Finiteness Theorems for Deformations of Complexes  [ Théorèmes de finitude pour déformations de complexes ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 2, p. 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 : https://doi.org/10.5802/aif.2770
Classification:  11F80,  20E18,  18E30
Mots clés: déformations verselles et universelles, catégories dérivées, questions de finitude, groupes fondamentaux modérés
@article{AIF_2013__63_2_573_0,
     author = {Bleher, Frauke M. and Chinburg, Ted},
     title = {Finiteness Theorems for Deformations of Complexes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {2},
     year = {2013},
     pages = {573-612},
     doi = {10.5802/aif.2770},
     mrnumber = {3112842},
     zbl = {06193041},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_2_573_0}
}
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. http://www.numdam.org/item/AIF_2013__63_2_573_0/

[1] Bleher, Frauke M.; Chinburg, Ted Deformations and derived categories, C. R. Math. Acad. Sci. Paris, Tome 334 (2002) no. 2, pp. 97-100 | Article | MR 1885087 | Zbl 1079.11027

[2] Bleher, Frauke M.; Chinburg, Ted Deformations and derived categories, Ann. Inst. Fourier (Grenoble), Tome 55 (2005) no. 7, pp. 2285-2359 | Article | Numdam | MR 2207385 | Zbl 1138.11020

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

[4] Brumer, Armand Pseudocompact algebras, profinite groups and class formations, J. Algebra, Tome 4 (1966), pp. 442-470 | Article | MR 202790 | Zbl 0146.04702

[5] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. France, Tome 90 (1962), pp. 323-348 | Numdam | MR 232821 | Zbl 0201.35602

[6] Gabriel, Pierre Étude infinitesimale des schémas en groupes, A. Grothendieck, SGA 3 (with M. Demazure), Schémas en groupes I, II, III, Springer-Verlag, Heidelberg (Lecture Notes in Mathematics, Vol. 151) (1970), pp. 476-562 | Zbl 0209.24201

[7] Grothendieck, Alexander; Murre, Jacob P. The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 208 (1971) | MR 316453 | Zbl 0216.33001

[8] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York (1977) (Graduate Texts in Mathematics, No. 52) | MR 463157 | Zbl 0531.14001

[9] Kisin, Mark Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), Tome 170 (2009) no. 3, pp. 1085-1180 | Article | MR 2600871 | Zbl 1201.14034

[10] Lazard, Michel Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math. (1965) no. 26, pp. 389-603 | Numdam | MR 209286 | Zbl 0139.02302

[11] Mazur, B. Deforming Galois representations, Galois groups over Q (Berkeley, CA, 1987), Springer, New York (Math. Sci. Res. Inst. Publ.) Tome 16 (1989), pp. 385-437 | MR 1012172 | Zbl 0714.11076

[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 1638481 | Zbl 0901.11015

[13] Schlessinger, Michael Functors of Artin rings, Trans. Amer. Math. Soc., Tome 130 (1968), pp. 208-222 | Article | MR 217093 | Zbl 0167.49503

[14] Schmidt, Alexander Tame coverings of arithmetic schemes, Math. Ann., Tome 322 (2002) no. 1, pp. 1-18 | Article | MR 1883386 | Zbl 1113.14022

[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 1638482 | Zbl 0907.13010

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