Alessandro De Stefani

Symbolic Powers of Ideals

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

Artinian Level Algebras of Low Socle Degree

In this article we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulays inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore, we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case, the extremal strata have been parametrized by Cho and Iarrobino.

A Zariski--Nagata theorem for smooth ℤ-algebras

Abstract In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra.

Frobenius Betti numbers and modules of finite projective dimension

Let

F-thresholds of graded rings

(R,\mathfrak{m},K)

PRODUCTS OF IDEALS MAY NOT BE GOLOD

be a local ring, and let

Globalizing F-invariants

M

On the existence of -thresholds and related limits

be an

Generalizing Serre’s Splitting Theorem and Bass’s Cancellation Theorem via free-basic elements

R

Cohomologically full rings.

-module of finite length. We study asymptotic invariants,