Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A is defined as the supremum of projective dimensions of modules having finite projective dimension. Are there exact formulas for the Gorenstein dimension and finitistic dimension of $A \otimes_K B$? I found that: http://at.yorku.ca/c/a/f/e/49.htm but no related paper at the moment. Im looking for a reference to quote if possible.

## 1 Answer

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For the finitistic dimension, this is Theorem 16 in

*Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky*, MR 98774 **On the dimension of modules and algebras. VIII. Dimension of tensor products**, *Nagoya Math. J.* **12** (1957), 71--93.