  Let $S$ be a regular local ring with residue field $k$ and let $M$ be a
finitely generated $S$-module. Suppose that $f_1,\dots ,f_c\in S$ 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.
