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On identifying sheaves and bundles of topological algebras

Kyriazis A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2003

[Z] 43(1)

95-104

In this paper we consider topological A-algebra sheaves on a topological space X, i.e., topological algebra sheaves on X with "coefficient sheaf" A. Thus, let Fi^(X) be the set of equivalence classes of locally free topological A-algebra sheaves of rank 1 on X, which is classified by H^1(X, A*), the 1-st cohomology set of X with coefficients in the topological group sheaf of units A* of A. Moreover,Fi_A(X) stands for the set of equivalence classes of topological algebra bundles on X of fibre type A, classified by H^1(X, Aut(A)), the 1-st cohomology set of X with coefficients in the topological group of automorphisms of A. Now, let A be a constant topological algebra sheaf whose fibre topological algebra A has continuous multiplication and inversion and also having locally equicontinuous Aut(A). Then, one gets the following bijections Fi(X) = H^1(X,A') =H^1(X,Aut(A))=Fi_A(X).