  Let $\lL(A)$ denote the coendomorphism left $R$-bialgebroid associated to a
left finitely generated and projective extension of rings $R \to A$ with
identities. We show that the category of left comodules over an epimorphic
image of $\lL(A)$ 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 B. Pareigis and D. Tambara for chain complexes of vector spaces over
fields. Our approach relies heavily on the non commutative theory of Tannaka
reconstruction, and the generalized faithfully flat descent for small additive
categories, or rings with enough orthogonal idempotents.
