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IPM
30
YEARS OLD

“School of Mathematics”

Paper   IPM / M / 2302
   School of Mathematics
  Title: Depth formulas, restricted Tor-dimension under base change
  Author(s):
1 . T. Sharif
2 . S. Yassemi
  Status: Published
  Journal: Rocky Mountain J. Math.
  Vol.: 34
  Year: 2004
  Pages: 1131-1146
  Supported by: IPM
  Abstract:
Let R be a commutative Noetherian ring and let M and N be R-modules. It is shown that
sup
{i|ToriR(M,N) ≠ 0}= sup
{depth R\frakpdepthR\frakpM\frakpdepthR\frakp N\frakp|\frakpSupp MSupp N}
provided that M has finite 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 that
sup
{i|ExtRi(N,M) ≠ 0}=depthRdepth N.
Set
TdRM= sup
{i ∈ \mathbbN0|ToriR(T,M) ≠ 0  for  some T  of  finite  flat  dimension}.
In addition, some results concerning TdR M under base change are given.

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