Commutative Algebra Seminar
Spring 2023, Friday 2:00–3:00 pm, LCB 222
Date  Speaker  Title — click for abstract 
January 27  Fellowship of the ring @2:30 Link here 
Matt Mastroeni (Iowa State University)
See The FoTR Website

February 3  Hanlin Cai University of Utah 
Perfectoid Signature and Perfectoid HilbertKunz Multiplicity
In this talk I'll talk about a (perfectoid) mixed characteristic version of Fsignature and HilbertKunz multiplicity by utilizing the perfectoidization functor of BhattScholze and Faltings' normalized length. These definitions coincide with the
classical theory in equal characteristic. We prove that a ring is regular if and only if either its perfectoid signature or perfectoid HilbertKunz multiplicity is 1 and we show that perfectoid HilbertKunz multiplicity characterizes BCM closure
and extended plus closure of ideals. We demonstrate that perfectoid signature detects BCM regularity and transforms similarly to Fsignature or normalized volume under quasiétale maps. As a consequence, we prove that BCMregular rings have
finite local étale fundamental group and torsion part of their divisor class groups. This is joint work with Seungsu Lee, Linquan Ma, Karl Schwede and Kevin Tucker.

February 17  Yotam Svoray University of Utah 
Invariants on nonisolated hypersurface singularities
A key tool in understanding (complex analytic) hypersurface singularities is to study what properties are preserved under special deformations. For example, the relationship between the Milnor number of an isolated singularity and the number of
A_1 points. In this talk we will discuss the transversal discriminant of a singular hypersurfaces whose singular locus is a smooth curve, and how it can be applied in order to generalize a classical result by Siersma, Pellikaan, and de Jong
regarding morsifications of such singularities. In addition, we will present some applications to the study of Yomdintype isolated singularities.

February 24  Eamon Quinlan University of Utah 
Fmodules for rings with FFRT
TBA

March 3  Fellowship of the ring @2:30 Link here 
Jan Draisma (Universität Bern)
TBA

March 17  Vaibhav Pandey Purdue University 
Linkage and Fregularity of generic determinantal rings
We prove that the generic link of a generic determinantal ring of maximal minors is Fregular. In the process, we strengthen a result of Chardin and Ulrich. Their result says that the generic residual intersections of a complete intersection ring with rational singularities again has rational singularities. We prove that they are, in fact, Fregular. In the mid 90s, Hochster and Huneke proved that generic determinantal rings are Fregular; however, their proof is quite involved. We give a new and simple proof of the Fregularity of determinantal rings of maximal minors. Time permitting, we will also give a new proof of the Fregularity of generic determinantal rings of minors of any size. This is joint work with Yevgeniya Tarasova. 
March 24  Fellowship of the ring @2:30 Link here 
Špela Špenko (Université Libre de Bruxelles)
TBA

March 31  Florian Enescu Georgia State University 
Rational twist with dominant eigenvalue and questions on Hilbert quasipolynomials
In this talk, we will highlight some classes of affine semigroups rings that have rational twist in positive characteristic. We will then relate this phenomenon to questions on the Hilbert quasipolynomial and, related to this, the Ehrhart polynomial associated to a certain polytope with integer vertices. This is joint work with Yongwei Yao. 
April 7  Andy Jiang University of Michigan 
Grothendieck duality without compactifications
I will discuss how a formula for the dualizing complex due to Avramov, Iyengar, Lipman, and Nayak (2010) can be as a foundation for Grothendieck duality for finite tor amplitude mapsgeneralizing results of Khusyairi (2017) in the flat case.

April 12 Different day Different location: JWB 333 
Ray Heitmann UT Austin 
Numbers of generators of perfect ideals
This talk will explore bounds on the number of generators of perfect ideals J in regular local rings
(R,m).
If J is sufficiently large modulo m^n, a bound is established depending only on n and the projective
dimension of R/J.
More ambitious conjectures are also introduced with some partial results.

April 21  Keller VandeBogert Notre Dame 
The Total Rank Conjecture in Characteristic 2
The total rank conjecture is a coarser version of the BuchsbaumEisenbudHorrocks conjecture which, loosely stated, predicts that modules with large annihilators must also have "large" syzygies. In 2017, Walker proved that the total rank conjecture holds over rings of odd characteristic, using techniques that heavily relied on the invertibility of 2. In this talk, I will talk about joint work with Mark Walker where we settle the total rank conjecture for rings of characteristic 2. The techniques used are very specialized to the characteristic 2 case, and imply an even stronger result showing that the counterexamples to the generalized total rank conjecture in odd characteristic constructed by IyengarWalker are not possible in characteristic 2.

April 28  Fellowship of the ring @2:30 Link here 
Karen Smith (University of Michigan)
TBA

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