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

Abstract

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.