Tor as a module over an exterior algebra

  • David Eisenbud

    University of California, Berkeley, USA
  • Irena Peeva

    Cornell University, Ithaca, USA
  • Frank-Olaf Schreyer

    Universität des Saarlandes, Saarbrücken, Germany
Tor as a module over an exterior algebra cover

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Let be a regular local ring with residue field and let be a finitely generated -module. Suppose that is a regular sequence that annihilates , and let be an exterior algebra over generated by elements. The homotopies for the on a free resolution of induce a natural structure of graded -module on Tor. In the case where is a high syzygy over the complete intersection we describe this -module structure in detail, including its minimal free resolution over .

Turning to Ext we show that, when is a high syzygy over , the minimal free resolution of Ext as a module over the ring of CI operators is the Bernstein–Gel’fand–Gel’fand dual of the -module Tor.

For the proof we introduce higher CI operators, and give a construction of a (generally non-minimal) resolution of over starting from a resolution of over and its higher CI operators.

Cite this article

David Eisenbud, Irena Peeva, Frank-Olaf Schreyer, Tor as a module over an exterior algebra. J. Eur. Math. Soc. 21 (2019), no. 3, pp. 873–896

DOI 10.4171/JEMS/853