Deformations and derived categories

Bleher, Frauke M.; Chinburg, Ted

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 ${\mu }_p$ 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 ${\mu }_p$ on certain affine CM ellitpic curves.

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

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

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

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

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

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

[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., Volume 14 (2001), pp. 843-939 | DOI | MR | Zbl

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

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

[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

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

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

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

[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.), Volume 151, Springer-Verlag, Heidelberg (1970), pp. 476-562

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

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

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

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

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

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

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

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

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

[24] Rickard, J. The abelian defect group conjecture, Volume II, Doc. Math., Extra Volume (1998), pp. 121-128 | MR | Zbl

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

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

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

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