In his paper [Gaussian maps and plethysm](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.12.771&rep=rep1&type=pdf), 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?



Total exterior Product
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 $p\_1^\*(\mathcal{F}\_1)\otimes p\_2^\*(\mathcal{F}\_2)$.


