Higher Koszul algebras and the $\mathbf{\text{(Fg)}}$-condition

Determining when a finite dimensional algebra satisfies the finiteness property known as the $(\mathbf{\text{Fg}})$-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 $(\mathbf{\text{Fg}})$-condition and higher homological algebra, which significantly extends the classes of algebras for which it is known whether the $(\mathbf{\text{Fg}})$-condition is satisfied. In particular, we show that the condition holds for an important class of algebras arising from consistent dimer models.