Commutative Algebra Seminar

A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Orthogonal subspace collections occur in many physical problems, but we'll show an example of the use of non-orthogonal ones, which when substituted in orthogonal ones greatly accelerate the convergence of Fast Fourier transform methods.

Hochschild cohomology was introduced in a 1945 paper by Hochschild, and Grothendieck duality dates back to the early 1960s. The fact that the two have some relation with each other is very new - it came up in papers by Avramov and Iyengar [2008], Avramov, Iyengar, and Lipman [2010] and Avramov, Iyengar, Lipman and Nayak [2011]. We will review this history, and the surprising formulas that come out. We will then discuss more recent progress. The remarkable feature of all this is the role played by Hochschild homology. One example, which we will discuss in some detail, comes about as follows. The new techniques permit us to write formulas giving trace and residue maps in Grothendieck duality in terms of expressions that are very Hochschild-homological - Alonso, Jeremias and Lipman gave such a formula, but couldn't prove that it agrees with the usual formula dating back to Verdier in the 1960s. The proof that these two agree, due to Lipman and the speaker, turns out to hinge on considering the action of ordinary Hochschild homology on the various objects in the formula.

Motivated by questions in modular representation theory, Carlson, Friedlander and the speaker instigated the study of projective varieties of abelian p-nilpotent subalgebras of a fixed dimension r for a p-Lie algebra g. These varieties are closely related to the much studied class of varieties of r-tuples of commuting p-nilpotent matrices which remain highly mysterious when r>2. In this talk, I shall present some of the representation-theoretic motivation behind the study of these varieties and describe their geometry in a very special case when it is well understood: namely, when r is the maximal dimension of an abelian p-nilpotent subalgebra of a semisimple Lie algebra g. This is joint work with J. Stark.

An important tool in the study of algebras is the representation variety, which parameterizes representations of a fixed dimension.  These varieties are useful for organizing the representation theory of wild algebras and also raise interesting geometric questions themselves.  It turns out that it is possible to define well-behaved representation varieties for commutative graded rings.  These varieties parameterize maximal Cohen-Macaulay (MCM) modules.  In my talk, I will explain some recent results both on representation varieties for a natural class of finite dimensional (non-commutative) algebras and on MCM modules over graded rings.

This is a report on joint work with P.H. Quy (Hanoi). I will talk about recent progress on the connection between F-singularities and non CM rings, using the Frobenius action on local cohomology modules. I will also discuss open problems.

Let Q be a power series ring, for example $\mathbb{Q}[\![x,y,z]\!]$, and let I be an ideal in Q. If the drop in depth from Q to the quotient R=Q/I is at most 3, then R can be classified based on a multiplicative structure on the finite free resolution of R as a Q-module. The identification of the possible multiplicative structures was done 25 years ago, but the question of which structures can actually be realized has only recently seen progress. In joint work with Veliche and Weyman we construct families of rings $\mathbb{Q}[\![x,y,z]\!]/I$ that realize multiplicative structures which had been conjectured not to occur. In fact, rings with this structure seem to be everywhere, and what emerges is a picture that describes generic local rings on a one-dimensional scale whose end points are Gorenstein rings and Golod rings.

The Castelnuovo-Mumford regularity of a module is a homological invariant that roughly measures complexity. Though straightforward to define, it is difficult to find ideals in polynomial rings with high Castelnuovo-Mumford regularity. I will demonstrate a method that takes as input well-understood modules and outputs ideals which cut out schemes supported on linear spaces with high Castelnuovo-Mumford regularity and other desirable homological properties.

A duality for representations of artin algebras was introduced in the 1970s by Auslander and Reiten. One can think of this as an analogue of Serre duality for projective varieties. The talk will present a version of Auslander-Reiten duality for Grothendieck abelian categories. This makes it possible to compare both dualities.

This talk will be a report on some results from a joint work in progress with Alessandro De Stefani and Yongwei Yao. Hilbert-Kunz multiplicity is a classical numerical invariant of study attached to local rings of prime characteristic. In this talk we will discuss how to naturally extend this numerical invariant to all rings which are not local and F-finite in a meaningful way.

We examine some usual notions of singularities in characteristic p > 0 such as F-finiteness, F-splitting, F-purity etc., for valuation rings. The novelty of this approach stems from the fact that most of the characteristic p notions of singularities have been extensively studied in the Noetherian setting, whereas valuation rings are usually highly non-Noetherian. There is a surprising connection between the well-studied class of Abhyankar valuations (these valuations may be viewed as higher dimensional analogues of valuations associated to prime divisors on a variety) and the class of F-finite valuations. The talk is based on joint work with Karen Smith, and our work attempts to answer some questions asked by Karl Schwede and Zsolt Patakfalvi.

In this talk I will define and investigate symmetric bicolored trees, the symmetric versions of the bicolored trees studied by Markwig and Yu. In particular, I'll show that a special subset of these symmetric bicolored trees, those with an "unbranched stem", form a pure-dimensional simplicial complex, and that this complex is shellable. I'll then describe a relationship between these symmetric bicolered trees with unbranched stems, or "happy trees", and the notion of the symmetric tropical rank of a matrix.

The space of m x n matrices admits a natural action of the group GL_m x GL_n via row and column operations on the matrix entries. The invariant closed subsets are the determinantal varieties defined by the (reduced) ideals of minors of the generic matrix. The minimal free resolutions of these ideals are well-understood by work of Lascoux and others. There are however many more invariant ideals which are non-reduced, and they were classified by De Concini, Eisenbud, and Procesi in the 80s. I will explain how to determine the syzygies of a large class of these ideals by employing a surprising connection with the representation theory of general linear Lie superalgebras. This is joint work with Jerzy Weyman.