Free resolutions and change of rings

Abstract

Projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimension ≤ 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings R with codimension ≤ 2. Explicit resolutions are given for cyclic modules over multigraded rings, and necessary and sufficient conditions are obtained for their minimality. © 1997 Academic Press.