# ac.commutative algebra – 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.