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A generalized Matlis duality for certain topological rings. (English) Zbl 0803.13013

Let \(R\) be a commutative, Noetherian, complete local ring with maximal ideal \(m\) and let \(E\) be the injective envelope of the \(R\)-module \(R/m\). Then the Matlis duality theorem says the following. If \(X\), \(Y\) are the categories of \(R\)-modules with ascending chain condition and descending chain condition, respectively, then the contravariant, exact functor \(\operatorname{Hom}_ R (.,E)\) establishes a one-to-one correspondence between \(X\) and \(Y\). The objective of this paper is to generalize this result to the case of some categories of modules over certain quasi-semi-local complete topological rings.

MSC:

13J10 Complete rings, completion
13E05 Commutative Noetherian rings and modules
18G05 Projectives and injectives (category-theoretic aspects)
13H99 Local rings and semilocal rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13D99 Homological methods in commutative ring theory
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References:

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