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- Title
Tor as a module over an exterior algebra.
- Authors
Eisenbud, David; Peeva, Irena; Schreyer, Frank-Olaf
- Abstract
Let S be a regular local ring with residue field k and let M be a finitely generated S-module. Suppose that f1;: ::; fc 2 S is a regular sequence that annihilates M, and let E be an exterior algebra over k generated by c elements. The homotopies for the fi on a free resolution of M induce a natural structure of graded E-module on TorS(M; k). In the case where M is a high syzygy over the complete intersection R := S(f1; . . ., fc) we describe this E-module structure in detail, including its minimal free resolution over E. Turning to ExtR(M, k) we show that, when M is a high syzygy over R, the minimal free resolution of ExtR(M, k) as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the E-module TorS(M, k). For the proof we introduce higher CI operators, and give a construction of a (generally nonminimal) resolution of Mover S starting from a resolution of Mover R and its higher CI operators.
- Subjects
AUSDEHNUNGSLEHRE; MODULES (Algebra); DIFFERENTIAL operators; DISCRETE element method; FREE resolutions (Algebra)
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2019, Vol 21, Issue 3, p873
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/853