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Abstract
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Let R denote a commutative Noetherian ring, a an ideal of R,
and M an a-cofinite R-module. The purpose of this article is to show
that for a positive integer t, the R-module Hia
(M) is finitely generated
for all i < t if and only if the Rp-module Hi
aRp(Mp) is finitely generated
for all i < t and all p ∈ Spec(R). As a consequence, we provide a generalization
and short proof of Faltings’ local–global principle for finiteness
dimensions; i.e., fa(M) = inf{faRp(Mp)| p ∈ Spec(R)}.
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