Finiteness of local homology modules
Archivum Mathematicum (2020)
- Volume: 056, Issue: 1, page 31-41
- ISSN: 0044-8753
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If
N R (0:_N {I_M})\ne 0 \operatorname{width}_{I_M}(N)= \inf \lbrace i\mid \operatorname{H}_i^{I_M}(N)\ne 0 \rbrace =\inf \lbrace i \mid \operatorname{H}_i^I(M,N)\ne 0 \rbrace \,. -
If
(R,\mathfrak {m}) N R \cup _{i<n}\operatorname{Cos}_R\big (\operatorname{H}_i^{I_M}(N)\big )=\cup _{i<n}\operatorname{Cos}_R\big (\operatorname{H}_i^I(M,N)\big )=\\ \cup _{i<n}\operatorname{Cos}_R\big (\operatorname{Tor}_i^R(M/IM,N)\big )\,, \inf \lbrace i \mid \operatorname{H}_i^{I_M}(N) \text{ is not Noetherian $R$-module\,} \rbrace =\\ \inf \lbrace i \mid \operatorname{H}_i^I(M,N) \mbox {\ is not Noetherian R-module\,}\rbrace \,.
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