# What is box tensor product?

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!

• It's not a standard notation and it can mean many different things depending on where you found it. Jun 8, 2014 at 13:21
• If you have two different tensor products (say from two different monoidal categories) then one might use the box product $\boxtimes$ to distinguish it from the usual circular product $\otimes$. Jun 8, 2014 at 13:36
• Can you supply a reference? Where did you find it? Book? lecturenotes?... Jun 8, 2014 at 18:10

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.
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).
• Nice answer! I had the impression that the author of a book I'm reading is assuming $E\boxtimes F=\displaystyle\bigcup_{(x,y)\in X\times Y}E_x\otimes F_y$ - is that wrong in general? Apr 28, 2021 at 9:29