Depth formulas, restricted Tor-dimension under base change

摘要

Let R be a commutative Noetherian ring, and let M and N be R-modules. It is shown thatTor (R)(i)(M,N) not equal 40}= sup{depthR(P)-depth(Rp)M(P)-depthR(p) N-p p is an element of Supp M boolean AND Supp N}provided that M has finite flat dimension. Assume that R is a complete local ring, M a finitely generated R-module, and N an R-module of finite flat dimension. It is then proved thatsup[i Ext (i)(R)(N, M) not equal 0} = depthR - depthN.SetTd(R)M = sup{i is an element of N-0 Tor(i)(R)(T, M) not equal 0 for some T of finite flat dimension}.In addition, some results concerning Td(R)M under base change are given.