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).