On the commutativity of the Clifford and “extension of scalars” functors

We introduce sheaves of A -modules of fractions (or just A -modules of fractions), on a topological space X, with denominator a monoid-subsheaf S of A ; as aside worth noting, we remark (Theorem 2.4) that there is an isomorphism between the functors S − 1 and ( S − 1 A ) ⊗ _ . Moreover, we discuss the classical problem related to the commutativity of the functors: Clifford functor Cl and algebra extension functor of the ground algebra K of a quadratic K-module ( M , q ) . As a particular case, we show (Corollary 3.5) that given a sheaf A of algebras on a topological space X and S as above, the functor C l S − 1 A commutes with the functor S − 1 C l A .