Commutative Algebra Seminar

Let L in k^n be a linear subspace. If we choose coordinates, we can readily write down its defining equations as an affine variety. These coordinates in turn suggest several projective closures of L in various projective varieties. I'll discuss joint work with Federico Ardila which computes one of these closures in great detail. This reveals a close interaction between several matroid invariants and invariants arising in commutative algebra.

This is joint work with Kazuhiko Kurano (Meiji University). Recently, Gabber proved a separable version of Cohen's structure theorem on complete local rings in positive characteristic. I will talk about a new proof of this theorem. Gabber used this theorem to provea refined alteration theorem. If time permits, I will talk about an application of Gabber's theorem to a local Bertini theorem for reduced quotients. The Bertini theorem has an application to the specialization method in Iwasawa theory

Commutative algebra can be studied in the general context of a symmetric monoidal abelian category. I will explain some joint work with Andrew Putman and Andrew Snowden in this direction on the category of functors from a category made out of sets or vector spaces to the category of modules over a ring. In some cases, there are strong analogies: a version of the Hilbert basis theorem, rationality of Hilbert series, etc. Time permitting, I will indicate some applications to twisted homological stability for families of groups such as finite linear groups.

Let G be a finite group and k an algebraically closed field of characteristic p > 0. Let H be a collection of p-subgroups of G. We investigate the relative stable category stmod_H(kG) of finitely generated modules modulo H-projective modules. Triangles in this category correspond to H-split sequences. Hence, compared to the ordinary stable category there are fewer triangles and more thick subcategories. Our interest is in the spectrum of this category and its relationship to the induction functor. The lecture will concentrate on the nature of the constructions with a few examples of what is now known about the subcategories in the spectrum.

In ongoing work with Willian Heinzer and Christel Rotthaus over the past twenty years we have been applying a construction for obtaining sometimes baffling, sometimes badly behaved, sometimes Noetherian, sometimes non-Noetherian, integral domains. This technique of interseting fields with power series goes back to Akizuki-Schmidt in the 1930s and Nagata in the 1950s, and since then has also been employed by Nishimuri, Heitmann, Ogoma, the authors and others. We are writing a book about our procedures and examples. We present some of the theory and techniques we use, and mention some examples.

Let H be an arbitrary positive normal affine semigroup (aka "Diophantine semigroup", aka "reduced finitely generated Krull semigroup"). Early in the millennium I proved that one can find a one-dimensional local domain R (commutative and Noetherian) and a finitely generated torsion-free R-module M such that add M is isomorphic to H. Now let S be the endomorphism ring of R. It is well-known and easy to prove that add M is isomorphic to proj-S, the semigroup of isomorphism classes of finitely generated projective right S-modules. Thus, strange direct-sum behavior over R leads to strange direct-sum behavior of finitely generated projective modules over the slightly non-commutative ring S. What is surprising is that this leads also to strange behavior of countably generated projective modules. In this talk I will indicated how to build a countably generated projective S-module with no non-zero finitely generated direct summands. Put another way, there is a direct summand N of M^{(\omega)} (the direct sum of countably many copies of M) such that N has no finitely generated direct summands. This is joint work with Dolors Herbera and Pavel Prihoda.

The F-pure threshold, the diagonal F-threshold and the a-invariant are three numerical invariants that can be attached to graded rings of positive characteristic. Hirose, Watanabe, and Yoshida conjectured some relations between these numbers for strongly F-regular rings. We prove their conjecture for a more general class of rings, namely, F-pure rings. Furthermore, we give an interpretation of the F-pure threshold of a standard graded Gorenstein ring in terms of the maximal length of a regular sequence that preserves F-purity at each step. This is joint work with Luis Nunez-Betancourt.

In this talk we reveal some connections between Koszul annihilators and residual intersection. After introducing the concepts, we show how sliding depth conditions provide non-trivial annihilators for Koszul homologies. To this end, we present a family of approximation complexes to determine the Koszul annihilators. The main theorem we’ll discuss is the following: Let (R, m) be a CM local ring of dimension d, I satisfy Sliding Depth, and depth (R/I)>􏰀 d − s-1. Let J = (a : I) be an s-residual intersection and use H_j(a) to denote the j-th Koszul homology module with respect to a minimal generating set of a. Then I annihilates all H_j(a) for j>0. Surprisingly, this result contradicts one of the old (unpublished) results of G. Levin!

This is an investigation of the ideal theoretic invariants of the gradient ideal of the determinant of some structural matrices, such as primary components, multiplicity, reductions and free resolution, along with a detailed study of homaloidal behavior. A homaloidal form is a homogeneous polynomial whose polar map is a birational transformation. This work was motivated by a question of Dolgachev regarding an upper bound for the degree of a homaloidal polynomial in terms of the dimension of the ring.

A long-standing conjecture of Lech states that the Hilbert-Samuel multiplicity does not drop for faithfully flat extensions of local rings. In this talk, we will prove this conjecture in dimension three, in characteristic p.

Holonomic D-modules play an important role in commutative algebra. Their definition and proofs of their elementary properties are quite simple over polynomial rings, but considerably more complicated over formal power series. In this talk, I will present the very recent work of my student Peyman Gharemani which considerably simplifies the proofs of some important aspects of the theory in the formal power series case.