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- W2896423083 abstract "Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,dots ,f_cin S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements. The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$. Turning to ${rm Ext}_{R}(M,, k)$ we show that, when $M$ is a high syzygy over $R$, the minimal free resolution of ${rm Ext}_{R}(M,, k)$ as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the $E$-module ${rm Tor}^{S}(M,,k)$. For the proof we introduce emph{higher CI operators}, and give a construction of a (generally non-minimal) resolution of $M$ over $S$ starting from a resolution of $M$ over $R$ and its higher CI operators." @default.
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- W2896423083 date "2018-11-29" @default.
- W2896423083 modified "2023-09-23" @default.
- W2896423083 title "Tor as a module over an exterior algebra" @default.
- W2896423083 doi "https://doi.org/10.4171/jems/853" @default.
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