On Gorenstein Homological Dimensions Over the Tensor Product of Algebras

Let k be a commutative ring, and let \(\Lambda \) $\mathrm{\Lambda }$ and \(\Gamma \) $\mathrm{\Gamma }$ be two k-algebras. In this paper, we give upper and lower bounds of Gorenstein global dimension and Gorenstein weak dimension over the tensor product \(\Lambda \otimes _k \Gamma \) $\mathrm{\Lambda }{\otimes }_k\mathrm{\Gamma }$ . As its applications, the Gorenstein dimensions of several special algebras such as group algebras, matrix algebras, triangular matrix algebras and polynomial algebras can be computed. Moreover, we compare the Gorenstein global dimension of the enveloping algebra \(\Lambda ^e\) ${\mathrm{\Lambda }}^e$ of \(\Lambda \) $\mathrm{\Lambda }$ with the Gorenstein projective dimension of \(\Lambda \) $\mathrm{\Lambda }$ . Some well-known results are also extended.