Deformations and derived categories  [ Déformations et catégories dérivées ]
Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2285-2359.

Dans cet article, nous généralisons la théorie des déformations de représentations d’un groupe profini dévélopée par Schlessinger et Mazur aux déformations d’objets d’une catégorie dérivée de complexes limités de modules pseudocompacts. Nous prouvons que de tels objets ont des déformations verselles selon certaines conditions naturelles, et nous déterminons une condition suffisante pour que ces déformations soient universelles. De plus, nous considérons des applications en des déformations de classes de cohomologie Galoisienne et de la hypercohomologie étale de μ p sur certaines CM courbes elliptiques affines.

In this paper we generalize the deformation theory of representations of a profinite group developed by Schlessinger and Mazur to deformations of objects of the derived category of bounded complexes of pseudocompact modules for such a group. We show that such objects have versal deformations under certain natural conditions, and we find a sufficient condition for these versal deformations to be universal. Moreover, we consider applications to deforming Galois cohomology classes and the étale hypercohomology of μ p on certain affine CM ellitpic curves.

DOI : https://doi.org/10.5802/aif.2162
Classification : 20CXX,  18E30,  18G40,  11F80
Mots clés: déformations verselles et universelles, catégories derivées, cohomologie Galoisienne, hypercohomologie, CM courbes elliptiques
@article{AIF_2005__55_7_2285_0,
     author = {Bleher, Frauke M. and Chinburg, Ted},
     title = {Deformations and derived categories},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     pages = {2285-2359},
     doi = {10.5802/aif.2162},
     mrnumber = {2207385},
     zbl = {05015290},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2005__55_7_2285_0/}
}
Bleher, Frauke M.; Chinburg, Ted. Deformations and derived categories. Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2285-2359. doi : 10.5802/aif.2162. https://aif.centre-mersenne.org/item/AIF_2005__55_7_2285_0/

[1] Alperin, J.L. Local Representation Theory, Cambridge Studies in Advanced Mathematics, Tome 11, Cambridge University Press, Cambridge, 1986 | MR 860771 | Zbl 0593.20003

[2] Bleher, F.M.; Chinburg, T. Universal deformation rings and cyclic blocks, Math. Ann., Tome 318 (2000), pp. 805-836 | Article | MR 1802512 | Zbl 0971.20004

[3] Bleher, F.M.; Chinburg, T. Applications of versal deformations to Galois theory, Comment. Math. Helv., Tome 78 (2003), pp. 45-64 | MR 1966751 | Zbl 1034.20005

[4] Bleher, F.M.; Chinburg, T. Deformations and derived categories, C. R. Acad. Sci. Paris Ser. I Math., Tome 334 (2002), pp. 97-100 | MR 1885087 | Zbl 01744359

[5] Bleher, F.M. Universal deformation rings and Klein four defect groups, Trans. Amer. Math. Soc., Tome 354-10 (2002), pp. 3893-3906 | Article | MR 1926858 | Zbl 1047.20006

[6] Boston, N.; Ullom, S.V. Representations related to CM elliptic curves, Math. Proc. Camb. Phil. Soc., Tome 113 (1993), pp. 71-85 | Article | MR 1188818 | Zbl 0795.14017

[7] Breuil, C.; Conrad, B.; Taylor, F. Diamond & R. On the modularity of elliptic curves over $\mathbb{Q}$: Wild $3$-adic exercises, J. Amer. Math. Soc., Tome 14 (2001), pp. 843-939 | Article | MR 1839918 | Zbl 0982.11033

[8] Broué, M. Isométries parfaites, types de blocs, catégories dérivées, Astérisque, Tome 181-182 (1990), pp. 61-92 | MR 1051243 | Zbl 0704.20010

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

[10] Cornell, G.; (eds.), J.H. Silverman & G. Stevens Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, Berlin-Heidelberg-New York, 1997 | MR 1638473

[11] Deligne, P. Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math., Tome 35 (1968), pp. 259-278 | Article | Numdam | MR 244265 | Zbl 0159.22501

[12] Smit, B. de; Jr., H.W. Lenstra Explicit Constructions of Universal Deformation Rings, Modular Forms and Fermat's Last Theorem' (Boston, 1995) (1997), pp. 313-326 | MR 1638482 | Zbl 0907.13010

[13] Gabriel, P. Des catégories abéliennes, Bull. Soc. Math. France, Tome 90 (1962), pp. 323-448 | Numdam | MR 232821 | Zbl 0201.35602

[14] Gabriel, P. A. Grothendieck, SGA 3 (with M. Demazure), `Schémas en groupes I, II, III', Étude infinitesimale des schémas en groupes, (Lecture Notes in Math.) Tome 151 (1970), pp. 476-562

[15] Griffiths, P.; Harris, J. Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978 | MR 507725 | Zbl 0408.14001

[16] Grothendieck, A. SGA 4 (with M. Artin and J.-L. Verdier), Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math., Tome 269, 270, 305, Springer-Verlag, 1972-1973

[17] Hartshorne, R. Residues and Duality, Lecture Notes in Math., Tome 20, Springer-Verlag, Berlin-Heidelberg-New York, 1966 | MR 222093 | Zbl 0212.26101

[18] Illusie, L. Complexe cotangent et déformations, I, II, Lecture Notes in Math., Tome 239, 283, Springer-Verlag, Berlin-New York, 1971, 1972 | MR 491680 | Zbl 0238.13017

[19] Mazur, B. `Galois groups over $\mathbb{Q}$' (Berkeley, CA, 1987), Deforming Galois representations (1989), pp. 385-437 | MR 1012172 | Zbl 0714.11076

[20] Mazur, B. `Modular Forms and Fermat's Last Theorem' (Boston, MA, 1995), Deformation theory of Galois representations (1997), pp. 243-311 | MR 1638481 | Zbl 0901.11015

[21] Milne, J.S. Étale cohomology, Princeton Univ. Press,, Princeton, 1980 | MR 559531 | Zbl 0433.14012

[22] Milne, J.S. Arithmetic Duality Theorems, Perspectives in Math. 1, Academic Press, Boston, 1986 | MR 881804 | Zbl 0613.14019

[23] Ribes, L.; Zalesskii, P. Profinite groups, Ergebnisse der Math. und ihrer Grenzgebiete 40, Springer-Verlag, Berlin-Heidelberg-New York, 2000 | MR 1775104 | Zbl 0949.20017

[24] Rickard, J. The abelian defect group conjecture, Tome II (1998), pp. 121-128 | MR 1648062 | Zbl 0919.20007

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

[26] Shimura, G. Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, 1971 | MR 314766 | Zbl 0221.10029

[27] Taylor, R.; Wiles, A. Ring-theoretic properties of certain Hecke algebras, Ann. of Math., Tome 141 (1995), pp. 553-572 | Article | MR 1333036 | Zbl 0823.11030

[28] Wiles, A. Modular elliptic curves and Fermat's last theorem, Ann. of Math., Tome 141 (1995), pp. 443-551 | Article | MR 1333035 | Zbl 0823.11029