Descent theory and Amitsur cohomology of triples

For a given triple (monad) in the category , we develop a theory of descent for U. We start by introducing the basic constructions associated to a triple: descent data, symmetry operators, and flat connections. The main result of this section asserts that the sets of these objects are bijectively equivalent. Next we construct a monoidal category such that U is an algebra in . If is abelian, we define Amitsur cohomology of U with coefficients in a functor . As an application of this construction, in the case where U is faithfully exact, we describe those morphisms that descend with respect to U. In the last part of the paper we classify all U-forms of a given object . We show that there is a one-to-one correspondence between the set of equivalence classes of U-forms and a certain noncommutative Amitsur cohomology. Let A/B be an extension of associative unitary rings and let be the category of right B-modules. Then is a triple which is faithfully exact if and only if the extension A/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings.