Dual isogenies, Cartier-dual kernels and canonical biduality

The kernel $H$ is a finite $k$-group scheme, and since $A\mathrm{/}H$ exists and the homomorphism $A\to A\mathrm{/}H$ has kernel $H$, the induced morphism $A\mathrm{/}H\to C$ is a closed immersion [F4] which is surjective because $f$ is surjective; since $C$ is reduced, a surjective closed immersion into $C$ has zero defining ideal and is therefore an isomorphism, so $f$ is the quotient map $A\to A\mathrm{/}H$ and in particular a faithfully flat $H$-torsor of finite presentation of degree $d=\mathrm{rank}H$.

For a $k$-scheme $T$ and a rigidified line bundle $N$ on $C_T$ define $f_T^{\vee }(N)=f_T^{\ast }N$; this is a rigidified line bundle on $A_T$ (pullback of the rigidification), and it is degree zero: for $N\in C^{\vee }(T)$ one has ${\phi }_N=0$, so by [F2] ${\phi }_{f_T^{\ast }N}=f_T^{\ast }\circ {\phi }_N\circ f_T=0$, hence $f_T^{\ast }N\in {\mathrm{Pic}}^0(A_T)=A^{\vee }(T)$. The rule is compatible with base change in $T$ and with tensor products, and pullback of bundles is contravariant, so $f\mapsto f^{\vee }$ is a contravariant additive functor; since source and target are represented by $C^{\vee }$ and $A^{\vee }$, Yoneda's lemma promotes $f^{\vee }$ to a homomorphism of $k$-group schemes.

The canonical morphism ${\kappa }_A:A\to A^{\vee \vee }$ is defined by the switched normalized Poincare bundle: for a test $T$, the pullback $Q=(\mathrm{swap}{)}^{\ast }\mathcal{P}$ is a rigidified line bundle on $A^{\vee }{\times }_{k}A$ which is degree zero along $A^{\vee }$, hence it represents a morphism ${\kappa }_A:A\to A^{\vee \vee }=(A^{\vee }{)}^{\vee }$ by the universal property [F1], and ${\kappa }_A$ is a homomorphism of group schemes because the biextension identities of $\mathcal{P}$ are multiplicative in the second variable, which is the theorem of the cube [F6].

We compute $\ker f^{\vee }$. Let $T$ be a $k$-scheme and let $N\in \ker f^{\vee }(T)$, so there is an isomorphism $\alpha :f_T^{\ast }N\to {\mathcal{O}}_{A_T}$ of rigidified line bundles. Since $f_T:A_T\to C_T$ is an $H_T$-torsor [step 1.1], the pair $(N,\alpha )$ is exactly a descent datum for the trivial line bundle ${\mathcal{O}}_{A_T}$ along $f_T$: an isomorphism $\theta :p_1^{\ast }\mathcal{O}\to p_2^{\ast }\mathcal{O}$ over $A_T{\times }_{C_T}{A}_T\cong A_T{\times }_T{H}_T$ satisfying the cocycle condition. Write $\theta (h)\in H^0(A_T,{\mathcal{O}}^{\ast })={\mathcal{O}}_T^{\ast }$ for the unit attached to $h\in H_T$; by [F6] every such unit is pulled back from $T$. The cocycle condition becomes $\theta (h_1{h}_2)=\theta (h_1)\theta (h_2)$ in ${\mathcal{O}}_T^{\ast }$, so $\theta$ is precisely an $H_T$-valued character, that is, an element of ${\mathrm{Hom}}_{T\text{-gr}}(H_T,{\mathbf{G}}_{m,T})=H^D(T)$ by [F3].

The character in step 2.1 is independent of the chosen trivialization: changing it by a base unit conjugates the scalar action trivially. Conversely, a character $\chi \in H^D(T)$ gives an $H_T$-linearization of ${\mathcal{O}}_{A_T}$; monoidal effective descent [F5] gives a line bundle $N$ on $C_T$ with $f_T^{\ast }N\cong {\mathcal{O}}_{A_T}$ and an induced rigidification. It remains to place $N$ in $C^{\vee }(T)$ on the entire test scheme. Since $[d{]}_H=0$ by [F3], ${\chi }^d=1$, so its $d$-th tensor-power descent datum is trivial and $N^d$ is rigidified-trivial. Multiplicativity of the normalized square family gives $d{\phi }_N={\phi }_{N^d}=0$ by [F2]. Thus the pointed morphism ${\phi }_N:C_T\to C_T^{\vee }$ factors through the finite affine $T$-group $C^{\vee }[d{]}_T$ supplied by [F6]. The universal structure-sheaf equality for $C_T$ in [F6] identifies maps to this relative affine target with algebra maps to ${\mathcal{O}}_T$, so the morphism factors through $T$; evaluation at the identity makes that factor the zero section. Hence ${\phi }_N=0$ on all tests, and [F2] gives $N\in {\mathrm{Pic}}^0(C_T)=C^{\vee }(T)$. The constructions are inverse and natural, and tensor products agree with products of characters. Therefore $\ker f^{\vee }\cong H^D$ as fppf group sheaves, and hence as group schemes by representability.

Consequently $f^{\vee }$ is finite (its kernel $H^D$ is finite) and $\dim C^{\vee }=\dim C=\dim A=\dim A^{\vee }$, so its image is a closed connected subgroup of dimension $\dim A^{\vee }$, hence all of $A^{\vee }$; thus $f^{\vee }$ is an isogeny. Applying the quotient-torsor argument of step 1.1 to it proves finite flatness, with $\mathrm{deg}{f}^{\vee }=\mathrm{rank}\ker f^{\vee }=\mathrm{rank}{H}^D=d=\mathrm{deg}f$, using that Cartier duality preserves ranks [F3].

To prove that ${\kappa }_A$ is an isogeny, choose an ample line bundle $L$ on $A$ and put $B=A^{\vee }$, $f={\phi }_L:A\to B$, an isogeny by [F2]. Let $P_A$ and $P_B$ be the normalized Poincare bundles of $A$ and $B$. By the definition of ${\kappa }_A$, $({\mathrm{id}}_B\times {\kappa }_A{)}^{\ast }{P}_B\cong {\mathrm{swap}}^{\ast }{P}_A$ on $B\times A$. The defining dual-pullback identity also gives $(f\times {\mathrm{id}}_{B^{\vee }}{)}^{\ast }{P}_B\cong ({\mathrm{id}}_A\times f^{\vee }{)}^{\ast }{P}_A$. Pulling these identities to $A\times A$ identifies $({\mathrm{id}}_A\times f^{\vee }{\kappa }_A{)}^{\ast }{P}_A$ with ${\mathrm{swap}}^{\ast }({\mathrm{id}}_A\times f{)}^{\ast }{P}_A={\mathrm{swap}}^{\ast }\mathrm{\Lambda }(L)$. The normalized bundle $\mathrm{\Lambda }(L)$ is symmetric, so the universal property of $P_A$ yields $f=f^{\vee }\circ {\kappa }_A$. Every pullback in this comparison has the displayed product domain; in particular $({\mathrm{id}}_A\times {\phi }_L{)}^{\ast }{P}_A$ lies on $A\times A$. Since $f$ has finite kernel, ${\kappa }_A$ does too, and equal dimensions make its proper image all of $A^{\vee \vee }$. The quotient-torsor argument makes it a finite flat isogeny.

Degrees multiply for compositions of isogenies, and by step 4.1 applied to ${\phi }_{\mathcal{L}}$ we have $\mathrm{deg}{\phi }_{\mathcal{L}}^{\vee }=\mathrm{deg}{\phi }_{\mathcal{L}}$. Taking degrees in ${\phi }_{\mathcal{L}}={\phi }_{\mathcal{L}}^{\vee }\circ {\kappa }_A$ gives $\mathrm{deg}{\phi }_{\mathcal{L}}=\mathrm{deg}{\phi }_{\mathcal{L}}\cdot \mathrm{deg}{\kappa }_A$, hence $\mathrm{deg}{\kappa }_A=1$; a finite flat morphism of degree one is an isomorphism, so ${\kappa }_A:A\to A^{\vee \vee }$ is an isomorphism.

Finally, $(\mathrm{id}{)}^{\vee }=\mathrm{id}$ and $(g\circ f{)}^{\vee }=f^{\vee }\circ g^{\vee }$ are immediate from $(g\circ f{)}_T^{\ast }=f_T^{\ast }{g}_T^{\ast }$. For homomorphisms $f,g:A\to C$ with common source and target, the cube identity gives $m_C^{\ast }N\cong p_1^{\ast }N\otimes p_2^{\ast }N$ for every degree-zero rigidified bundle $N$ on $C_T$ [F2]; hence $(f+g{)}_T^{\ast }N\cong f_T^{\ast }N\otimes g_T^{\ast }N$, which is $f^{\vee }+g^{\vee }$ under the tensor group law of $A^{\vee }$. The identity $f^{\vee \vee }\circ {\kappa }_A={\kappa }_C\circ f$ follows from the definition of $\kappa$ by the switched normalized Poincare bundles and uniqueness of representing morphisms, since both sides are represented by the same pullback of the switched bundle.