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Change of rings in deformation theory of modules


Author: Runar Ile
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4873-4896
MSC (2000): Primary 13D10, 14B10; Secondary 13D07
DOI: https://doi.org/10.1090/S0002-9947-04-03516-0
Published electronically: January 29, 2004
MathSciNet review: 2084403
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Abstract: Given a $B$-module $ M $ and any presentation $ B=A/J $, the obstruction theory of $ M $ as a $ B $-module is determined by the usual obstruction class $ \mathrm{o}_{ \scriptscriptstyle{A}}^{\scriptscriptstyle{}}$ for deforming $ M $ as an $ A $-module and a new obstruction class $ \mathrm{o}_{ \scriptscriptstyle{J}}^{\scriptscriptstyle{}} $. These two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor. We obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the Yoneda complex. Calculation of the deformation functor of $ M $ as a $ B $-module, including the (generalised) Massey products, is thus possible within any $ A $-free $ 2 $-presentation of $ M $.


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  • 1. Michael Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165-189. MR 53:2945
  • 2. Inger Christin Borge, A cohomological approach to the modular isomorphism problem, Preprint in Pure Math. No. 15, Dept. of Math., University of Oslo, August 2002, www.math.uio.no/eprint/pure_math/2002/15-02, submitted to J. Pure Appl. Algebra.
  • 3. Inger Christin Borge and Olav Arnfinn Laudal, The modular isomorphism problem, Preprint in Pure Math. No. 25, Dep. of Math., University of Oslo, July 2003, www.math.uio.no/eprint/pure_math/2003/25-03, submitted to Invent. Math.
  • 4. David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. MR 82d:13013
  • 5. Renée Elkik, Solutions d'équations a coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553-604. MR 49:10692
  • 6. Barbara Fantechi and Marco Manetti, Obstruction calculus for functors of Artin rings, I, J. Algebra 202 (1998), 541-576. MR 99f:14004
  • 7. Gunnar Fløystad, Determining obstructions for space curves, with applications to non-reduced components of the Hilbert scheme, J. Reine Angew. Math. 439 (1993), 11-44. MR 94e:14004
  • 8. Runar Ile, Obstructions to deforming modules, Ph.D. thesis, University of Oslo, 2001.
  • 9. -, Deformation theory of rank $1$ maximal Cohen-Macaulay modules on hypersurface singularities and the Scandinavian complex, 2002, To appear in Compositio Math.
  • 10. Luc Illusie, Complexe cotangent et déformations I, Lecture Notes in Math., no. 239, Springer-Verlag, 1971. MR 58:10886a
  • 11. -, Complexe cotangent et déformations II, Lecture Notes in Math., no. 283, Springer-Verlag, 1972. MR 58:10886b
  • 12. Akira Ishii, Versal deformation of reflexive modules over rational double points, Math. Ann. 317 (2000), 239-262. MR 2001i:14005
  • 13. Yujiro Kawamata, Unobstructed deformations. II, J. Algebraic Geom. 4 (1995), 277-279. MR 96a:14014
  • 14. Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture Notes in Math., no. 754, Springer-Verlag, 1979. MR 82h:14009
  • 15. -, Matric Massey products and formal moduli I, Algebra, Algebraic Topology and Their Interactions, Lecture Notes in Math., no. 683, Springer-Verlag, 1986, pp. 218-240. MR 87k:55023
  • 16. Barry Mazur, Deforming Galois representations, Galois Groups over $\mathbb{Q} $ (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), MSRI Publications, no. 16, Springer-Verlag, 1989, pp. 385-437. MR 90k:11057
  • 17. -, Deformation theory of Galois representations, Modular Forms and Fermat's Last Theorem (Gary Cornell, Joseph H. Silvermann, and Glenn Stevens, eds.), Springer-Verlag, 1997, pp. 243-311.
  • 18. Victor P. Palamodov, Deformations of complex spaces, Russian Math. Surveys 31 (1976), no. 3, 129-194. MR 58:22671
  • 19. Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222. MR 36:184
  • 20. Arvid Siqveland, Global matric Massey products and the compactified Jacobian of the $\mathbf{E_6}$-singularity, J. Algebra 241 (2001), 259-291. MR 2002d:14001
  • 21. -, The method of computing formal moduli, J. Algebra 241 (2001), 292-327. MR 2002g:16050
  • 22. Hartwig von Essen, Nonflat deformations of modules and isolated singularities, Math. Ann. 287 (1990), no. 3, 413-427. MR 91j:14007
  • 23. Charles H. Walter, Some examples of obstructed curves in $\mathbb{P} ^3$, Complex Projective Geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser., no. 179, Cambridge Univ. Press, 1992, pp. 324-340. MR 94a:14033

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Additional Information

Runar Ile
Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
Email: ile@math.uio.no

DOI: https://doi.org/10.1090/S0002-9947-04-03516-0
Keywords: Obstruction, Massey product, spectral sequence, mixed characteristic
Received by editor(s): April 20, 2003
Published electronically: January 29, 2004
Additional Notes: This article is based on parts of the author’s 2001 Ph.D. Thesis at the Department of Mathematics, University of Oslo.
Article copyright: © Copyright 2004 American Mathematical Society

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