Absolute homology theory of stereotype algebras (Q1576935)

A locally convex space \(X\) is stereotypical if the canonical inclusion \(i\colon X\to(X'_c)'_c\), where \(X'_c\) is its dual equipped with the topology of uniform convergence on completely bounded sets, is a topological isomorphism. The author introduced and studied these spaces in [\textit{S. S. Akbarov}, Mat. Zametki 57, No. 3, 463-466 (1995; Zbl 0836.46002); translation in Math. Notes 57, No. 3-4, 319-322 (1995); Funkts. Anal. Prilozh. 29, No. 4, 68-72 (1995; Zbl 0883.46046) translation in Funct. Anal. Appl. 29, No. 4, 276-279 (1996) and Funkts. Anal. Prilozh. 33, No. 2, 68-73 (1999; Zbl 0951.46004); translation in Funct. Anal. Appl. 33, No. 2, 137-140 (1999)]. Let \(A\) be a topological unital algebra over the complex numbers that is a stereotypical space, and let \(_AS\) [resp. \(S_A\)] be the category of locally convex stereotypical left (resp. right) topological \(A\)-modules and continuous \(A\)-linear maps. This short note sketches the homology theory of \(_AS\) [resp. \(S_A\)]: injective and projective stereotypical \(A\)-modules, as well as projective and injective resolutions are introduced, so that every stereotypical \(A\)-module has an injective and a projective resolution. This defines, in a standard way, injective dimension and projective dimension of stereotypical \(A\)-modules. The author introduces also the absolute homological dimension of a stereotypical algebra and gives several examples: the algebra of complex numbers has absolute homological dimension zero, whereas the absolute homological dimension of the algebra of continuous functions on a compact space \(T\) is zero if and only if the space \(T\) is discrete (hence finite). No proofs are given.