Monica Lewis (University of Minnesota)Title: Finiteness properties of local cohomology in characteristic p > 0Abstract: Local cohomology modules are (typically) very large algebraic objects that encode rich geometric information about the structure of a commutative ring. For large classes of regular rings, the local cohomology of R is known to become finitely generated over a suitable noncommutative ring of endomorphisms -- either the ring of differential operators over R in characteristic 0, or the ring generated by R and its Frobenius homomorphism in characteristic p > 0. The extent to which the results of this finiteness theory can be extended when R has singularities remains an open question. In this talk, we will review the classic finiteness theory over a regular ring, will discuss what is known to hold (and known to fail) in the singular setting, and will discuss recent work searching for a "Bernstein class" of Frobenius modules in characteristic p > 0.
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