Proper resolutions and their applications

Homological techniques provide potent tools in commutative algebra. For example, successive approximation by projective modules results in a projective resolution, and the minimal length is an invariant known as the projective dimension of the module. Auslander and Buchsbaum and Serre demonstrate the utility of this approach, characterizing regular local rings as those over which every module has finite projective dimension. This settled Krull\u27s conjecture that the localization of a regular local ring is regular. Auslander and Bridger enlarged the class of resolving modules from projectives to the class of totally reflexives, giving rise to a refinement of projective dimension. This G-dimension characterizes Gorenstein rings: a local ring is Gorenstein if and only if every finitely generated module has finite G-dimension. This thesis focuses on the structure of modules and complexes, usually in the context of related homological invariants. For resolving classes containing modules other than the projectives, certain types of resolutions---those with particularly good lifting properties---are crucial to my research. Some of my results address the existence of these proper resolutions and other parts of my work study their associated relative cohomology functors