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Abstract
This dissertation consists of two parts, both under the overarching theme of resolutions over a commutative Noetherian ring R. In particular, we use complete resolutions to study stable local cohomology and cotorsion-flat resolutions to investigate cosupport.
In Part I, we use complete (injective) resolutions to define a stable version of local cohomology. For a module having a complete injective resolution, we associate a stable local cohomology module; this gives a functor to the stable category of Gorenstein injective modules. We show that this functor behaves much like the usual local cohomology functor. When there is only one non-zero local cohomology module, we show there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.
In Part II, we utilize minimal cotorsion-flat resolutions (both on the left and right) to compute cosupport. We first develop a criterion for a cotorsion-flat resolution to be minimal. For a module having an appropriately minimal resolution by cotorsion-flat modules, we show that its cosupport coincides with those primes ``appearing'' in such a resolution---much like the dual notion that minimal injective resolutions detect (small) support. This gives us a method to compute the cosupport of various modules, including all flat modules and all cotorsion modules. Moreover, if R is either a 1-dimensional domain that is not a complete local ring or any ring of the form k[x,y] for an uncountable field k, we show that the cosupport of R is all of Spec(R), and consequently that the cosupport of a finitely generated module over such a ring is the same as its support. Finally, we show that when k is any countable field the cosupport of k[x,y] is not closed in Spec(R).
