## 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 $M$.

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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
- 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