Question about sheaves on projective varieties.

Question

I am new to algebraic geometry, and really can't get idea of this:

For any product $X_{1} \times X_{2}$ of a projective varieties, with projections $p:X_{1}\times X_{2} \rightarrow X_{1}, q:X_{1}\times X_{2}\rightarrow X_{2}$ and let $L_{1},L_{2}$ be sheaves on $X_{1},X_{2}$.

We set $L_{1}\boxtimes L_{2}:= p^{*}L_1 \otimes q^{*}L_2$

i found this in a paper "Betti numbers of graded modules and cohomolgy of vector bundles", page $25$ http://arxiv.org/pdf/0712.1843v3.pdf

can anyone tell me what are $p^{*}$ and $q^{*}$?

Answer 1

The usual notation is $L_1 \boxtimes L_2$. It is called the external tensor product of $L_1$ with $L_2$.

To your question: Every morphism of ringed spaces $f : X \to Y$ induces a pullback functor $f^* : \mathsf{Mod}(Y) \to \mathsf{Mod}(X)$. It is (defined to be) the left adjoint functor to the direct image functor $f_* : \mathsf{Mod}(X) \to \mathsf{Mod}(X)$ (with $(f_* F)(U)=F(f^{-1}(U))$ for $U \subseteq Y$).