1 Background In our report on the first “Derived Category Methods in Commutative Algebra”
workshop (2008) we wrote:“Derived category methods have proved to be very successful in …

We prove ap-adic analog of Kunz's theorem: ap-adically complete noetherian ring is regular
exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a …

It is well-known that for a large class of local rings of positive characteristic, including
complete intersection rings, the Frobenius endomorphism can be used as a test for finite …

We explore the equimultiplicity theory of the F-invariants Hilbert–Kunz multiplicity, F-
signature, Frobenius Betti numbers, and Frobenius Euler characteristic in strongly F-regular …

Abstract Let (R,m,K) be a local ring, and let M be an R-module of finite length. We study
asymptotic invariants, β^F_i(M,R), defined by twisting with Frobenius the free resolution of M …

Let $(R,\mathfrak m) $ be a local ring of characteristic $ p> 0$ and $ M $ a finitely generated
$ R $-module. In this note we consider the limit: $\lim_ {n\to\infty}\frac {\ell (H^ 0_ {\mathfrak …

This note grew from the lectures I delivered at ICTP during the Summer School in honor of
Hochster and Huneke. Its purpose is to provide an introduction to the notion of …

We extend the notion of Frobenius Betti numbers to large classes of finitely generated
modules over rings of prime characteristic, which are not assumed to be local. To do so, we …

We show that $ R $ is regular if $ Tor_ {i}^{R}(R^{+}, k)= 0$ for some $ i\geq 1$ assuming
further that $ R $ is a $\mathbb {N} $-graded ring of dimension $2 $ finitely generated over …

The notion of Frobenius Betti numbers generalizes the Hilbert-Kunz multiplicity theory and
serves as an invariant that measures singularity. However, the explicit computation of the …