Free resolutions and change of rings
The author constructs a projective resolution \(F(X,Y)\) of a module \(M\) over a ring \(R\), starting from appropriate projective resolutions, \(X\) of \(R\) and \(Y\) of \(M\), over a ring \(Q\) mapping to \(R\). It is shown that such resolutions may be chosen to be minimal in codimension \(\leq 2\), but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings \(R\) with codimension \(\leq 2\). Explicit resolutions are given for cyclic modules over multigraded rings, and necessary and sufficient conditions are obtained for their minimality.
scientific article; zbMATH DE number 1013971
scientific article