Change of rings in deformation theory of modules
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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 $M$.References
- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI 10.1007/BF01390174
- 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.
- 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.
- 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 570778, DOI 10.1090/S0002-9947-1980-0570778-7
- Renée Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603 (1974) (French). MR 345966, DOI 10.24033/asens.1258
- Barbara Fantechi and Marco Manetti, Obstruction calculus for functors of Artin rings. I, J. Algebra 202 (1998), no. 2, 541–576. MR 1617687, DOI 10.1006/jabr.1997.7239
- Gunnar Fløystad, Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme, J. Reine Angew. Math. 439 (1993), 11–44. MR 1219693, DOI 10.1515/crll.1993.439.11
- Runar Ile, Obstructions to deforming modules, Ph.D. thesis, University of Oslo, 2001.
- —, Deformation theory of rank $1$ maximal Cohen-Macaulay modules on hypersurface singularities and the Scandinavian complex, 2002, To appear in Compositio Math.
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680, DOI 10.1007/BFb0059052
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680, DOI 10.1007/BFb0059052
- Akira Ishii, Versal deformation of reflexive modules over rational double points, Math. Ann. 317 (2000), no. 2, 239–262. MR 1764236, DOI 10.1007/s002089900092
- Yujiro Kawamata, Unobstructed deformations. II, J. Algebraic Geom. 4 (1995), no. 2, 277–279. MR 1311350
- Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture Notes in Mathematics, vol. 754, Springer, Berlin, 1979. MR 551624, DOI 10.1007/BFb0065055
- O. A. Laudal, Matric Massey products and formal moduli. I, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 218–240. MR 846451, DOI 10.1007/BFb0075462
- B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
- —, 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.
- V. P. Palamodov, Deformations of complex spaces, Uspehi Mat. Nauk 31 (1976), no. 3(189), 129–194 (Russian). MR 0508121
- Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 217093, DOI 10.1090/S0002-9947-1968-0217093-3
- Arvid Siqveland, Global matric Massey products and the compactified Jacobian of the $E_6$-singularity, J. Algebra 241 (2001), no. 1, 259–291. MR 1838853, DOI 10.1006/jabr.2001.8758
- Arvid Siqveland, The method of computing formal moduli, J. Algebra 241 (2001), no. 1, 292–327. MR 1838854, DOI 10.1006/jabr.2001.8757
- Hartwig von Essen, Nonflat deformations of modules and isolated singularities, Math. Ann. 287 (1990), no. 3, 413–427. MR 1060684, DOI 10.1007/BF01446903
- Charles H. Walter, Some examples of obstructed curves in $\textbf {P}^3$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324–340. MR 1201393, DOI 10.1017/CBO9780511662652.024
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
- 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.
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2084403