Let R be a commutative Noetherian ring and let M and N be
R-modules. It is shown that
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sup
| {i|ToriR(M,N) ≠ 0}= |
sup
| {depth R\frakp−depthR\frakpM\frakp−depthR\frakp N\frakp|\frakp ∈ Supp M∩Supp N} |
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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
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sup
| {i|ExtRi(N,M) ≠ 0}=depthR−depth N. |
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Set
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TdRM= |
sup
| {i ∈ \mathbbN0|ToriR(T,M) ≠ 0 for some T of finite flat dimension}. |
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In addition, some
results concerning TdR M under base change are given.
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