Phantom Homology

Provides insight into many problems previously not recognized as related

The authors use the powerful new technique - tight closure - to show that certain elements inthe homology of complexes must vanish when mapped to well-behaved rings. Ideally suited for an advanced graduate course on commutative algebra.

This book uses a powerful new technique, tight closure, to provide insight into many different problems that were previously not recognized as related. The authors develop the notion of weakly Cohen-Macaulay rings or modules and prove some very general acyclicity theorems. These theorems are applied to the new theory of phantom homology, which uses tight closure techniques to show that certain elements in the homology of complexes must vanish when mapped to well-behaved rings.
These ideas are used to strengthen various local homological conjectures. Initially, the authors develop the theory in positive characteristic, but it can be extended to characteristic 0 by the method of reduction to characteristic p. The book is suitable for an advanced graduate course in commutative algebra.