A counterexample to a conjecture of Ding

We give a counterexample to a conjecture posed by S. Ding regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and generalized L\"oewy length 6.Comment: 9 pages, a few changes from the first versio

Products of ideals may not be Golod

We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for weak Golodness of monomial ideals. Along the way, we ask some related questions.Comment: 18 pages, minor changes from first versio

Price Transmission in Three Italian Food Chains: A Structural Break Approach

Recently a wide instability of food prices has been observed in world and European agricultural and food markets. Both media and policy makers have dealt with the unsatisfactory patterns of marketing margins and price transmission along the food chain which may bring about distributive issues and affect inflationary trends. Although price transmission and margins dynamics have attracted so much interest at the policy level, few Italian studies deal with this topic. Our aim is to provide a first analysis of the price transmission mechanism in three Italian agri-food chains (lamb, pork and pasta), within a structural change framework. Results show that structural breaks in the price transmission mechanism are an issue in the food chain of pasta and pork with the regime change arising in occasion of the price bubble of 2007-2008.price transmission, cointegration, structural breaks, Agribusiness, Q13, L11,

A sufficient condition for strong FF-regularity

Let $(R,\mathfrak{m},K)$ be an $F$-finite Noetherian local ring which has a canonical ideal $I \subsetneq R$. We prove that if $R$ is $S_2$ and $H^{d-1}_{\mathfrak{m}}(R/I)$ is a simple $R\{F\}$-module, then $R$ is a strongly $F$-regular ring. In particular, under these assumptions, $R$ is a Cohen-Macaulay normal domain.Comment: 9 pages, to appear in Proceedings of the American Mathematical Societ

On the existence of FF-thresholds and related limits

The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard graded algebras over a field for which $F$-pure threshold and $F$-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the $a$-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of $F$-thresholds and $F$-pure thresholds, obtaining existence of related limits in certain cases.Comment: 21 page

An algorithm for constructing certain differential operators in positive characteristic

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $\delta$, i.e., the least integer $e$ such that $\delta$ is $R^{p^e}$-linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.Comment: 23 pages. Comments are welcom