By Yves André

ISBN-10: 3034883366

ISBN-13: 9783034883368

ISBN-10: 3034895224

ISBN-13: 9783034895224

This is a learn of algebraic differential modules in different variables, and of a few in their family with analytic differential modules. allow us to clarify its resource. the assumption of computing the cohomology of a manifold, particularly its Betti numbers, by way of differential types is going again to E. Cartan and G. De Rham. in terms of a tender complicated algebraic kind X, there are 3 editions: i) utilizing the De Rham advanced of algebraic differential varieties on X, ii) utilizing the De Rham complicated of holomorphic differential kinds at the analytic an manifold X underlying X, iii) utilizing the De Rham advanced of Coo complicated differential kinds at the vary 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 final result of the Poincare lemma, which identifies either vector areas with the advanced cohomology H (XtoP, C) of the topological area underlying X, the 1st isomorphism is a deeper results of A. Grothendieck, which indicates particularly that the Betti numbers will be computed algebraically. This end result has been generalized by means of P. Deligne to the case of nonconstant coeffi cients: for any algebraic vector package deal .M on X endowed with an integrable usual connection, one has canonical isomorphisms The concept of standard connection is the next dimensional generalization of the classical proposal of fuchsian differential equations (only typical singularities).

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**Example text**

OXj Let us remark that for any A E Frac Ox,z, A f= 0, f3j E K, the simultaneous equations x j a~ . A = f3 j A imply that all f3 j are integers. We conclude that if some ) Ak f= 0, then a jk = a j,r+1 for all j, so that Ak E K. This shows Ak E K for all k, a contradiction. Let us show that in our situation, condition i) of the lemma is satisfied. We fix (j, k). If ek does not vanish in Ez ® Oz},Z' we are done. Otherwise, there exists i f= j such that q does not vanish in Ez ® Oz;,z (by construction of V).

X :::;:: N. Incase 1), the structural morphism (X, M) ----+ K is strict. Therefore x is a smooth point of X and any smooth morphism ofa neighborhood ofx toAk \{t = O} gives what we need. In case 2), we pick an element pEP such that p t---+ 1, underthe canonical map P ----+ Cx. x :::;:: N. The corresponding morphism SpecK[P] ----+ SpecK[N] is then log smooth while the composition X ----+ SpecK[P] ----+ SpecK[N] is strict. Now, etale locally, the log structure of (X, M) is the inverse image of the log structure of A ~og, hence it coincides with the log structure M z, where Z is the smooth divisor cp-I (t = 0).

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. If[F: g] < 00, we have Exp(£') = e(vlw)Exp(g£') + Zfor any FI K -differential module £'. Proof: Let x be a generator of mv such that y = xe(vlw) is a generator of m w , so that F ~ C«x», g ~ C'«y», with C' = C n g. 3), it suffices to consider the case when £ ~ c' «y» ®C' Ker E (y y - ex)/L. By reason of dimension, we then have Eft i ~ C(x» ®c KerEp(yt - ex)/L.

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