Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field \(\Bbbk \) $\mathbb{k}$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When \(A=\Bbbk \langle \Delta \rangle \) $A=\mathbb{k}⟨\mathrm{\Delta }⟩$ is the exterior Stanley–Reisner algebra associated to a finite simplicial complex \(\Delta \) $\mathrm{\Delta }$ , we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.