We study the generation of the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring over a field with positive characteristic. In particular, by extending the ideas used by M. Katzman to give a counterexample to a question raised by G. Lyubeznik and K. E. Smith about the finite generation of Frobenius algebras, we prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring can be only principally generated or infinitely generated. Also, by using our explicit description of the generators of such algebra and applying the recent work by M. Blickle about Cartier algebras and generalized test ideals, we are able to show that the set of Fjumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete subset inside the non-negative real numbers.

Let ν be a valuation of arbitrary rank on the polynomial ring K[x] with coefficients in a field K. We prove comparison theorems between MacLane-Vaquié key polynomials for valuations µ ≤ ν and abstract key polynomials for ν.Also, some results on invariants associated to limit key polynomials are obtained. In particular, if char(K) = 0 we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over Fp in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g ≥ 2 has level 2. We provide a good number of examples and raise a conjecture.

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.2010 Mathematics Subject Classification. Primary 13D45; Secondary 13A35, 13N10.

This note describes a Macaulay2 package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, p −e -linear and p e -linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given p −e -linear map.