Question

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?

Answer 1

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

Answer 2

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.