## Algebra Seminar: Spring 2018

If you are interested in giving a talk, please contact the organizers Emily Witt and Jonathan Montaño.

### February 26, 2018, 3:00 PM (Unusual day and time)

• Name: Alessandra Costantini, Purdue University
• Title: The Cohen-Macaulay property of the Rees algebra of a module.

• Abstract:

Rees algebras of ideals and modules appear frequently in Algebraic Geometry, for instance in connection with blow-up constructions, equisingularity theory and birational study of projective varieties. For an ideal I in a Cohen-Macaulay ring R, the Cohen-Macaulay property of the Rees algebra R(I) is deeply connected with the Cohen-Macaulayness of the associated graded ring gr(I). In fact, gr(I) is Cohen-Macaulay whenever R(I) is; and although the converse is not true in general, the Cohen-Macaulay property of gr(I) combined with other assumptions often implies that R(I) is Cohen-Macaulay. However, for an R-module E there is no analogue of the associated graded ring, so the study of the Cohen-Macaulay property of R(E) is more complicated. In this talk, using the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos we will provide classes of modules whose Rees algebra is Cohen-Macaulay. Our results generalize known results of Jonhson and Ulrich, and of Goto, Nakamura and Nishida for the Rees algebra of an ideal.

### April 19, 2018

• Name: Jack Jeffries, University of Michigan
• Title: The Smith-Van den Bergh functors.

• Abstract:

In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive characteristic. We will discuss formulas for the ring of differential operators as well as these derived functors of differential operators in terms of local cohomology. One can use this description to relate questions on the behavior of differential operators under base change to questions on p-torsion in local cohomology. We will also connect the vanishing of these functors to some interesting properties of singularities.

### April 23, 2018, 3:00 PM (Unusual day and time)

• Name: Daniel Katz, University of Kansas
• Title: Talks in honor of the 100th birthday of D. Rees: An overview of the work of David Rees.

• Abstract:

### May 1, 2018

• Name: Justin Lyle, University of Kansas
• Title: Hom and Ext, Revisited.

• Abstract:

Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated R-modules. We prove a number of results of the form: if $\text{Hom}_R(M,N)$ has some nice properties and $\text{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$, then $M$ (and sometimes $N$) must be close to free. Using these results, we are able to extend and unify a number of results in the literature.

### May 3, 2018, 2:00 PM (Unusual time)

• Name: Hailong Dao, University of Kansas
• Title: Talks in honor of the 100th birthday of D. Rees: m-full ideals.

• Abstract: