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.
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$\begingroup$ May be Corollary 3.2 in arxiv.org/abs/1512.02804 will help. Note that all rings in that paper are assumed to be commutative (and noetherian). $\endgroup$– SlavaMCommented Jan 13, 2017 at 20:40
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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.
answered Jan 13, 2017 at 11:37