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Deformation theory of rank one maximal Cohen–Macaulay modules on hypersurface singularities and the Scandinavian complex

Published online by Cambridge University Press:  04 December 2007

Runar Ile
Affiliation:
Solheimgata 2, 0267 Oslo, Norwayile@math.ntnu.no Norwegian University of Science and Technology, Department of Mathematical Sciences, 7491 Trondheim, Norway
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Abstract

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We show that the deformation functor of a maximal Cohen–Macaulay module M = coker($\phi$) over the hypersurface singularity det($\phi$) is given by deformations of the presenting matrix which keep the determinant constant. A simplified expression for an edge map in the canonical five-term exact sequence to a change of rings spectral sequence is obtained, including the tangent and obstruction spaces (H1 and H2). We relate the edge map to the Scandinavian complex$\mathcal{S}$ of $\phi$ which yields relations between the homology of $\mathcal{S}$ and Hi for i = 1, 2. This gives (infinitesimal) rigidity and non-rigidity results and a dimension estimate for the formally (mini-)versal formal hull H of the deformation functor.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2004