1. Descent theory and Amitsur cohomology of triples
- Author
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Menini, Claudia and Ştefan, Dragoş
- Subjects
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MONADS (Mathematics) , *THEORY of descent (Mathematics) - Abstract
For a given triple (monad)
U :C→C in the categoryC , we develop a theory of descent forU . 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 categoryC(U) such thatU is an algebra inC(U) . IfC is abelian, we define Amitsur cohomology ofU with coefficients in a functorF :C(U)→D . As an application of this construction, in the case whereU is faithfully exact, we describe those morphisms that descend with respect toU . In the last part of the paper we classify allU -forms of a given objectC0∈C . We show that there is a one-to-one correspondence between the set of equivalence classes ofU -forms and a certain noncommutative Amitsur cohomology. LetA/B be an extension of associative unitary rings and letC be the category of rightB -modules. Then(−)⊗BA :C→C is a triple which is faithfully exact if and only if the extensionA/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings. [Copyright &y& Elsevier]- Published
- 2003
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