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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.

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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).

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  • $\begingroup$ Thank you! This paper is very readable, and was exactly the type of thing I was looking for. $\endgroup$
    – Rellek
    Commented Apr 5, 2021 at 13:49
  • $\begingroup$ @Rellek: you might be interested in Question 4.5 from that paper, even the case $K=m$ is not known. $\endgroup$
    – Hailong Dao
    Commented Apr 5, 2021 at 14:21
  • $\begingroup$ Is question 4.5 known to be false for an arbitrary regular local ring of dimension 3, or is this also open? $\endgroup$
    – Rellek
    Commented Apr 14, 2021 at 14:10
  • $\begingroup$ @Rellek: Also open, as far as I know. $\endgroup$
    – Hailong Dao
    Commented Apr 14, 2021 at 14:16

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