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Local Jacobian of a relative formal curve. (Jacobienne locale d’une courbe formelle relative.) (French. English summary) Zbl 1317.14100
The author constructs the local Jacobian of a relative formal curve and proves a relative duality formula.
Let \({\mathcal F}\) be the \(S\)-group extension of the completion \(\check W\), of the universal \(S\)-Witt vector group \(W\), by the group of units \({\mathcal O}_S [[T]]^*\).
The author proves the following theorem: Let \(S= \text{Spec} (A)\) be a noetherian affine scheme. Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Let \(B\) be an \(A\)-algebra adic isomorphic to \( A[[T]]\), \({\mathcal X} = \text{Spf} (B)\), \({\mathcal U}= \text{Spec} (A[[T]][T^{-1}])\). In the case then for any section \(\sigma \in G({\mathcal U})\) there exists unique homomorphism \( h: {\mathcal F} \to G\) such that \(\sigma = h_{\text{omb}} \circ f\) where \(f: {\mathcal U} \rightarrow {\mathcal F}_{\text{omb}}\) is an Abel-Jacobi morphism of \(({\mathcal X}, {\mathcal F})\), i.e. the arrow \(\mathrm{Hom}_{S\text{-gr}} ({\mathcal F},G) \rightarrow G({\mathcal U})\) is bijective.
The proof of the theorem is reduced to the proof of the theorem 1.4.4 (see below).
This paper is the next in a sequence of papers by the author in which he is involved with the Grothendieck program concerning global and local dualities with continuous coefficients. The relative generalized Jacobian of the smooth curve \(X - D\) and an Abel-Jacobi morphism \(X - D \rightarrow J\) are constructed in the author’s papers [C. R. Acad. Sci., Paris, Sér. A 289, 203–206 (1979; Zbl 0447.14005); Prog. Math. 87, 69–109 (1990; Zbl 0752.14023)].
Let \({\mathfrak X}\) be the formal completion of \(X\) along \(D\), \(\text{omb}({\mathfrak X}) = \text{Spec} (\Gamma({\mathfrak X}, {\mathcal O}_{\mathfrak X}) = \text{Spec} (A[[T]])\).
Let \(\rho: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}^{0}, G) \to F(G)\) and \(\rho^{+}: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}, G) \to F^{+} (G)\).
Theorem 1.4.4. The notations are taken above. Let \(A\) be a Noetherian ring and \(S = \text{Spec} (A)\). Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Then \(\rho \) and \(\rho^{+}\) are isomorphisms.
The proof of the theorem is given in sections 2–5.
This interesting article is done (is presented) by the author in the spirit of the algebraic geometry by Grothendieck, Verdier, Artin, Deligne, Saint-Donat [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)], by Grothendieck and Demazure [Schémas en groupes. I: Propriétés générales des schémas en groupes. Exposés I à VIIb. Séminaire de Géométrie Algébrique 1962/64, dirigé par Michel Demazure et Alexander Grothendieck. Revised reprint. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0207.51401)] together with some new developments by A. Beilinson [Fields Institute Communications 56, 15–82 (2009; Zbl 1186.14019)], by K. Rülling [J. Algebr. Geom. 16, No. 1, 109–169 (2007; Zbl 1122.14006)], by M. Kapranov and É. Vasserot [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113–133 (2007; Zbl 1129.14022)] and by A. N. Parshin [in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 362–392 (2011; Zbl 1266.11118)].
The paper under review closes with some results on the autoduality of \({\mathcal F}\) in the sense of Cartier.

MSC:
14L15 Group schemes
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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