The rigidified relative Picard functor and the dual abelian variety

Assume AC for the site-level sheafification construction. Let $f:A\to S$ be an abelian scheme (Abelian schemes over a base) with unit section $e:S\to A$. For an $S$-scheme $T$ write $A_T=A{\times }_{S}T$ and $e_T=e{\times }_S{\mathrm{id}}_T$; a rigidified line bundle on $A_T$ is a pair $(\mathcal{L},\alpha )$ consisting of an invertible sheaf $\mathcal{L}$ on $A_T$ (Invertible sheaves) and a trivialisation $\alpha :{\mathcal{O}}_T\to e_T^{\ast }\mathcal{L}$ along the unit section.

The rigidified relative Picard functor ${\mathcal{P}}_{A\mathrm{/}S,e}$ is the fppf sheaf on the category of $S$-schemes associated (Fppf sheaves of sets and sheafification, Sheafification exists for the fppf site) with the functor $T⟼\{\text{isomorphism classes of rigidified line bundles on }{A}_T\},$ with group law given by tensor product of rigidified line bundles, identity the trivially rigidified structure sheaf and inverse by the dual pairing. An $S$-scheme representing ${\mathcal{P}}_{A\mathrm{/}S,e}$ is called a relative Picard scheme of $A\mathrm{/}S$; its identity component is written ${\mathrm{Pic}}_{A\mathrm{/}S}^0$, and when this exists and is an abelian scheme over $S$ it is called the dual abelian variety $\widehat{A}$ of $A$. A Poincare sheaf is the universal rigidified invertible sheaf $\mathcal{P}$ on $A{\times }_S\widehat{A}$.

The algebraically trivial subfunctor of ${\mathcal{P}}_{A\mathrm{/}S,e}$ consists of the classes whose geometric-fibre restrictions are algebraically equivalent to zero, where algebraic equivalence is the equivalence relation generated by differences of fibres in connected finite-type families of line bundles; this definition of the subfunctor does not presuppose a representing scheme, and once a Picard scheme exists the appropriate identity-component theorem identifies the subfunctor with ${\mathrm{Pic}}_{A\mathrm{/}S}^0$. The notation "degree zero" here refers to this subfunctor and is not numerical degree on $A$ when $\dim A\ge 2$. For $A$ over a field $k$ the dual $\widehat{A}$ means a representing abelian variety for this subfunctor; representability of ${\mathcal{P}}_{A\mathrm{/}S,e}$, the existence of ${\mathrm{Pic}}^0$ and the Poincare sheaf remain claims of the commissioned theorem, and no link from this definition to that theorem is a dependency. The functor is considered with the fppf topology; all test objects are arbitrary $S$-schemes, and flatness and local finite presentation enter only through the definitions of abelian schemes and invertible sheaves used above (Flat morphism of schemes, Locally finite presentation morphisms, Group schemes over a base scheme).

Sheafification is taken on a fixed set-sized big fppf site containing the test schemes in use, as in the cited construction; the convention imposes no finite-type or reducedness restriction on those tests. AC selects representatives and equality refinements in the plus construction, not line bundles or a representing Picard scheme.