Commutative Algebra Seminar
Fall 2025
Regular Day: Friday
Regular Time: 2:00PM - 3:00PM
Regular Location: LCB 222
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August 29 |
Nawaj KC
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If R is a local ring of dimension d and x = x_1, ..., x_d is a maximal regular sequence, then R/(x) is an R-module of finite length and finite projective dimension. There exist at least three open questions which stipulate that such quotients of regular sequences are the "simplest" or "smallest" modules amongst all modules of finite length and finite projective dimension. I will talk about some evidence supporting this philosophy. I will also sketch a proof from a joint work with Josh Pollitz where we solve the Loewy length version of this problem over strict Cohen-Macaulay rings.
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September 5 |
Mark Walker
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I discuss joint work with Briggs, Grifo, and Pollitz, in which we investigate a question due to Avramov: Does every central element of degree two in the homotopy Lie algebra of a local ring come from an embedded deformation? I'll define the terms appearing in this question and answer it. (Spoiler alert: No)
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September 19 |
JJ Garzella
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We will survey some recent results about a class of rings called *perfectoid Tate algebras*. These rings do not satisfy Hilbert's nullstellensatz, as shown by Gleason. We suggest an alternative, based on the theory of Berkovich spaces. We conclude by showing our statement in the case of two variables.
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September 26 |
Manav Batavia
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The arithmetic rank of a variety is the minimal number of equations needed to define it set-theoretically, i.e., the smallest number of polynomials generating the defining ideal upto radical. Computing this invariant is notoriously difficult: the minimal generators up to radical often bear little relation to the given ideal generators and can vary unpredictably across characteristics.
Residual intersections provide a natural extension of the classical notion of algebraic links. We establish a general upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian ring, and we show that this bound is sharp under specific characteristic assumptions. This work is joint with Kesavan Mohana Sundaram, Taylor Murray, and Vaibhav Pandey.
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October 3 |
Desiree Laurel Martin
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In this talk we discuss the existence of a splitting morphism between \(\text{Tor}^{R}_{*}(R/I,R/I)\) and the
exterior algebra of the conormal module \(\bigwedge I/I^2\) using properties of differential graded algebras. If time permits, we
will chat about some interesting examples.
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October 17 |
Prashanth Sridhar
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In this talk, I'll discuss novel reconstruction results for projective varieties using differential graded
sheaves.
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October 24 |
Vijaylaxmi Trivedi
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In the first part of the talk we discuss Hilbert-Kunz density function and its applications
for positive characteristic invariants, namely Hilbert-Kunz multiplicities and \(F\)-thresholds.
Next, based on a joint work with Suprajo Das and Sudeshna Roy, we introduce the *adic* and
*saturated* density functions. These were introduced to give numerical characterizations for integral
dependence of ideals in a graded set up, where we recall that two ideals \(I \subset J\) in a commutative
Noetherian ring are *integrally dependent* if they have the same integral closure.
Attempts to give a numerical characterization for ideals which might not necessarily be of
finite colength led to numerical invariants like \(j\)-multiplicity, \(\varepsilon\)-multiplicity which require (even
in graded set up) computing the invariant at several localizations, hence not readily amenable
to computations. Also there exists a notion of multiplicity sequence which gives a numerical
characterization of integral dependence.
As an application of these density functions we show that any of the multiplicities, namely, the
polar multiplicities, the \(\varepsilon\)-multiplicities or the \(j\)-multiplicities of the truncated ideals \(I[Y]_{\geq c}\) and
\(J[Y]_{\geq c}\) in \(R[Y]\) characterizes the integral dependence of \(I\) and \(J\). A novelty of this approach is that
it does not involve localization and only requires checking computable and well-studied invariants
like Hilbert-Samuel multiplicities.
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October 31 |
Paul Balmer
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This is joint work with Beren Sanders (UCSC). Inspired by work of
Benson-Iyengar-Krause-Pevtsova, we show how the perfect complexes over the completion of a reasonable
commutative ring along an ideal can be obtained from the derived category of the ring, using an abstract
tensor-triangular construction. We shall then discuss the effect of this construction on the spectrum,
and prove what we call the Tate Intermediate Value Theorem.
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November 14 |
Maria Akter
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The study of singularities under perturbation is classical, with origins dating back to the 50's and 60's through the work of Samuel
and Hironaka. In this talk, we introduce the theory of *F*-singularities in prime characteristic rings and examine continuity problems in
relation to perturbation theory. Our main result characterizes the continuity of the *F*-splitting ratio.
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November 21 |
Yairon Cid-Ruiz
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In this talk, we introduce the mixed Segre zeta function associated with a sequence of homogeneous ideals in a polynomial ring. This power series encodes information about the mixed Segre classes that arise when the ideals are extended to projective spaces of arbitrarily large dimension. Our construction generalizes and unifies classical results by Kleiman and Thorup on mixed Segre classes and by Aluffi on Segre zeta functions. We show that this function is always rational, with poles determined by the degrees of the generators of the ideals, and that it depends only on the integral closures of these ideals. Finally, we explain how the numerator of a modified form of the mixed Segre zeta function exhibits a remarkable combinatorial property: its homogenization is denormalized Lorentzian in the sense of Brändén and Huh.
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November 28 No Seminar |
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December 5 |
TBA
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TBA
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