De Rham Cohomology of Differential Modules on Algebraic by Yves André

By Yves André

This is a examine of algebraic differential modules in different variables, and of a few in their kinfolk with analytic differential modules. allow us to clarify its resource. the belief of computing the cohomology of a manifold, specifically its Betti numbers, by way of differential types is going again to E. Cartan and G. De Rham. with regards to a delicate advanced algebraic style X, there are 3 editions: i) utilizing the De Rham advanced of algebraic differential types on X, ii) utilizing the De Rham complicated of holomorphic differential varieties at the analytic an manifold X underlying X, iii) utilizing the De Rham complicated of Coo advanced differential varieties at the range­ entiable manifold Xdlf underlying Xan. those editions tum out to be similar. particularly, one has canonical isomorphisms of hypercohomology: whereas the second one isomorphism is a straightforward sheaf-theoretic end result of the Poincare lemma, which identifies either vector areas with the advanced cohomology H (XtoP, C) of the topological house underlying X, the 1st isomorphism is a deeper results of A. Grothendieck, which indicates specifically that the Betti numbers may be computed algebraically. This outcome has been generalized via P. Deligne to the case of nonconstant coeffi­ cients: for any algebraic vector package .M on X endowed with an integrable usual connection, one has canonical isomorphisms The suggestion of standard connection is a better dimensional generalization of the classical suggestion of fuchsian differential equations (only commonplace singularities).

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Ii) Exp(£] ®C«x)) £2) = Exp(£d+Exp(£2) and Exp(Homc«x))(£l, -Exp(£]) + EXP(£2). 4 Let F/ K be a function field in one variable (K not necessarily algebraically closed), and let v be a non-trivial valuation of F trivial on K. Then C := k(v) Regularity in several variables 34 is a finite extension of K, and there is an element x of mv such that (F, 1) (C(x», x-adic valuation), Rv = C[[x]]. If E is an FIC-differential module, regular at v, we set Expv(E) = Exp(£). 3), where £ is a gI K -differential module, we have Exp(£j:) = e(vlw)Exp(£) + Z.

We prove iii) ::::} i) We must prove two things: that the canonical map (f*wl y,W)/K)I/Z ~ wlX,Z)/K,I/z is injective and that the OX,I/z-module wtx,Z)/(y,W),I/z is free. The first statement holds because both the source and the target are free OX,I/z-modules that inject into K(X) ®K(Y) Q;(Y)/K and Q;(X)/K' respectively. On the other hand, as all extensions are separably generated, the sequence of K(X)-vector spaces o ~ K(X) ®K(Y) Q;(Y)/K ~ Q;(X)/K ~ Q;(X)/K(Y) ~ 0 is exact. As for the second statement, it is clear, by definition, that 1 ' / .

3 Let (X, Z)~(Y, W) be a morphism of models such that feZ) c W. Let (£, V) be a coherent sheaf with integrable connection on Y \ W, regular along W. Then Expz(f*(V» = ef Expw(V) + Z. 5). ii) Let us next assume that X is a curve. We may replace X by any connected etale neighborhood of Z. 7), and the previous case, we are reduced to the case where f is a closed immersion of models. 11). iii) In the general case, we can find a closed immersion (C, P)~(X, Z) with C a curve. Then e f = ei o f and the result follows from ii).

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