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.



Is Koszul homology of a monomial ideal always generated by the "obvious" things?
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"](https://arxiv.org/abs/1902.00806) (but probably known to experts before).


