Higher Koszul algebras and the $\textbf{(Fg)}$-condition
Johanne Haugland, Mads Hustad Sandøy
TL;DR
This work characterizes the $(\mathrm{Fg})$-condition for finite dimensional algebras in the broad setting of $n$-$T$-Koszul algebras by linking $(\mathrm{Fg})$ to the graded center of the $n$-$T$-Koszul dual $\Lambda^!$ and to the finiteness of $\Lambda^!$ over that center. Central to the approach is showing that the dg-endomorphism algebra $\Gamma=\mathrm{REnd}_\Lambda(T)$ is formal, so $\Lambda^! \simeq \mathrm{Ext}^*_{\Lambda}(T,T) \simeq H^*(\Gamma)$ carries a canonical $A_\infty$-structure with $m_d=0$ for $d\ge3$, enabling a clean reduction to center-based finiteness criteria. The main theorem extends Erdmann–Solberg to the $n$-$T$-Koszul setting and yields a concrete criterion to verify $(\mathrm{Fg})$ via $Z_{\mathrm{gr}}(\Lambda^!)$ and module finiteness. The paper further connects to higher Auslander–Reiten theory, showing that for graded symmetric highest-degree-1 algebras with $\Lambda_0$ $n$-representation infinite, $(\mathrm{Fg})$ holds exactly when $\Lambda_0$ is $n$-representation tame, with significant applications to trivial extensions of dimer algebras and related noncommutative geometric contexts.
Abstract
Determining when a finite dimensional algebra satisfies the finiteness property known as the $(\textbf{Fg})$-condition is of fundamental importance in the celebrated and influential theory of support varieties. We give an answer to this question for higher Koszul algebras, generalizing a result by Erdmann and Solberg. This allows us to establish a strong connection between the $(\textbf{Fg})$-condition and higher homological algebra, which significantly extends the classes of algebras for which it is known whether the $(\textbf{Fg})$-condition is satisfied. In particular, we show that the condition holds for an important class of algebras arising from consistent dimer models.