What is box tensor product?

Question

I have seen the symbol $\boxtimes$ (LaTeX: \boxtimes) in a few places, but I couldn't find a good reference for that. What does it mean and how is it different from the usual tensor product? Thanks!

Answer 1

Let $R,S$ be $k$-algebras. If $M$ is an $R$-module and $N$ is an $S$-module, then the $k$-module $M|_k \otimes_k N|_k$ carries the structure of an $R \otimes_k S$-module. This is sometimes denoted by $M \boxtimes_k N$.

More generally, if $X \to S$ and $Y \to S$ are morphisms of schemes, $M$ is an $\mathcal{O}_X$-module and $N$ is an $\mathcal{O}_Y$-module, then the external tensor product $M \boxtimes_{\mathcal{O}_S} N$ is by definition the $\mathcal{O}_{X \times_S \, Y}$-module $\mathrm{pr}_X^*(M) \otimes_{\mathcal{O}_{X \times_S \, Y}} \mathrm{pr}_Y^*(N)$.

Depending on the context, $\boxtimes$ can also denote something different.

Answer 2

If $E \to X$ and $F \to Y$ are vector bundles, and if $p_X : X \times Y \to X$ and $p_Y : X \times Y \to Y$ are the natural projections, then $E \boxtimes F$ is the vector bundle $p_X ^* E \otimes p_Y ^* F$ (the factors in the tensor product being the natural pull-back bundles).