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Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$.

Then the exterior product $M \boxtimes N $ is defined as $p^*M \otimes q^*N$ on $X \times Y$.

Is this the same as

$\large{\mathcal O_{X\times Y} \otimes_{p^{-1}\mathcal O_X \otimes_k q^*\mathcal O_Y}(p^{-1}M \otimes_k q^{-1}N)}$ and why?

My main problem is the tensor product within the tensor product. How does one resolve such a thing?

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1 Answer 1

up vote 0 down vote accepted

Use $p^* M = \mathcal{O}_{X \times Y} \otimes_{p^{-1} \mathcal{O}_X} p^{-1} M$ and the similar formula for $q^* N$.

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