Hochschild cohomology, finiteness conditions and a generalisation of $d$-Koszul algebras

Given a finite-dimensional algebra $\mathrm{\Lambda }$ and $A⩾1$, we construct a new algebra ${\tilde{\mathrm{\Lambda }}}_A$, called the stretched algebra, and relate the homological properties of $\mathrm{\Lambda }$ and ${\tilde{\mathrm{\Lambda }}}_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $\mathrm{\Lambda }$ has (Fg) if and only if ${\tilde{\mathrm{\Lambda }}}_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $\mathrm{\Lambda }$ is a $d$-Koszul algebra for some $d \geqslant 2$.