In his paper Gaussian maps and plethysm, Manivel uses the term "total exterior product" of line bundles, appearing for the first time on page 3. Given two (projective) varieties $V_1$ and $V_2$ and sheaves of $\mathcal{O}_{V_i}$-modules $\mathcal{F}_i$ (possibly line bundles), there should be some way to construct a sheaf $\mathcal{F}$ on $V_1\times V_2$ which we call the total exterior product of $\mathcal{F}_1$ and $\mathcal{F}_2$. However, I was unable to find a formal definition anywhere in the literature. So, my question is, what is the total exterior product?
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2 Answers
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I believe what is meant by this is the usual external tensor product: if $p_1$ and $p_2$ are projections from $V_1\times V_2$ to $V_1$ and $V_2$ respectively, the external tensor product $\mathcal{F}_1\boxtimes\mathcal{F}_2$ is nothing but $p_1^*(\mathcal{F}_1)\otimes p_2^*(\mathcal{F}_2)$.
answered May 3, 2012 at 15:08
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$\begingroup$ Oh! Well of course, that seems perfectly reasonable. I was thinking in the completely wrong direction. Thanks! $\endgroup$ Commented May 3, 2012 at 15:42
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The word "total" may refer to the situation when you are dealing with complexes of sheaves (then the exterior tensor product is a bicomplex and you take the total complex associated with it). If these are line bundles, it is likely that "total" does not mean much.
answered May 7, 2012 at 4:20