Global dimension of tensor product of algebras

Question

I am looking for a reference or proof of the following fact:

If $K$ is algebraically closed field and $A$ and $B$ are finite dimensional $K$-algebras then $\text{gl.dim}(A\otimes_KB)= \text{gl.dim}(A)+ \text{gl.dim}(B)$.

Answer 1

It seems surprisingly difficult to find an explicit reference, but it follows from the much more general Theorem 16 in

Auslander, Maurice, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9, 67-77 (1955). ZBL0067.27103.

which states that $\text{gl.dim}(A\otimes_KB)= \text{gl.dim}(A)+ \text{gl.dim}(B)$ if $A$ and $B$ are semiprimary algebras over K (i.e., have nilpotent Jacobson radical with the quotient by the Jacobson radical being semisimple) and $(A/\text{rad}A)\otimes_K (B/\text{rad}B)$ is semisimple (which is true for finite dimensional algebras over an algebraically closed, or even perfect, field).

The proof there is essentially the same (a bit more complicated, but not immensely so) as the proof I would give for finite dimensional algebras.