no. 2
Finiteness Theorems for Deformations of Complexes
[Théorèmes de finitude pour déformations de complexes]
Bleher, Frauke M.1 ; Chinburg, Ted2
1 University of Iowa Department of Mathematics Iowa City, IA 52242-1419 (U.S.A.)
2 University of Pennsylvania Department of Mathematics Philadelphia, PA 19104-6395 (U.S.A.)
Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 573-612.

Nous considérons les déformations de complexes bornés de G-modules, sur un corps de caractéristique positive lorsque G est un groupe profini. Nous démontrons un théorème de finitude qui fournit des conditions suffisantes pour que la déformation verselle d’un tel complexe puisse être représentée par un complexe de G-modules strictement parfait sur l’anneau de déformation verselle associé.

We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex to be represented by a complex of G-modules that is strictly perfect over the associated versal deformation ring.

DOI : 10.5802/aif.2770
Classification : 11F80, 20E18, 18E30
Keywords: Versal and universal deformations, derived categories, finiteness questions, tame fundamental groups
Mot clés : déformations verselles et universelles, catégories dérivées, questions de finitude, groupes fondamentaux modérés

Bleher, Frauke M. 1 ; Chinburg, Ted 2

1 University of Iowa Department of Mathematics Iowa City, IA 52242-1419 (U.S.A.)
2 University of Pennsylvania Department of Mathematics Philadelphia, PA 19104-6395 (U.S.A.) 
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Bleher, Frauke M.; Chinburg, Ted. Finiteness Theorems for Deformations of Complexes. Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 573-612. doi : 10.5802/aif.2770. https://aif.centre-mersenne.org/articles/10.5802/aif.2770/

[1] Bleher, Frauke M.; Chinburg, Ted Deformations and derived categories, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 2, pp. 97-100 | DOI | MR | Zbl

[2] Bleher, Frauke M.; Chinburg, Ted Deformations and derived categories, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 7, pp. 2285-2359 | DOI | Numdam | MR | Zbl

[3] Bleher, Frauke M.; Chinburg, Ted Obstructions for deformations of complexes, Ann. Inst. Fourier (Grenoble), Volume 63 (2013) no. 2, pp. 613-654 | DOI

[4] Brumer, Armand Pseudocompact algebras, profinite groups and class formations, J. Algebra, Volume 4 (1966), pp. 442-470 | DOI | MR | Zbl

[5] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. France, Volume 90 (1962), pp. 323-348 | Numdam | MR | Zbl

[6] Gabriel, Pierre Étude infinitesimale des schémas en groupes, A. Grothendieck, SGA 3 (with M. Demazure), Schémas en groupes I, II, III (Lecture Notes in Mathematics, Vol. 151), Springer-Verlag, Heidelberg, 1970, pp. 476-562 | Zbl

[7] Grothendieck, Alexander; Murre, Jacob P. The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Springer-Verlag, Berlin, 1971 | MR | Zbl

[8] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[9] Kisin, Mark Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), Volume 170 (2009) no. 3, pp. 1085-1180 | DOI | MR | Zbl

[10] Lazard, Michel Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math. (1965) no. 26, pp. 389-603 | Numdam | MR | Zbl

[11] Mazur, B. Deforming Galois representations, Galois groups over Q (Berkeley, CA, 1987) (Math. Sci. Res. Inst. Publ.), Volume 16, Springer, New York, 1989, pp. 385-437 | MR | Zbl

[12] Mazur, Barry An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 243-311 | MR | Zbl

[13] Schlessinger, Michael Functors of Artin rings, Trans. Amer. Math. Soc., Volume 130 (1968), pp. 208-222 | DOI | MR | Zbl

[14] Schmidt, Alexander Tame coverings of arithmetic schemes, Math. Ann., Volume 322 (2002) no. 1, pp. 1-18 | DOI | MR | Zbl

[15] de Smit, Bart; Lenstra, Hendrik W. Jr. Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 313-326 | MR | Zbl

[16] Verdier, J.-L. Catégories derivées, P. Deligne, SGA 4.5, Cohomologie étale (Lecture Notes in Mathematics, Vol. 569), Springer-Verlag, Heidelberg, 1970, pp. 262-311

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