Finite-field descent of the dual and the Poincare bundle

Work over an algebraic closure and write $G$ for the represented full rigidified Picard functor and $B=G^0$ for its identity component. By [F1], $B$ is smooth proper connected of dimension $\dim A$, hence an abelian variety and projective. We establish the algebraically trivial identification directly. The connected components of the locally finite-type scheme $G$ are open and closed, and translation identifies them with cosets of $B$. A rigidified line-bundle family on a connected finite-type parameter scheme gives a morphism to $G$, whose image lies in one connected component; therefore differences of its geometric fibre classes lie in $B$. A chain of such differences has the same property. Conversely, restricting the universal rigidified bundle on $A\times G$ to $A\times B$ gives a connected finite-type family whose identity fibre is trivial and whose fibre at any geometric point $b$ has class $b$, proving that every $B$-class is algebraically trivial. For an arbitrary test scheme $T$, its classifying morphism $T\to G$ factors through the open subscheme $B$ exactly when all geometric fibre classes lie in $B$: the inverse image of the complementary open-and-closed components is empty if it has no geometric point. This argument also retains every nilpotent of $T$, since factorization through an open imposes no reduction. Thus $B$ represents the entire algebraically trivial rigidified subfunctor on all tests. Restricting the universal bundle and normalizing on both axes now gives the Poincare bundle.

For arbitrary $k$, spread $B$, a projective embedding, its group operations and the rigidified Poincare bundle from $\bar{k}$ to a finite extension $K\mathrm{/}k$ by [F2]. The resulting natural transformation to the algebraically trivial rigidified functor is an isomorphism after base change to $\bar{k}$. For an affine $K$-test $T$ and a rigidified algebraically trivial bundle on $A_T$, its unique classifying morphism over $T_{\bar{k}}$ and the isomorphism with the pulled-back Poincare bundle descend to $T_{K^{\mathrm{\prime }}}$ for some finite extension $K^{\mathrm{\prime }}\mathrm{/}K$ inside $\bar{k}$, by the finite-presentation statements of [F2]. Uniqueness is detected after faithful field extension: two classifying morphisms for the same bundle become equal over $\bar{k}$ by [F1], hence were equal already. This also applies on the finite cover's double overlap, including its nilpotents, by tensoring that overlap with $\bar{k}$ over $K$ and using the all-test universal property over $\bar{k}$. Thus the local classifying morphism has equal pullbacks and descends along the finite fppf cover $T_{K^{\mathrm{\prime }}}\to T$ by [F2]; the bundle isomorphism descends by module descent and rigidity. Affine-test extensions glue uniquely, proving the universal property over $K$ on every test scheme.

Consequently $B_K$ carries a canonical descent datum over $K{\otimes }_{k}K$ from the uniqueness of representatives of the same base-changed functor; the datum includes the entire nonreduced tensor algebra and satisfies the cocycle by uniqueness. The scheme $B_K$ is smooth, proper, connected, projective over $K$, hence as a $k$-scheme projective too, since $\mathrm{Spec}K\to \mathrm{Spec}k$ is finite and projective. Every finite descent orbit lies in an affine open of this projective scheme: choose a closed specialization of each of its finitely many points; for a high power of a very ample line, Serre vanishing makes the map to the fibres at those finitely many closed points surjective, and a section nonzero at each of them has an affine nonvanishing locus; nonvanishing at a specialization implies nonvanishing at the original point. The exact finite-field descent lemma [F3] now descends $B_K$, including inseparable $K$.

Module descent gives the Poincare bundle $\mathcal{P}$, compatible morphism descent gives the group law and the rigidifications, and the sheaf isomorphism descends, proving the full universal property over $k$ and its compatibility with field extension.