Does this fact remain true for *graded-commutative* rings:

Let $R$ be a commutative ring and $(x_1, \cdots, x_d)$ a sequence of elements in $R$, such that

$R$ is Noetherian;

each $x_i$ is contained in the Jacobson radical of $R$

then the following are equivalent:

the cochain cohomology of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;

the Koszul complex $K(x_1, \cdots, x_d)$ is a free resolution of the quotient ring $R/(x_1, \cdots, x_d)$, hence its cochain cohomology vanishes in all degrees $\leq -1$;

Does this remain true for $R$ a $\mathbb{Z}$-graded-commutative ring?

]]>I gave *Koszul complex* and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of $R/(x_1, \cdots, x_n)$ is that $R$ is Noetherian, the $x_i$ are in the Jacobson radical, and the cohomology in degree -1 vanishes.

Finally I stated the special case of this (here) where $R$ is a formal power series algebra over a field and the elements $x_i$ are formal power series with vanishing constant term.

(I have added the relevant facts as citable numbered examples at *Noetherian ring* and at *Jacobson radical*.)

This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at *A first idea of quantum field theory* (here).