Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, p. 581-633

Building on Schlessinger’s work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger’s functors of Artin rings, we proceed in two steps: we replace functors to sets by categories fibered in groupoids, allowing us to keep track of automorphisms, and we work with deformation problems naturally associated to a scheme X, and which naturally localize on X, so that we can formalize the local behavior. The first step is already carried out by Rim in the context of his homogeneous groupoids, but we develop the theory substantially further. In this setting, many statements known for a range of specific deformation problems can be proved in full generality, under very general stack-like hypotheses.

Classification:  14D15,  14D23
@article{ASNSP_2010_5_9_3_581_0,
     author = {Osserman, Brian},
     title = {Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     pages = {581-633},
     zbl = {1200.14028},
     mrnumber = {2722657},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_581_0}
}
Osserman, Brian. Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, pp. 581-633. http://www.numdam.org/item/ASNSP_2010_5_9_3_581_0/

[1] M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. | MR 399094 | Zbl 0317.14001

[2] M. Artin, A. Grothendieck and J.-L. Verdier, “Théorie des Topos et Cohomologie Étale des Schémas”, Tome 2, SGA, no. 4, Springer-Verlag, 1972.

[3] M. Artin, A. Grothendieck and J.-L. Verdier, “Théorie des Topos et Cohomologie Étale des Schémas”, Tome 3, SGA, no. 4, Springer-Verlag, 1973. | MR 354652

[4] T. Beke, Higher Čech theory, K-theory 32 (2004), 293–322. | MR 2112899 | Zbl 1070.18008

[5] B. Fantechi and L. Göttsche, Local properties and Hilbert schemes of points, In: “Fundamental Algebraic Geometry, Mathematical Surveys and Monographs”, Vol. 123, American Mathematical Society, 2005, pp. 139–178. | MR 2223408 | Zbl 0951.14017

[6] A. Grothendieck and J. Dieudonné, “Éléments de Géométrie Algébrique: IV. Étude Locale des Schémas et des Morphismes de Schémas, quatriéme partie”, Publ. Math. Inst. Hautes Études Sci., Vol. 32, Institut des Hautes Études Scientifiques, 1967. | Numdam | MR 217086 | Zbl 0153.22301

[7] A. Grothendieck, “Catégories Cofibrées Additives et Complexe Cotangent Relatif”, Lecture Notes in Mathematics, Vol. 79, Springer-Verlag, 1968. | MR 241495 | Zbl 0201.53803

[8] R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, 1977. | MR 463157 | Zbl 0367.14001

[9] R. Hartshorne, “Deformation Theory”, Graduate Texts in Mathematics, Vol. 257, Springer-Verlag, 2010. | MR 2583634 | Zbl 1186.14004

[10] D. Huybrechts and M. Lehn, “The Geometry of Moduli Spaces of Sheaves”, Fried. Vieweg & Sohn, Braunschweig, 1997. | MR 1450870 | Zbl 0872.14002

[11] L. Illusie, “Complexe Cotangent et Déformations. I”, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, 1971. | MR 491680 | Zbl 0224.13014

[12] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967), 41–70. | MR 209339 | Zbl 0156.27201

[13] H. Matsumura, “Commutative Ring Theory”, Cambridge University Press, 1986. | MR 879273

[14] M. Olsson and J. Starr, Quot functors for Deligne-Mumford stacks, Comm. Algebra 31 (2003), 4069–4096, special volume in honor of S. Kleiman’s 60th birthday. | MR 2007396 | Zbl 1071.14002

[15] M. C. Olsson, Crystalline cohomology of stacks and Hyodo-Kato cohomology, Astérisque 316, (2007). | Zbl 1199.14006

[16] B. Osserman, Deformations and automorphisms: triangles of deformation problems, in preparation.

[17] D. S. Rim, “Formal Deformation Theory”, Groupes de monodromie en géométrie algébrique, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, 1972, expose VI. | Zbl 0246.14001

[18] M. Schlessinger, Functors of Artin rings, Tran. Amer. Math. Soc. 130 (1968), 208–222. | MR 217093 | Zbl 0167.49503

[19] J.-P. Serre, “Local Fields”, Springer-Verlag, 1979. | MR 554237

[20] A. Vistoli, The deformation theory of local complete intersections, preprint.

[21] V. Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra 214 (2010), 1384–1398. | MR 2593670 | Zbl 1194.55020

[22] V. Voevodsky, Unstable motivic homotopy categories in Nisnevich and cdh-topologies, J. Pure Appl. Algebra 214 (2010), 1399–1406. | MR 2593671 | Zbl 1187.14025