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

We introduce sheaves of AA-modules of fractions (or just AA-modules of fractions), on a topological space X  , with denominator a monoid-subsheaf SS of AA; as aside worth noting, we remark (Theorem 2.4) that there is an isomorphism between the functors S−1S−1 and (S−1A)⊗_(S−1A)⊗_. 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)(M,q). As a particular case, we show (Corollary 3.5) that given a sheaf AA of algebras on a topological space X   and SS as above, the functor ClS−1AClS−1A commutes with the functor S−1ClAS−1ClA.