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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!

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    $\begingroup$ It's not a standard notation and it can mean many different things depending on where you found it. $\endgroup$ Jun 8, 2014 at 13:21
  • $\begingroup$ 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 $\otimes$. $\endgroup$
    – Dan Rust
    Jun 8, 2014 at 13:36
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    $\begingroup$ Can you supply a reference? Where did you find it? Book? lecturenotes?... $\endgroup$ Jun 8, 2014 at 18:10

2 Answers 2

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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.

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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).

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  • $\begingroup$ 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? $\endgroup$
    – Filippo
    Apr 28, 2021 at 9:29
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    $\begingroup$ @Filippo: As a set, that's fine. Keep in mind, though, that a fiber bundle is more than just a set: it also means its trivializations, its topological structure etc. In order to obtain these "invisible" structures, those pull-backs are very useful. $\endgroup$
    – Alex M.
    Apr 28, 2021 at 12:48
  • $\begingroup$ I understand, thank you very much! $\endgroup$
    – Filippo
    Apr 28, 2021 at 12:52

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