Commutative Algebra Seminar

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In this talk I'll talk about a (perfectoid) mixed characteristic version of F-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze 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 Hilbert-Kunz multiplicity is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM closure and extended plus closure of ideals. We demonstrate that perfectoid signature detects BCM regularity and transforms similarly to F-signature or normalized volume under quasi-étale maps. As a consequence, we prove that BCM-regular 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.

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 Yomdin-type isolated singularities.

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We prove that the generic link of a generic determinantal ring of maximal minors is F-regular. 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, F-regular.

In the mid 90s, Hochster and Huneke proved that generic determinantal rings are F-regular; however, their proof is quite involved. We give a new and simple proof of the F-regularity of determinantal rings of maximal minors. Time permitting, we will also give a new proof of the F-regularity of generic determinantal rings of minors of any size. This is joint work with Yevgeniya Tarasova.

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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 quasi-polynomial and, related to this, the Ehrhart polynomial associated to a certain polytope with integer vertices. This is joint work with Yongwei Yao.

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 maps--generalizing results of Khusyairi (2017) in the flat case.

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.

The total rank conjecture is a coarser version of the Buchsbaum-Eisenbud-Horrocks 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 Iyengar-Walker are not possible in characteristic 2.

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