A of matrix algebras over algebraically closed fields, and their generalizations (matrix algebras with bimodules). These algebras can also be viewed as path algebras of quivers with no oriented cycles. We study categories of modules, mod A, over such algebras, and torsion classes in mod A. For these algebras, torsion classes always form a lattice. Particularly important torsion classes among those are, so called, functorially finite torsion classes. We show that functorially finite torsion classes form lattice precisely when the algebra A has only finitely many isomorphism classes of indecomposable modules, or the algebra is related to 2x2 matrices (with bimodules). It is known that such torsion classes are closely related to tilting modules.

Julianna Tymoczko
Equivariant cohomology and generalized splines

Recent constructions of equivariant cohomology have used algebra and combinatorics to develop explicit algorithms for computing cohomology rings. These approaches emerged independently in algebraic topology/symplectic geometry (where they are often called GKM theory, after seminar work by Goresky, Kottwitz, and MacPherson) and in algebraic geometry (where they're associated with work of Brion, Payne, and Bahri-Franz-Ray, among others).
It turns out that these constructions also extend a classical problem in applied mathematics and in algebraic combinatorics: the problem of identifying a ring of splines.
We will define splines classically and show how they relate to equivariant cohomology rings of suitable varieties. We then describe how to generalize the definition of splines to a more natural algebraic and combinatorial setting. We end with recent results (due to various authors) that show how powerful generalized splines can be. Time permitting, we will also include open questions.

Bart Van Steirteghem
Weight monoids and spherical roots of affine spherical varieties

A normal affine variety $X$ equipped with an action of a complex connected reductive group $G$ is called spherical if its ring of regular functions $O(X)$ is a multiplicity free representation of $G$. A basic invariant of $X$ is its weight monoid $\Gamma(X)$: the set of irreducible representations of $G$ that occur in $O(X)$. A more subtle invariant of $X$ is its set $\Sigma(X)$ of spherical roots, which "measures" how far $O(X)$ is from being $\Gamma(X)$-graded. By a theorem of I. Losev, the two invariants together determine $X$ up to equivariant isomorphism.
The "interaction" between the two invariants is related to the geometry of Alexeev and Brion's moduli scheme of affine spherical varieties with a given weight monoid. After recalling examples of this relationship in the work of S. Jansou, P. Bravi and S. Cupit-Foutou, I will discuss joint work in progress with P. Bravi on the tangent space to the moduli scheme at its "most degenerate" point.