Commutative Algebra Seminar
Fall 2015, Friday 3:10−4:00, LCB 222
Date  Speaker  Title — click for abstract 
September 4  Adam Boocher
University of Utah 
On the closure of a linear space
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.

September 11  Kazuma Shimomoto
Nihon University 
A new proof of a theorem of Cohen and Gabber in the positive characteristic case
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

September 18  Steven Sam
University of Wisconsin, Madison 
Commutative algebra of functor categories
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.

September 25  Jon F. Carlson
University of Georgia, Athens 
The relative stable category
Let G be a finite group and k an algebraically closed field of characteristic p > 0. Let H be a collection of psubgroups of G. We investigate the relative stable category stmod_H(kG) of finitely generated modules modulo Hprojective modules. Triangles in this category correspond to Hsplit 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.

October 2, 3:10 PM  Sylvia Wiegand
University of NebraskaLincoln 
Building examples using power series over Noetherian rings
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 nonNoetherian, integral domains. This technique of interseting fields with power series goes back to AkizukiSchmidt 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.

October 2, 4:10 PM  Roger Wiegand
University of NebraskaLincoln 
Big projective modules over slightly noncommutative rings
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 onedimensional local domain R (commutative and Noetherian) and a finitely generated torsionfree Rmodule M such that add M is isomorphic to H. Now let S be the endomorphism ring of R. It is wellknown and easy to prove that add M is isomorphic to projS, the semigroup of isomorphism classes of finitely generated projective right Smodules. Thus, strange directsum behavior over R leads to strange directsum behavior of finitely generated projective modules over the slightly noncommutative 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 Smodule with no nonzero 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.

October 9  Alessandro De Stefani
University of Virginia 
Fthresholds of graded rings
The Fpure threshold, the diagonal Fthreshold and the ainvariant 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 Fregular rings. We prove their conjecture for a more general class of rings, namely, Fpure rings. Furthermore, we give an interpretation of the Fpure threshold of a standard graded Gorenstein ring in terms of the maximal length of a regular sequence that preserves Fpurity at each step. This is joint work with Luis NunezBetancourt.

October 23  Hamid Hassanzadeh
University of Utah/Federal University of Rio de Janeiro 
Annihilators of Koszul homologies
In this talk we reveal some connections between Koszul annihilators and residual intersection. After introducing the concepts, we show how sliding depth conditions provide nontrivial 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 − s1. Let J = (a : I) be an sresidual intersection and use H_j(a) to denote the jth 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! 
October 30  Maral Mostafazadehfard
University of Utah 
On the singular locus of Hankel determinants and some of their degenerations
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.

November 6  Linquan Ma
University of Utah 
Lech's conjecture
A longstanding conjecture of Lech states that the HilbertSamuel 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.

November 20  Gennady Lyubeznik
University of Minnesota 
Holonomic Dmodules over formal power series
Holonomic Dmodules 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.

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