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.

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
Osserman, Brian 1

1 Department of Mathematics, University of California at Davis, One Shields Ave., Davis, CA 95616 USA
@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},
     pages = {581--633},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     mrnumber = {2722657},
     zbl = {1200.14028},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2010_5_9_3_581_0/}
}
TY  - JOUR
AU  - Osserman, Brian
TI  - Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 581
EP  - 633
VL  - 9
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_3_581_0/
LA  - en
ID  - ASNSP_2010_5_9_3_581_0
ER  - 
%0 Journal Article
%A Osserman, Brian
%T Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 581-633
%V 9
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_3_581_0/
%G en
%F 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://archive.numdam.org/item/ASNSP_2010_5_9_3_581_0/

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

[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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

[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

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

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

[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 | Zbl

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