# Total exterior Product

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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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 $p_1^*(\mathcal{F}_1)\otimes p_2^*(\mathcal{F}_2)$.

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Oh! Well of course, that seems perfectly reasonable. I was thinking in the completely wrong direction. Thanks! –  Jesko Hüttenhain May 3 '12 at 15:42

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.

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