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Frauke M. Bleher; Ted Chinburg
Deformations and derived categories
(Déformations et catégories dérivées)
Annales de l'institut Fourier, 55 no. 7 (2005), p. 2285-2359
Article: subscription required | Reviews MR 2207385 | Zbl 05015290
Class. Math.: 20CXX, 18E30, 18G40, 11F80
Keywords: Versal and universal deformations, derived categories, hypercohomology, CM elliptic curves
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] J.L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Cambridge University Press, 1986 MR 860771 | Zbl 0593.20003
[2] F.M. Bleher & T. Chinburg, “Universal deformation rings and cyclic blocks”, Math. Ann. 318 (2000), p. 805-836 MR 1802512 | Zbl 0971.20004
[3] F.M. Bleher & T. Chinburg, “Applications of versal deformations to Galois theory”, Comment. Math. Helv. 78 (2003), p. 45-64 MR 1966751 | Zbl 1034.20005
[4] F.M. Bleher & T. Chinburg, “Deformations and derived categories”, C. R. Acad. Sci. Paris Ser. I Math. 334 (2002), p. 97-100 MR 1885087 | Zbl 01744359
[5] F.M. Bleher, “Universal deformation rings and Klein four defect groups”, Trans. Amer. Math. Soc. 354-10 (2002), p. 3893-3906 MR 1926858 | Zbl 1047.20006
[6] N. Boston & S.V. Ullom, “Representations related to CM elliptic curves”, Math. Proc. Camb. Phil. Soc. 113 (1993), p. 71-85 MR 1188818 | Zbl 0795.14017
[7] C. Breuil, B. Conrad & F. Diamond & R. Taylor, “On the modularity of elliptic curves over $\mathbb{Q}$: Wild $3$-adic exercises”, J. Amer. Math. Soc. 14 (2001), p. 843-939 MR 1839918 | Zbl 0982.11033
[8] M. Broué, “Isométries parfaites, types de blocs, catégories dérivées”, Astérisque 181-182 (1990), p. 61-92 MR 1051243 | Zbl 0704.20010
[9] A. Brumer, “Pseudocompact algebras, profinite groups and class formations”, J. Algebra 4 (1966), p. 442-470 MR 202790 | Zbl 0146.04702
[10] G. Cornell & J.H. Silverman & G. Stevens (eds.), Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, 1997 MR 1638473
[11] P. Deligne, “Théorème de Lefschetz et critères de dégénérescence de suites spectrales”, Inst. Hautes Études Sci. Publ. Math. 35 (1968), p. 259-278
Numdam | MR 244265 | Zbl 0159.22501
[12] B. de Smit & H.W. Lenstra Jr., Explicit Constructions of Universal Deformation Rings, Springer-Verlag, 1997, p. 313-326 MR 1638482 | Zbl 0907.13010
[13] P. Gabriel, “Des catégories abéliennes”, Bull. Soc. Math. France 90 (1962), p. 323-448
Numdam | MR 232821 | Zbl 0201.35602
[14] P. Gabriel, Étude infinitesimale des schémas en groupes,, Lecture Notes in Math., Springer-Verlag, 1970, p. 476-562
[15] P. Griffiths & J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978 MR 507725 | Zbl 0408.14001
[16] A. Grothendieck, 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] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer-Verlag, 1966 MR 222093 | Zbl 0212.26101
[18] L. Illusie, Complexe cotangent et déformations, I, II, Lecture Notes in Math. 239, 283, Springer-Verlag, 1971, 1972 MR 491680 | Zbl 0238.13017
[19] B. Mazur, Deforming Galois representations, Springer-Verlag, 1989, p. 385-437 MR 1012172 | Zbl 0714.11076
[20] B. Mazur, Deformation theory of Galois representations, Springer-Verlag, 1997, p. 243-311 MR 1638481 | Zbl 0901.11015
[21] J.S. Milne, Étale cohomology, Princeton Univ. Press,, 1980 MR 559531 | Zbl 0433.14012
[22] J.S. Milne, Arithmetic Duality Theorems, Perspectives in Math. 1, Academic Press, 1986 MR 881804 | Zbl 0613.14019
[23] L. Ribes & P. Zalesskii, Profinite groups, Ergebnisse der Math. und ihrer Grenzgebiete 40, Springer-Verlag, 2000 MR 1775104 | Zbl 0949.20017
[24] J. Rickard, The abelian defect group conjecture, in Proceedings of the International Congress of Mathematicians, (Berlin, 1998), Doc. Math., Extra Volume, 1998, p. 121-128 MR 1648062 | Zbl 0919.20007
[25] M. Schlessinger, “Functors of Artin Rings”, Trans. Amer. Math. Soc. 130 (1968), p. 208-222 MR 217093 | Zbl 0167.49503
[26] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971 MR 314766 | Zbl 0221.10029
[27] R. Taylor & A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Ann. of Math. 141 (1995), p. 553-572 MR 1333036 | Zbl 0823.11030
[28] A. Wiles, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. 141 (1995), p. 443-551 MR 1333035 | Zbl 0823.11029
Frauke M. Bleher; Ted Chinburg
Deformations and derived categories
(Déformations et catégories dérivées)
Annales de l'institut Fourier, 55 no. 7 (2005), p. 2285-2359
Article: subscription required | Reviews MR 2207385 | Zbl 05015290
Class. Math.: 20CXX, 18E30, 18G40, 11F80
Keywords: Versal and universal deformations, derived categories, hypercohomology, CM elliptic curves
Résumé - Abstract
Bibliography
[2] F.M. Bleher & T. Chinburg, “Universal deformation rings and cyclic blocks”, Math. Ann. 318 (2000), p. 805-836 MR 1802512 | Zbl 0971.20004
[3] F.M. Bleher & T. Chinburg, “Applications of versal deformations to Galois theory”, Comment. Math. Helv. 78 (2003), p. 45-64 MR 1966751 | Zbl 1034.20005
[4] F.M. Bleher & T. Chinburg, “Deformations and derived categories”, C. R. Acad. Sci. Paris Ser. I Math. 334 (2002), p. 97-100 MR 1885087 | Zbl 01744359
[5] F.M. Bleher, “Universal deformation rings and Klein four defect groups”, Trans. Amer. Math. Soc. 354-10 (2002), p. 3893-3906 MR 1926858 | Zbl 1047.20006
[6] N. Boston & S.V. Ullom, “Representations related to CM elliptic curves”, Math. Proc. Camb. Phil. Soc. 113 (1993), p. 71-85 MR 1188818 | Zbl 0795.14017
[7] C. Breuil, B. Conrad & F. Diamond & R. Taylor, “On the modularity of elliptic curves over $\mathbb{Q}$: Wild $3$-adic exercises”, J. Amer. Math. Soc. 14 (2001), p. 843-939 MR 1839918 | Zbl 0982.11033
[8] M. Broué, “Isométries parfaites, types de blocs, catégories dérivées”, Astérisque 181-182 (1990), p. 61-92 MR 1051243 | Zbl 0704.20010
[9] A. Brumer, “Pseudocompact algebras, profinite groups and class formations”, J. Algebra 4 (1966), p. 442-470 MR 202790 | Zbl 0146.04702
[10] G. Cornell & J.H. Silverman & G. Stevens (eds.), Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, 1997 MR 1638473
[11] P. Deligne, “Théorème de Lefschetz et critères de dégénérescence de suites spectrales”, Inst. Hautes Études Sci. Publ. Math. 35 (1968), p. 259-278
Numdam | MR 244265 | Zbl 0159.22501
[12] B. de Smit & H.W. Lenstra Jr., Explicit Constructions of Universal Deformation Rings, Springer-Verlag, 1997, p. 313-326 MR 1638482 | Zbl 0907.13010
[13] P. Gabriel, “Des catégories abéliennes”, Bull. Soc. Math. France 90 (1962), p. 323-448
Numdam | MR 232821 | Zbl 0201.35602
[14] P. Gabriel, Étude infinitesimale des schémas en groupes,, Lecture Notes in Math., Springer-Verlag, 1970, p. 476-562
[15] P. Griffiths & J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978 MR 507725 | Zbl 0408.14001
[16] A. Grothendieck, 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] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer-Verlag, 1966 MR 222093 | Zbl 0212.26101
[18] L. Illusie, Complexe cotangent et déformations, I, II, Lecture Notes in Math. 239, 283, Springer-Verlag, 1971, 1972 MR 491680 | Zbl 0238.13017
[19] B. Mazur, Deforming Galois representations, Springer-Verlag, 1989, p. 385-437 MR 1012172 | Zbl 0714.11076
[20] B. Mazur, Deformation theory of Galois representations, Springer-Verlag, 1997, p. 243-311 MR 1638481 | Zbl 0901.11015
[21] J.S. Milne, Étale cohomology, Princeton Univ. Press,, 1980 MR 559531 | Zbl 0433.14012
[22] J.S. Milne, Arithmetic Duality Theorems, Perspectives in Math. 1, Academic Press, 1986 MR 881804 | Zbl 0613.14019
[23] L. Ribes & P. Zalesskii, Profinite groups, Ergebnisse der Math. und ihrer Grenzgebiete 40, Springer-Verlag, 2000 MR 1775104 | Zbl 0949.20017
[24] J. Rickard, The abelian defect group conjecture, in Proceedings of the International Congress of Mathematicians, (Berlin, 1998), Doc. Math., Extra Volume, 1998, p. 121-128 MR 1648062 | Zbl 0919.20007
[25] M. Schlessinger, “Functors of Artin Rings”, Trans. Amer. Math. Soc. 130 (1968), p. 208-222 MR 217093 | Zbl 0167.49503
[26] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971 MR 314766 | Zbl 0221.10029
[27] R. Taylor & A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Ann. of Math. 141 (1995), p. 553-572 MR 1333036 | Zbl 0823.11030
[28] A. Wiles, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. 141 (1995), p. 443-551 MR 1333035 | Zbl 0823.11029
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