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Approximation of versal deformations. (English) Zbl 1087.14004

Let \(S\) be an excellent scheme and \(F\) a category cofibred in groupoids over the category of \(S\)-schemes. Roughly means that \(F(T)\) is the category of geometric structures over a \(S\)-scheme \(T\), which behave well with respect to the base change. \(F\) is locally of finite presentation over \(S\) if for any directed system \((A_i)\) of \({\mathcal O}_S\)-algebras with direct limit \(A\), the natural transformation of categories from the inductive limit of \(F(A_i)\) to \(F(A)\) is fully faithful and essential surjective. Let \({\bar F}(T)\) be the set of isomorphism classes of objects in \(F(T)\) and suppose that \(F\) satisfies the so called Schlessinger-Rim criteria and the natural map from \({\bar F}(B)\) to the projective limit of \({\bar F}(B/m_B^{n+1})\) has dense image for all complete local noetherian \({\mathcal O}_S\)-algebras \((B,m_B)\) with \(B/m_B\) residually finite over \(S\). For any residually finite \({\mathcal O}_S\)-field \(k\) and any object \(\rho\) in \(F(k)\), any formal versal deformation of \(\rho\) in \(F(k)\) is algebraizable.
A similar result was claimed by the reviewer and M. Roczen in [Rev. Roum. Math. Pures Appl. 33, 251–260 (1988; Zbl 0661.14003)] but the proof has a gap.

MSC:

14B12 Local deformation theory, Artin approximation, etc.
14B20 Formal neighborhoods in algebraic geometry
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14B10 Infinitesimal methods in algebraic geometry
14B07 Deformations of singularities
13B40 Étale and flat extensions; Henselization; Artin approximation

Citations:

Zbl 0661.14003
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References:

[1] Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. Inst. Hautes Études Sci., 36, 23-58 (1969) · Zbl 0181.48802
[2] Artin, M., Algebraization of formal moduli: I, (Global Analysis (Papers in honor of K. Kodaira) (1969), Univ. of Tokyo Press: Univ. of Tokyo Press Tokyo), 21-71 · Zbl 0205.50402
[3] Artin, M., Versal deformations and algebraic stacks, Invent. Math., 27, 165-189 (1974) · Zbl 0317.14001
[4] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron models. Néron models, Ergeb. Math. Grenzgeb. (3), 21 (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0705.14001
[5] Bourbaki, N., Algèbre (1980), Masson: Masson Paris
[6] Dieudonné, J.; Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci., 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967) · Zbl 0203.23301
[7] Eisenbud, D., Adic approximation of complexes, and multiplicities, Nagoya Math. J., 54, 61-67 (1974) · Zbl 0299.13010
[8] Grothendieck, A., Groupes de monodromie en géométrie algébrique. Groupes de monodromie en géométrie algébrique, Lecture Notes in Math., 288 (1972), Springer-Verlag: Springer-Verlag New York, (with M. Raynaud, D.S. Rim)
[9] Lazard, D., Autour de la platitude, Bull. Soc. Math. France, 97, 81-128 (1969) · Zbl 0174.33301
[10] Laumon, G.; Moret-Bailly, L., Champs algébriques. Champs algébriques, Ergeb. Math. Grenzgeb. (3), 39 (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0945.14005
[11] Popescu, D., General Néron desingularization, Nagoya Math. J., 100, 97-126 (1985) · Zbl 0561.14008
[12] Popescu, D., General Néron desingularization and approximation, Nagoya Math. J., 104, 85-115 (1986) · Zbl 0592.14014
[13] Popescu, D., Letter to the editor: General Néron desingularization and approximation, Nagoya Math. J., 118, 45-53 (1990) · Zbl 0685.14009
[14] Popescu, D.; Roczen, M., Algebraization of deformations of exceptional couples, Rev. Roumaine, XXXIII, 3, 251-260 (1988) · Zbl 0661.14003
[15] Schlessinger, M., Functors on Artin rings, Trans. Amer. Math. Soc., 130, 208-222 (1968) · Zbl 0167.49503
[16] Spivakovsky, M., A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Amer. Math. Soc., 12, 381-444 (1999) · Zbl 0919.13009
[17] Swan, R., Néron-Popescu desingularization, Lectures in Algebra and Geometry, 2 (1998), International Press: International Press Cambridge, 135-192 · Zbl 0954.13003
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