Doctor of Philosophy

dissertationIn Chapter 2, we compute a semi-free resolution of the Koszul complex over its endomorphism ring. This computation leads to an expression of the local cohomology of a commutative ring with respect to a finitely generated ideal as the homology of a set of polynomials with coefficients in the Koszul complex associated to that ideal endowed with a differential. This expression is quasi-isomorphic to the extended ˇ Cech complex and is comprised of free modules, a property that yields a computation of derived completion via adjunction. In Chapter 3, we establish an extrinsic theory of support and cosupport via the action of a ring with an abelian-group grading. The process extends a theorem of Gruson-Raynaud to the category of graded rings, and we prove the costratification of the derived category of graded modules over skew-commutative group-graded rings. We also prove that cosupport descends along finite maps for this class of rings, a class that contains traditional graded, commutative rings