Duality theory for Grothendieck categories and linearly compact rings

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l’he main new definition is that of linear compactness (more general than in the literature): An R-module X is called algebraically linearly compact if for each decreasing family (Xi ; i E I), I directed, of finitely generated submodules Xi of X the intersection nj Si is again finitely generated and the canonical map X lim X/Xi is surjective. A topological R-left module X is called topologically linearly compact if it has a basis of neighborhoods of 0 consisting of submodules X’ such that S/X’ is algebraically linearly compact. A comp!ete topologically coherent R-module is called strict (Jan-Erik Roos suggests the term “proper”) if each topologically coherent closed submodule Iof -\I, such that (X/Y)dis (X/Y with the discrete topology) is coherent, is open. Strictness is a notion of a technical nature which is necessary to insure that the relevant categories of topological modules are Abelian. As a consequence of Theorem A one sees that a category \!I is a locally nocthcrian Grothendieck category i f f it is dual to the category of complete topologically coherent R-left modules over some complete topologically left