Topological Morita contexts

Abstract

In synthesis, this paper presents a generalization of the theory of Morita contexts from the case of abstract modules over abstract rings to that of complete l. t. (linearly topological) modules over complete l. t. rings. To begin with, given three complete right l. t. rings ( R, ρ), ( S, σ) and ( T, τ), and two complete l. t. bimodules ( R A S , α) and ( S B T , β) satisfying suitable hypotheses, we introduce the “topological tensor product” ( A, α)⊗ u S ( B, β). Next, we define a topological Morita context to be a family made up of two complete l. t. rings ( R, ρ) and ( S, σ), two bimodules ( S A R , α) and ( R B S , β) of the above kind, and two continuous bilinear maps μ :(B,β)⊗ u S(A,α)→(R,ρ) and ν :(A,α)⊗ u R(B,β)→(S,σ) ; the context is called dense if both μ and ν have dense image. We then prove that such a dense Morita context yields an equivalence of categories between CLT -( R, ρ) and CLT -( S, σ), in such a way to induce an equivalence between Mod-( R, ρ) and Mod-( S, σ). Finally, we give a “topological” parallel of the notion of progenerator, and we show that such a “topological progenerator” gives rise to a dense context, and hence to an equivalence of the above-mentioned kind. Conversely, we show that every such equivalence arises in this way.