56 research outputs found
Generalized Lyubeznik numbers
Given a local ring containing a field, we define and investigate a family of
invariants that includes the Lyubeznik numbers, but that captures finer
information. These "generalized Lyubeznik numbers" are defined as lengths of
certain iterated local cohomology modules in a category of D-modules, and in
order to define them, we develop the theory of a functor Lyubeznik utilized in
proving that his original invariants are well defined. In particular, this
functor gives an equivalence of categories with a category of D-modules. These
new invariants are indicators of F-regularity and F-rationality in
characteristic p>0, and have close connections with characteristic cycle
multiplicities in characteristic zero. We compute the generalized Lyubeznik
numbers associated to monomial ideals using interpretations as lengths in a
category of straight modules, as well as provide examples of these invariants
associated to certain determinantal ideals.Comment: 25 pages; comments welcom
A sufficient condition for strong -regularity
Let be an -finite Noetherian local ring which has a
canonical ideal . We prove that if is and
is a simple -module, then is a
strongly -regular ring. In particular, under these assumptions, is a
Cohen-Macaulay normal domain.Comment: 9 pages, to appear in Proceedings of the American Mathematical
Societ
Properties of Lyubeznik numbers under localization and polarization
We exhibit a global bound for the Lyubeznik numbers of a ring of prime
characteristic. In addition, we show that for a monomial ideal, the Lyubeznik
numbers of the quotient rings of its radical and its polarization are the same.
Furthermore, we present examples that show striking behavior of the Lyubeznik
numbers under localization. We also show related results for generalizations of
the Lyubeznik numbers.Comment: 17 page
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