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Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular we present examples of polynomials $g$ and $f$ such that there is no differential operator raising $g/f$ to its $p$th power., 14 pages, comments are welcome

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/ D 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 $\mathbb{F}_p$ 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 $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture., 14 pages, comments are welcome. This version: errata corrected, main results unchanged

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, { I j } , and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to C r -rings, for 2 ≤ r ≤ ∞ . As an auxiliary step we establish the surjectivity of the completion map for a very broad class of filtrations.

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster., Comment: 63 pages, comments are welcome. To appear in International Mathematics Research Notices

This note describes a \emph{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., Comment: 21 pages, minor improvements throughout. This paper is the companion to the the latest version of the package (https://github.com/kschwede/Workshop-2017-Berkeley/tree/master/Fsing/TestIdealsPackage), which should be included in 1.15 of Macaulay2 and later

The purpose of this report is to introduce a formalism to produce two collections of spectral sequences. On one hand, a collection is made up by spectral sequences which involve in their second page the left derived functors of the colimit on a certain finite poset. On the other hand, the other is made up by spectral sequences which involve in their second page the right derived functors of the limit on a certain finite poset. In both cases, we provide sufficient conditions to ensure their degeneration at the second page. Finally, we see how to use our second collection of spectral sequences to produce a decomposition of local cohomology modules which can be regarded as a generalization of the classical Hochster formula for the local cohomology of a Stanley–Reisner ring.

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., Comment: 15 pages, comments are welcome. This paper recovers and extends the results obtained by the second named author in arXiv:1501.00718 [math.AC]

We construct a Koszul complex in the category of left skew polynomial rings associated with a flat endomorphism that provides a finite free resolution of an ideal generated by a Koszul regular sequence