Finiteness Properties of Local Cohomology.

Abstract

Local cohomology modules have played an important role in commutative algebra. These modules are usually not finitely generated; however, they have finiteness properties over local regular rings of equal characteristic. Namely, the associated primes and the Bass numbers of local cohomology modules are finite in this case. This work shows these properties for certain non-local regular rings, flat extensions with regular fibers, rings of mixed characteristic and direct summands. In addition, the F-Jacobian ideal, the generalized Lyubeznik numbers and the Lyubeznik numbers in mixed characteristic are defined by using finiteness properties of local cohomology. Furthermore, it is shown that these invariants relate to singularities of rings.