Lyubeznik numbers, 𝐹-modules and modules of generalized fractions

Abstract

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 R of prime characteristic. The methods used employ Lyubeznik’s F F -modules over R R , particularly his F F -finite F 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 R have natural structures as F F -modules; these lead to F F -module structures on certain local cohomology modules over R R , which are exploited, in conjunction with F F -module structures on injective R 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.