Finiteness of associated primes of local cohomology modules

In this thesis we investigate when the set of primes of a local cohomology module is finite. We show that there are only finitely many primes associated to the local cohomology of any finitely generated module over a three-dimensional ring whose prime cyclic modules have S2-fications with respect to an ideal whose height is at least two on that module. We show that a polynomial ring, R, over either an unramified regular local ring of mixed characteristic, or a two or three dimensional ring with a resolution of singularities Y0, formed by blowing up an ideal of depth at least two where the sheaf cohomology of OY0 has finite length over the base ring, has AssHiIR (R) finite for any ideal I ⊂ R.;We also define a new class of extensions R → S, called calm, where the associated primes of S ⊗R M over S are controlled by the associated primes of M over R for any R-module M. We show that calm extensions have many good properties including that compositions of calm maps are calm, calmness can be checked locally on open covers of Spec(R), and calmness persists after localization. We give various classes of rings whose extensions are all calm as well as some examples of extensions which are not calm.