On Poincaré series

By analogy with the well-known notion of the Poincaré series of a local ring, the author introduces the notion of Poincaré series of a module over a local ring. Some theorems on the behavior of Poincaré series of rings and modules under faithfully flat extensions of the base ring are proved. Applying these theorems, the author proves in a uniform way some old and new results on the behavior of various homological properties under faithfully flat extensions; among these are results on Gorenstein modules, canonical modules, Gorenstein rings, complete intersections, and injective modules. Theorems on the canonical module of a tensor product of algebras over a regular ring and on homological isomorphisms between the Koszul complexes of a local ring and its quotient ring are also proved. In this paper the author gives complete proofs of his previously announced results.
Bibliography: 22 titles.