Descent theory and Amitsur cohomology of triples

✍ Scribed by Claudia Menini; Dragoş Ştefan

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 394 KB
• Volume: 266
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00145-5

✦ Synopsis

For a given triple (monad) U : C → C in the category C, 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 C(U ) such that U is an algebra in C(U ). If C is abelian, we define Amitsur cohomology of U with coefficients in a functor F : C(U ) → D. 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 C 0 ∈ C. 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 C be the category of right B-modules. Then (-) ⊗ B A : C → C 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.