Given a finite-dimensional algebra $\mathrm{\Lambda }$ and $A\ge 1$, 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 $d\ge 2$.

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Hochschild 上同调、有限性条件和 d-Koszul 代数的推广

给定一个有限维代数$\mathrm{\Lambda }$和$\mathrm{一个}\ge 1$, 我们构造一个新的代数${\tilde{\mathrm{\Lambda }}}_{\mathrm{一个}}$, 称为拉伸代数, 并将其同调性质联系起来$\mathrm{\Lambda }$和${\tilde{\mathrm{\Lambda }}}_{\mathrm{一个}}$. 我们研究 Hochschild 上同调和有限性条件(Fg)，并使用分层理想来证明$\mathrm{\Lambda }$有(Fg)当且仅当${\tilde{\mathrm{\Lambda }}}_{\mathrm{一个}}$有(Fg)。我们还考虑投影分辨率并将我们的结果应用于以下情况$\mathrm{\Lambda }$是一个$d$-Koszul 代数$d\ge 2$.