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Let $R$ be a regular local ring (or polynomial ring over a field). Let $I$ be an ideal of $R$ and $F_\bullet$ a minimal free resolution of $R/I$. Let $K_\bullet$ denote the Koszul complex resolving the residue field, denoted $k$.

It is a well-known fact that $F_i \otimes k \cong H_i (K_\bullet \otimes R/I)$, which follows by the balancing of Tor, since both of these are isomorphic to $\textrm{Tor}_i (R/I , k)$. However, if one knows the minimal free resolution explicitly, is it possible to use this isomorphism to compute explicit generators of the Koszul homology?

For example, if $I$ and $J$ are two ideals with $\textrm{Tor}_{>0} (R/I , R/J) = 0$, then it seems like one should be able to deduce the Kunneth formula for the Koszul homology of $R/(I+J)$ from the fact that the minimal free resolution of $R/(I+J)$ is the tensor product of the minimal free resolutions for $R/I$ and $R/J$.

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  • $\begingroup$ Choose a basis for $F_i \otimes k$, take representing cycles in $F_*\otimes K_*$, and project to $R/I\otimes K_*$. $\endgroup$ Apr 2, 2021 at 22:10
  • $\begingroup$ How does one choose the representing cycles? $\endgroup$
    – Rellek
    Apr 3, 2021 at 1:33
  • $\begingroup$ Applying the axiom of choice. They have isomorphic homology. $\endgroup$ Apr 3, 2021 at 7:11

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