# Is Koszul homology of a monomial ideal always generated by the "obvious" things?

Let $$R = k[x_1 , \dots , x_n]$$ be a polynomial ring over a field and $$I$$ a monomial ideal in $$R$$. Then, is it true that the Koszul homology of $$R/I$$ is always generated by elements of the form $$r e_{i_1} \wedge \cdots \wedge e_{i_k} \quad \textrm{where} \ x_{i_\ell} r \in I \ \textrm{for all} \ 1 \leq \ell \leq k ?$$ These elements are certainly contained in the Koszul homology. Moreover, this does constitute a generating set, for example, for stable ideals, since one can show that the Koszul homology is actually minimally generated by a subset of elements of the above form. I have computed a fair amount of examples and it seems true more generally that this is a generating set.

I'm not sure if this is well-known or perhaps false, and any help or references for this would be greatly appreciated.

This holds for $$n\leq 3$$ but may fail for $$n=4$$ and higher. See Proposition 2.6 and Example 2.9 in the paper "On monomial Golod ideals" (but probably known to experts before).
• @Rellek: you might be interested in Question 4.5 from that paper, even the case $K=m$ is not known. Apr 5, 2021 at 14:21