A Poincaré–Birkhoff–Witt (PBW) basis is a particularly nice basis of a quadratic algebra that can be used to prove that it is Koszul (see Priddy's 1970 paper "Koszul resolutions", Trans. Amer. Math. Soc.).

I found in the 2005 book Quadratic Algebras by Polishchuk and Positselski a notion of commutative PBW basis, that can be used to prove that quadratic commutative algebras (i.e. polynomial algebras modded out by quadratic relations) are Koszul, a result due to Kempf.

Is there a version of this written somewhere for graded commutative quadratic algebras? Here I mean algebras with a presentation $A = S(V) / (R)$ where $V$ is a graded vector space (let's say positively graded), $S(V)$ is the free graded commutative algebra on $V$, and $R \subset S^{(2)}(V)$ is a set of quadratic relations?

I guess one just needs to introduce relations of the form $x_i x_j = \pm x_j x_i$ instead of simply $x_i x_j = x_j x_i$. Theorem 8.1 in Chapter 4 of Quadratic Algebras already allows for general relations of the form $x_i x_j = q_{ij} x_j x_i$ for scalars $q_{ij}$, and in Section 10 it is proved that a $\mathbb{Z}$-algebra (generalization of graded associative algebras) which admits a PBW basis is Koszul. But it'd be nice to have something precise to reference...


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