Commutative Algebra Seminar

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 D-module H^t_I(R), or the D-module 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_2-page, collapses.

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 m-primary 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 quasi-polynomial. We will also discuss when the various components of this quasi-polynomial 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 m-primary 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.

There are a number of invariants defined via Frobenius in the study of singularities in characteristic. One such is the F-signature, which can be viewed as a quantitative measure of F-regularity - 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 F-rationality - another important class of F-singularity closely related to rational singularities in characteristic zero. These include the F-rational signature (Hochster-Yao), relative F-rational signature (Smirnov-Tucker), and dual F-signature (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 F-rational signature and dual F-signature coincide, while also verifying that the dual F-signature limit converges.

We shall discuss a natural generalization of the classical Hilbert-Kunz multiplicity theory when the underlying objects are graded. More precisely, given a graded ring $R$ and a finite co-length 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 Hilbert-Kunz multiplicity of the pair $(R,I)$: the value of the limiting function at the origin is the Hilbert-Kunz multiplicity of the pair $(R,I)$. We name this limiting function the \textit{Frobenius-Poincare function} of $(R,I)$. We shall establish that Frobenius-Poincare functions are holomorphic everywhere in the complex plane. We shall discuss properties of Frobenius-Poincare 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 Frobenius-Poincare functions.

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 F-purity conjecture in commutative algebra. We will discuss historical developments around this conjecture, recent progress, and relations with the problem of deforming F-purity. This talk is based on collaborative efforts with Austyn Simpson and Kevin Tucker.

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel 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 Hilbert-Kunz 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 Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example 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.