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This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring R of prime characteristic. The methods used employ Lyubeznik's F-modules over R, particularly his F-finite F-modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over R have natural structures as F-modules; these lead to F-module structures on certain local cohomology modules over R, which are exploited, in conjunction with F-module structures on injective R-modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2. [ABSTRACT FROM AUTHOR]

We extend a result by Huneke and Watanabe bounding the multiplicity of F-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of F-injective, generalized Cohen–Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen–Macaulay ring of prime characteristic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen–Macaulay rings in characteristic zero which have dense F-injective type. [ABSTRACT FROM AUTHOR]

Abstract We establish a criterion for the strong F -regularity of a (non-Gorenstein) Cohen–Macaulay reduced complete local ring of dimension at least 2 and prime characteristic p. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an n × (n − 1) matrix X of indeterminates. For a perfect field k of characteristic p ≥ 5 , these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring over k given by the maximal minors of X is strongly F -regular. [ABSTRACT FROM AUTHOR]

We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal primes of a local ring. [ABSTRACT FROM AUTHOR]

We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in R = k[x1,…, xn] with [k : kp] < ∞ or in R = k[[x1,…, xn]] with an arbitrary field k of characteristic p > 0. As a consequence of this result, we establish an upper bound on the number of F-jumping coefficients of a principal ideal with an isolated singularity. [ABSTRACT FROM AUTHOR]

This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view towards providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings that includes all quasi-Gorenstein complete local rings. [ABSTRACT FROM PUBLISHER]

Abstract: We prove that the F-jumping coefficients of a principal ideal of an excellent regular local ring of characteristic are all rational and form a discrete subset of . [Copyright &y& Elsevier]