Categories of comodules and chain complexes of modules

Let y (A) denote the coendomorphism left R-bialgebroid associated to a left finitely generated and projective extension of rings R → A with identities. We show that the category of left comodules over an epimorphic image of y(A) is equivalent to the category of chain complexes of left R-modules. This equivalence is monoidal whenever R is commutative and A is an R-algebra. This is a generalization, using entirely new tools, of results by Pareigis and Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the noncommutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.