Layered resolutions of Cohen–Macaulay modules

  • David Eisenbud

    University of California, Berkeley and MSRI, USA
  • Irena Peeva

    Cornell University, Ithaca, USA
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Let SS be a Gorenstein local ring and suppose that MM is a finitely generated Cohen–Macaulay SS-module of codimension cc. Given a regular sequence f1,,fcf_1, \ldots, f_c in the annihilator of MM we set R=S/(f1,,fc)R = S/(f_1, \ldots, f_c) and construct layered SS-free and RR-free resolutions of MM. The construction inductively reduces the problem to the case of a Cohen–Macaulay module of codimension c1c-1 and leads to the inductive construction of a higher matrix factorization for MM. In the case where MM is a sufficiently high RR-syzygy of some module of finite projective dimension over SS, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP]. Our results provide a characterization of all MCM modules over a complete intersection in terms of higher matrix factorizations.

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David Eisenbud, Irena Peeva, Layered resolutions of Cohen–Macaulay modules. J. Eur. Math. Soc. 23 (2021), no. 3, pp. 845–867

DOI 10.4171/JEMS/1024