Rigidification and effective descent of line bundles

For an affine test $T=\mathrm{Spec}R$ the universal-sections statement [F1] gives $\mathrm{\Gamma }(A_T,\mathcal{O})=R$ compatibly with base change; on a finite affine Cech cover of $A_T$ the cohomology complex computing $H^0$ is obtained by tensoring the field cohomology complex, whose $H^0$ is $k$, and hence has $H^0=R$. It follows that the normalization $\mathcal{L}\mapsto \mathcal{L}\otimes p_T^{\ast }{e}_T^{\ast }{\mathcal{L}}^{-1}$ is well defined on isomorphism classes and identifies rigidified classes with $\mathrm{Pic}(A_T)\mathrm{/}{p}_T^{\ast }\mathrm{Pic}(T)$: tensoring by constants is exactly the ambiguity removed by the trivialisation along $e_T$.

A rigidified line bundle has no nontrivial automorphism: an automorphism of $(\mathcal{L},\alpha )$ is a unit of ${\mathcal{O}}_T$ acting on $\mathcal{L}$, and compatibility with the rigidification forces it to restrict to $1$ along $e_T$; since $e_T$ is a section, the unit is $1$. Consequently isomorphism data on overlaps of an fppf cover are unique and therefore automatically satisfy the cocycle condition.

Let $T^{\mathrm{\prime }}\to T$ be an fppf cover and suppose a rigidified line bundle is given on $A_{T^{\mathrm{\prime }}}$ together with an isomorphism of its two pullbacks to $A_{T^{\mathrm{\prime }}{\times }_T{T}^{\mathrm{\prime }}}$; by step 2.1 this isomorphism is unique and satisfies the cocycle condition, so the usual effective descent for invertible modules [F2] produces an invertible sheaf on $A_T$; the rigidification descends because it is a morphism whose pullbacks agree. Hence the rigidified functor is already an fppf sheaf, without invoking representability, and the identification of step 1.1 is compatible with the sheaf structure.

For general $k$ one first verifies the assertions over an algebraic closure using the field-compatibility of global sections in [F2] and then descends the resulting identifications along the faithfully flat field extension; the rigidification data are defined over $k$ and descend by [F2]. No representability of the Picard functor is used anywhere.