Local Comology, Local Duality and Basic Notions of Tight Closure Theory

This lectures were given by Florian Enescu at the mini-course on classical problems in commutative algebra held at University of Utah, June 2004. The references listed were used extensively in preparing these notes and the author makes no claim of originality. Moreover, he encourages the reader to consult these references for more details and many more results that had to be omitted due to time constraints. This version has been typed and prepared by Bahman Engheta. 1. Injective modules, essential extensions, and local cohomology Throughout, let R be a commutative Noetherian ring. Definition. An R-module I is called injective if given any R-module monomorphism f : N → M , every homomorphism u : N → I can be extended to a homomorphism g : M → I, i.e. g f = u. 0 N M