Commutative Algebra Seminar
Spring 2023, Friday 2:00–3:00 pm, LCB 222
Date  Speaker  Title — click for abstract 
February 4  Uli Walther Purdue University 
Lyubeznik and Cechde Rham numbers
If Y is an affine variety inside CC^n cut out by the ideal I inside, and m a distinguished maximal ideal of, CC[x_1,...,x_n], one can attach two sets of numbers to them, either by applying the de Rham functor the Dmodule H^t_I(R), or the
Dmodule restriction functor for the inclusion Spec(R/m)\into Spec R. It turns out that these numbers are in fact functions of Y and not of the embedding into an affine space.
In the talk we discuss known facts as well as some recent insights on these double arrays of numbers. This will include some general vanishing results, as well as a discussion on when the associated spectral sequence for which these arrays
are the E_2page, collapses.

March 18  Swaraj Pande University of Michigan 
Multiplicities of Jumping Numbers
Multiplier ideals are refined invariants of singularities of algebraic varieties. They give rise to other numerical invariants, for example, the log canonical threshold and more generally, jumping numbers. This talk is about another related
invariant, namely multiplicities of jumping numbers. For an mprimary ideal I in the local ring of a smooth complex variety, multiplicities of jumping numbers measure the difference between successive multiplier ideals of I. The main result
is that these multiplicities naturally fit into a quasipolynomial. We will also discuss when the various components of this quasipolynomial have the highest possible degree, relating it to the Rees valuations of I. As a consequence, we
derive some formulas for a subset of jumping numbers of mprimary ideals. Time permitting, we will consider the special case of monomial ideals where these invariants have a combinatorial description in terms of the Newton polyhedron.

March 25  Kevin Tucker University of Illinois at Chicago 
The Theory of Frational Signature
There are a number of invariants defined via Frobenius in the study of singularities in characteristic. One such is the Fsignature, which can be viewed as a quantitative measure of Fregularity  an important class of singularities central
to the celebrated theory of tight closure pioneered by Hochster and Huneke, and closely related to KLT singularities via standard reduction techniques from characteristic zero. Recently, similar invariants have been introduced as a
quantitative measures of Frationality  another important class of Fsingularity closely related to rational singularities in characteristic zero. These include the Frational signature (HochsterYao), relative Frational signature
(SmirnovTucker), and dual Fsignature (Sannai). In this talk, I will discuss new results in joint work with Smirnov relating each of these invariants. In particular, we show that the relative Frational signature and dual Fsignature
coincide, while also verifying that the dual Fsignature limit converges.

April 1  Alapan Mukhopadhyay University of Michigan 
FrobeniusPoincare Function and HilbertKunz Multiplicity
We shall discuss a natural generalization of the classical HilbertKunz multiplicity theory when the underlying objects are graded. More precisely, given a graded ring $R$ and a finite colength homogeneous ideal $I$ in a positive
characteristic $p$ and for any complex number $y$, we shall show that the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(R)}\sum \limits_{j= \infty}^{\infty}\lambda \left( (\frac{R}{I^{[p^n]}R})_j\right)e^{iyj/p^n}$$
exists. This limit as a function in the complex variable $y$ is a natural refinement of the HilbertKunz multiplicity of the pair $(R,I)$: the value of the limiting function at the origin is the HilbertKunz multiplicity of the pair
$(R,I)$. We name this limiting function the \textit{FrobeniusPoincare function} of $(R,I)$. We shall establish that FrobeniusPoincare functions are holomorphic everywhere in the complex plane. We shall discuss properties of
FrobeniusPoincare functions, give examples and describe these functions in terms of the sequence of graded Betti numbers of $\frac{R}{I^{[p^n]}R}$. On the way, we shall mention some questions on the structure and properties of
FrobeniusPoincare functions.

April 22  Thomas Polstra University of Virginia 
Inversion of Adjunction of Fpurity
Critical to the inductive treatment of the complex minimal model program are theorems which compare the singularities of a complex variety with the
singularities of a codimension $1$ subvariety. Such theorems, when viewed through the lens of reduction to prime characteristic, produce conjectures on the
behavior of Noetherian rings of prime characteristic. In particular, Kawakita's Inversion of Adjunction of Log Canonical Singularities Theorem inspires the
similarly named Inversion of Adjunction of Fpurity conjecture in commutative algebra. We will discuss historical developments around this conjecture, recent
progress, and relations with the problem of deforming Fpurity. This talk is based on collaborative efforts with Austyn Simpson and Kevin Tucker.

April 29  Kriti Goel University of Utah 
HilbertKunz multiplicity of powers of an ideal
We provide suitable conditions under which the asymptotic limit of the HilbertSamuel coefficients of the Frobenius powers of an $m$primary ideal exists in a Noetherian local ring $(R,m)$ with prime characteristic $p>0.$ This, in turn, gives an expression of the HilbertKunz
multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized HilbertKunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized
HilbertKunz multiplicity of powers of $J$ in terms of HilbertSamuel multiplicity of $J.$ We conclude by giving a counterexample to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of
the ideal.

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