Theta extensions, splitting and isotropic descent

Define the theta functor $\mathrm{\Theta }(N)$ on $k$-schemes by letting $\mathrm{\Theta }(N)(T)$ be the set of pairs $(x,\alpha )$ with $x\in A(T)$ and $\alpha :t_x^{\ast }{N}_T\to N_T$ an isomorphism of line bundles on $A_T$, where $t_x:A_T\to A_T$ is translation. The product $(x,\alpha )(y,\beta )=(x+y,\alpha \circ t_x^{\ast }\beta )$ makes $\mathrm{\Theta }(N)$ a group sheaf for the fppf topology, with unit $(0,\mathrm{id})$; the projection $\pi :\mathrm{\Theta }(N)\to A$, $(x,\alpha )\mapsto x$ is a homomorphism onto $K(N)$, as an fppf sheaf: for $x\in K(N)(T)$ the difference $t_x^{\ast }{N}_T\otimes N_T^{-1}$ is pulled back from a line bundle on $T$, which becomes trivial on an fppf cover, and conversely such a local isomorphism makes its rigidified Picard class zero.

The scalars act on each pair by $\lambda \cdot (x,\alpha )=(x,\lambda \circ \alpha )$, exhibiting ${\mathbf{G}}_m\subseteq \mathrm{\Theta }(N)$ as a central subgroup with ${\pi }^{-1}(0)={\mathbf{G}}_m$; conjugation by $(x,\alpha )$ on ${\mathbf{G}}_m$ is trivial because ${\mathbf{G}}_m$ is commutative, and the commutator $[(x,\alpha ),(y,\beta )]=(x,\alpha )(y,\beta )(x,\alpha {)}^{-1}(y,\beta {)}^{-1}$ lies in ${\pi }^{-1}(0)={\mathbf{G}}_m$ since $K(N)$ is commutative. Its value is the scalar $\alpha t_x^{\ast }\beta (t_y^{\ast }\alpha {)}^{-1}{\beta }^{-1}$ (with the evident identifications), which depends only on the classes of $\alpha ,\beta$ modulo scalars by [F2]; thus it defines a pairing $e_N:K(N)\times K(N)\to {\mathbf{G}}_m$, alternating because $[(x,\alpha ),(x,\alpha )]=1$, and bilinear because the commutator is multiplicative in each variable. The identity $e_{f^{\ast }L}=e_L\circ (f,f)$ is immediate from pullback of automorphisms, $e_{L\otimes M}=e_L{e}_M$ from the tensor product of automorphisms, and $e_L=1$ for $[L]\in {\mathrm{Pic}}^0$ since then $K(L)=A$ and the pairing $e_L:A\times A\to {\mathbf{G}}_m$ is constant on the complete variety $A$.

We prove (b). Let $0\to {\mathbf{G}}_m\to E\stackrel{\pi }{\to }H\to 0$ be a commutative extension with $H$ finite commutative of rank $n$, and let $k$ be algebraically closed. Since $H$ is killed by $n$ [F3], the endomorphism $[n{]}_E$ lands in ${\mathbf{G}}_m=\ker \pi$, so $[n{]}_E$ is a homomorphism $E\to {\mathbf{G}}_m$ whose restriction to ${\mathbf{G}}_m$ is $t\mapsto t^n$. Let $F=\ker ([n{]}_E:E\to {\mathbf{G}}_m)$, a subgroup sheaf of $E$ containing ${\mu }_n={\mathbf{G}}_m[n]$. For an fppf-local point $h\in H$ lift to $e\in E$; then $[n{]}_{E}e=t\in {\mathbf{G}}_m$ and $x\mapsto x^n$ is an fppf epimorphism on ${\mathbf{G}}_m$ [F6], so locally $t=s^{-n}$ and $es\in F$ maps to $h$; hence $F\to H$ is an fppf epimorphism with kernel ${\mu }_n$, i.e. $F$ is a ${\mu }_n$-torsor over the finite scheme $H$ and is therefore representable and finite.

Restricting along the closed immersion $H\subseteq K(N)$ gives the central extension $0\to {\mathbf{G}}_m\to \mathrm{\Theta }(N{)}_H={\pi }^{-1}(H)\to H\to 0$ of fppf group sheaves whose commutator is the restriction of $e_N$; this is statement (a).

By Cartier exactness [F3] the dual $F^D\to ({\mu }_n{)}^D\cong \mathbb{Z}\mathrm{/}n$ is an fppf epimorphism of finite commutative group schemes. As $F$ is nonempty finite over the algebraically closed field $k$, it has a rational point, so $F^D(k)\mathrm{\ne }\mathrm{\varnothing }$, and surjectivity on $k$-points (a morphism of finite type schemes over an algebraically closed field which is fppf surjective is surjective on closed points) provides a character $\chi :F\to {\mathbf{G}}_m$ whose restriction to ${\mu }_n$ is the tautological character.

Consider the multiplication morphism $\mu :{\mathbf{G}}_m\times F\to E$, $(t,f)\mapsto tf$. It is an fppf epimorphism: for $e\in E$ with $t=[n{]}_{E}e$ and locally $t=s^n$, the element $s^{-1}e$ lies in $F$, so $e=s(s^{-1}e)$; its kernel is $\{(t,f):tf=1\}=\{(s,s^{-1}):s\in {\mu }_n\}$, so $\mu$ is a ${\mu }_n$-torsor. The morphism $\psi (t,f)=t\chi (f)$ satisfies $\psi (t\zeta ,{\zeta }^{-1}f)=t\zeta \chi (\zeta {)}^{-1}\chi (f)=t\chi (f)$ for $\zeta \in {\mu }_n$, hence is invariant under the kernel and descends along the fppf cover $\mu$ to a morphism $r:E\to {\mathbf{G}}_m$ [F5]. On ${\mathbf{G}}_m\subset E$ one has $r(t)=t\chi (1)=t$, and $r$ is a homomorphism because $\psi$ is multiplicative and $E$, hence $F$, is commutative. Therefore $E\to {\mathbf{G}}_m\times \ker r$, $e\mapsto (r(e),er(e{)}^{-1})$, is an isomorphism with inverse $(t,u)\mapsto tu$, and $\ker r\to H$ is an isomorphism; the extension splits.

We prove (c). If $e_N$ is trivial on $H\times H$, then every commutator in $\mathrm{\Theta }(N{)}_H$ is trivial, so $\mathrm{\Theta }(N{)}_H$ is commutative; by (b) the extension $0\to {\mathbf{G}}_m\to \mathrm{\Theta }(N{)}_H\to H\to 0$ splits. A homomorphic section $\sigma :H\to \mathrm{\Theta }(N{)}_H$ assigns to each $h\in H(T)$ an isomorphism ${\alpha }_h:t_h^{\ast }{N}_T\to N_T$ with ${\alpha }_{h+h^{\mathrm{\prime }}}={\alpha }_h\circ t_h^{\ast }{\alpha }_{h^{\mathrm{\prime }}}$, which is precisely an $H$-linearization of $N$ covering the translation action of $H$ on $A$.

With this linearization, $N$ is an $H$-equivariant line bundle on the $H$-torsor $f:A\to A\mathrm{/}H$ [F4]; the fppf descent equivalence of modules [F5] produces a line bundle $L$ on $A\mathrm{/}H$ with $f^{\ast }L\cong N$. The descended module is invertible of rank one because $A\to A\mathrm{/}H$ is faithfully flat and rank-one local freeness is checked after such base change; this proves (c).