# Can we write the isomorphism $F_i \otimes_R k \cong H_i (K_\bullet \otimes R/I)$ explicitly?

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 $$R/J$$.

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