Mathematics

We show that F -purity deforms in local Q -Gorenstein rings of prime characteristic p>0 . Furthermore, we show that F -purity is m -adically stable in local Cohen-Macaulay Q -Gorenstein rings.

It is known that a two-dimensional F -rational ring has a rational singularity. However a two-dimensional ring with a rational singularity is not F -rational in general. In this paper, we investigate F -rationality of a two-dimensional graded ring with a rational singularity in terms of the multiplicity. Moreover, we determine when a two-dimensional graded ring with a rational singularity and a small multiplicity is F -rational.

We provide a formula for F -thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the F -pure threshold of a diagonal hypersurface. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our result to find hypersurfaces where the log canonical thresholds equals the F -pure thresholds for infinitely many prime numbers.

We prove that deformation of F-injectivity holds for local rings (R,m) that admit secondary representations of H i m (R) which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when (R,m) is sequentially Cohen-Macaulay (or more generally when all the local cohomology modules H i m (R) have no embedded attached primes). We obtain some additional cases if R/m is perfect or if R is N -graded.

We consider ring extensions whose set of all subextensions is stable under the formation of sums, the so-called Delta extensions and exhibit new examples of these extensions.

Let S⊆ Z m ⊕T be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in S having at least two factorizations of the same length, namely the ideal L S . To this end, we work with a certain (lattice) ideal associated to the monoid S . Our study can be seen as a new approach generalizing \cite{chapman:2011}, which only studies the case of numerical semigroups. When S is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal L S when S is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that L S is a principal ideal; (3) we classify the computational problem of determining the largest integer not in L S as an NP -hard problem.

We study families of cellular resolutions by looking at them as a category and applying tools from representation stability. We obtain sufficient conditions on the structure of the family to have a noetherian representation category and apply this to concrete examples of families. In the study of syzygies we make use of defining the syzygy module as a representation and find conditions for the finite generation of this representation. We then show that many families of cellular resolutions coming from powers of ideals satisfy these conditions and have finitely generated syzygies, including the maximal ideals and edge ideals of paths.

In this article, we present FastLinAlg, a package in Macaulay2 designed to introduce new methods focused on computations in function field linear algebra. Some key functionality that our package offers includes: finding a submatrix of a given rank in a provided matrix (when present), verifying that a ring is regular in codimension n, recursively computing the ideals of minors in a matrix, and finding an upper bound of the projective dimension of a module.

We consider a rational map ϕ: P m k ⇢ P n k that is a parameterization of an m -dimensional variety. Our main goal is to study the (m−1) -dimensional fibers of ϕ in relation to the m -th local cohomology modules of the Rees algebra of its base ideal.

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.