Suppose I have a list of monomials $f_1, \dots, f_m \in R = K[x_1, \dots, x_n]$.

Is there a nice description of the cohomology of the Koszul complex

\begin{equation} \cdots \rightarrow\bigwedge^{r+1}M^* \rightarrow \bigwedge^{r}M^* \rightarrow \cdots \end{equation}

with $M = R^{\oplus m}$ and differential given by contracting with the vector $(f_1,\dots,f_m)\in M$?

By nice, mean something that might glue together well in a global setting (over a toric variety).