In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{Corr}(T) := \textrm{Pic}(T \times T)/ (\pi\_1^\*\textrm{Pic}(T) \oplus\pi\_2^\*\textrm{Pic}(T))$$ Why is this true? A reference would suffice.


I can see why this is true for an Abelian variety $A = T$. It would also be enough to show that there is no $2$-torsion.



Reference for torsion-freeness of the group of correspondences on a smooth projective variety
It's not just torsion-free, we can actually compute it in terms of the Picard and Albanese variety of $T$, by the following classical result:


**Lemma.** *If $X$ and $Y$ are smooth projective varieties over a field $k$, then there is a canonical short exact sequence of group schemes*
$$0 \to \mathbf{Pic}\_X \times \mathbf{Pic}\_Y \to \mathbf{Pic}\_{X \times Y} \to \mathbf{Hom}\big(\mathbf{Alb}^1\_X,(\mathbf{Pic}^0\_Y)^\text{red}\big) \to 0.$$
Here, $(\mathbf{Pic}^0\_X)^\text{red}$ is an abelian variety; see for instance [FGA Explained, Thm. 9.5.4]. Its dual is denoted $\mathbf{Alb}^0\_X$, and without choosing a base point this gives a torsor $\mathbf{Alb}^1\_X$ under $\mathbf{Alb}^0\_X$ together with a morphism $X \to \mathbf{Alb}^1\_X$ that is initial in the category of maps from $X$ to torsors under abelian varieties. Finally, $\mathbf{Hom}$ denotes the (ind-étale) group scheme of morphisms of abelian varieties (or probably morphisms of pairs of an abelian variety with a torsor under it, up to translation ― once you choose a base point on $X$, this should be the same thing).


Torsion-freeness then follows since $\mathbf{Hom}$ schemes between abelian varieties are torsion-free.


For a proof of the lemma, see for instance my dissertation [vDdB, §4.4], although I claim no originality here! (That said, I also didn't find this precise statement in the literature...)


A different argument goes by choosing base points $x \in X(k)$ and $y \in Y(k)$ (first enlarge the field if necessary and worry about descent later). These give a section $\mathbf{Pic}\_{X \times Y} \to \mathbf{Pic}\_X \times \mathbf{Pic}\_Y$. To compute the kernel, consider a line bundle $\mathscr L$ on $X \times Y \times T$ for some $k$-scheme $T$ that is trivial along $x \times Y \times T$ and $X \times y \times T$. The second trivialisation gives a morphism $X \times T \to \mathbf{Pic}\_Y$ via the rigidified Picard functor, and the first says that the image lands in $\mathbf{Pic}\_Y^0$. This gives a morphism of $T$-schemes $X \times T \to \mathbf{Pic}\_Y^0 \times T$, which by the universal property of $\mathbf{Alb}^1\_X$ corresponds to a $T$-point of $\mathbf{Hom}(\mathbf{Alb}^1\_X,(\mathbf{Pic}^0\_Y)^\text{red})$. Now check that this doesn't depend on the choice of section and hence descends down to $k$.




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**References.**


[FGA Explained] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, A. Vistoli, [*Fundamental algebraic geometry: Grothendieck’s FGA explained*](https://bookstore.ams.org/surv-123-s). Mathematical Surveys and Monographs **123**. American Mathematical Society, 2005. [ZBL1085.14001](https://zbmath.org/?q=an:1085.14001).


For Kleiman's chapter, see also arXiv:[math/0504020](https://arxiv.org/abs/math/0504020)


[vDdB] R. van Dobben de Bruyn, [*Dominating varieties by liftables ones*](https://webspace.science.uu.nl/%7Edobbe012/doc/Dominating%20varieties%20by%20liftable%20ones.pdf). PhD thesis, Columbia University, 2018.


