Noncommutative descent and non-abelian cohomology
This paper gives an elegant presentation of descent for modules over extensions of non-commutative rings. Let \(\phi:R\to S\) be a \(k\)-algebra homomorphism, \(k\) a commutative ring. Descent theory answers two questions: I. Given a right \(S\)-module \(M\), decide if there is a right \(R\)-module \(N\) so that \(M\cong N\otimes S\) (\(\otimes=\otimes_R\)). II. Given an \(R\)-module \(N_0\), classify the twisted forms of \(N_0\), that is, right \(R\)-modules \(N\) so that \(N_0\otimes S\cong N\otimes S\). Question I is answered by trying to attach to \(M\) a descent datum: that is, finding an \(S\)-linear map \(f:M\to M\otimes S\) which is a section of the multiplication map \(M\otimes S\to M\) and satisfies a certain cocycle conditition. If \(\phi:R\to S\) makes \(S\) into a faithfully flat left \(R\)-module, then \textit{M. Cipolla} [Rend. Circ. Math. Palermo, II. Ser. 25, 43-46 (1976; Zbl 0391.16022)] proved that the functor \(N\mapsto N\otimes S\) yields an equivalence from the category of right \(R\)-modules to the category of right \(S\)-modules with descent data. The author elicidates this result by defining two other categories of right \(S\)-modules. To do so, the author identifies \(S\otimes S\) with \(S\oplus\Omega(S/R)\) where \(\Omega(S/R)\), the kernel of the multiplication map \(S\otimes S\to S\), is called the module of differential 1-forms of \(S\) over \(R\): if \(\Omega(S/R)\) is viewed as an ideal of square 0, then the resulting multiplication makes \(S\otimes S\) into a \(k\)-algebra. An \(R\)-linear map \(\Delta\) from \(M\) to \(M\otimes_S\Omega(S/R)\) is called a connection; \(\Delta\) is flat if the induced ``curvature'' map \(\Delta^2\) from \(M\) to \(M\otimes_S\Omega(S/R)\otimes_S\Omega(S/R)\) is zero. The collection of right \(S\)-modules with flat connections forms a category. The author then defines two different \(S\otimes S\)-module structures on \(M\otimes S\), subscripted \(_1\) and \(_2\). A twist is an \(S\otimes S\)-module isomorphism \(\xi\) from \((M\otimes S)_1\) to \((M\otimes S)_2\) which commutes with the scalar multiplication map \(M\otimes S\to M\); \(\xi\) is a symmetry if \(\xi\) is a twist which satisfies a Yang-Baxter equation. The set of right \(S\)-modules with symmetries forms a category. The main theorem of the paper is that the three categories of right \(S\)-modules: with descent data, with flat connections, and with symmetries, are equivalent. The formalism is specialized to the case where \(\phi:R\to S\) is an \(H\)-Galois extension where \(H\) is a \(k\)-bialgebra and \(S\) a right \(H\)-comodule algebra: in that case the three categories are equivalent to the category of right \((H,S)\)-bialgebra modules. The theory is applied to generalize to a non-commutative setting a theorem of \textit{M.-A. Knus} [``Quadratic and Hermitian forms over rings'', Grundlehren Math. 294 (1991; Zbl 0756.11008)] on descent of Hermitian right \(S\)-modules, where \(R, S\) are rings with involution. Concerning question II, classification, the formalism of non-abelian Amitsur cohomology is extended to yield an isomorphism of pointed sets between \(H^1(S/R,\Aut(N_0))\) and the set of isomorphism classes of \(S/R\)-twisted forms of \(N_0\), that is, pairs \((N,\psi)\) where \(N\) is a right \(R\)-module and \(\psi:N_0\otimes S\to N\otimes S\) is an \(S\)-module isomorphism. Any twisted form yields an involutive symmetry on \(N_0\otimes S\otimes S\); the isomorphism between twisted forms and cohomology is defined using the symmetry.
scientific article; zbMATH DE number 1059651