Coherent Kunneth, the tangent bound and the proper-image dual

By [F1] compute $H^{\ast }(A,{\mathcal{O}}_A)$ by the double Cech complex of two finite separated affine covers: on products of affine intersections the sections are tensor products, the two augmented Cech directions compute cohomology because affine quasi-coherent higher cohomology vanishes, and field Kunneth gives $H^{\ast }(A{\times }_{k}A,\mathcal{O})=H^{\ast }(A,\mathcal{O}){\otimes }_k{H}^{\ast }(A,\mathcal{O})$ with Koszul signs. The addition law makes $H^{\ast }(A,{\mathcal{O}}_A)$ a connected graded Hopf algebra in which every element of $H^1$ is primitive. If $v_1,\dots ,v_r\in H^1$ are linearly independent, apply the $r$-fold coproduct to $v_1\cdots v_r$ and project to $(H^1{)}^{\otimes r}$: the result is the signed sum over permutations of the independent tensors $v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (r)}$, all coefficients being $\pm 1$, so it is nonzero even in characteristic two. Hence $H^r(A,{\mathcal{O}}_A)\mathrm{\ne }0$ and therefore $r\le g$ by the vanishing above dimension $g$; no Borel structure theorem is needed.

The exponential sequence $1\to 1+\epsilon {\mathcal{O}}_A\to {\mathcal{O}}_{A[\epsilon ]}^{\ast }\to {\mathcal{O}}_A^{\ast }\to 1$ on the dual numbers identifies the rigidified Picard tangent space with $H^1(A,{\mathcal{O}}_A)$, so its dimension is at most $g$ by step 1.1.

It remains to justify finiteness of $K=\ker {\phi }_{\mathcal{L}}$. The kernel is represented by the Picard scheme just constructed, and the normalized family $\mathrm{\Lambda }(\mathcal{L})$ is trivial on $A{\times }_{k}K$ by the universal property of the kernel. Over the algebraic closure, $Y=K_{\mathrm{r}\mathrm{e}\mathrm{d}}^0$ is a smooth connected proper subgroup, hence an abelian subvariety: a reduced finite-type group over a perfect field is smooth by translating its nonempty smooth locus. Restricting $\mathrm{\Lambda }(\mathcal{L})$ to $Y{\times }_{k}Y$ and pulling back by $(\mathrm{id},-1)$ makes $\mathcal{L}{\mathrm{\mid }}_Y\otimes [-1{]}^{\ast }\mathcal{L}{\mathrm{\mid }}_Y$ trivial; it is ample because $\mathcal{L}{\mathrm{\mid }}_Y$ is ample, inversion is an automorphism and tensor products of ample sheaves are ample. A trivial ample line bundle on a proper integral variety forces dimension zero: a high power embeds it, but all sections of the trivial bundle are scalars, so the embedding is constant. Hence $\dim Y=0$, so $K$ is zero-dimensional proper finite type and therefore finite, including its nonreduced structure. This is the EGM argument of the cited chapter; no pre-existing ample-kernel theorem is presumed.

For an ample $\mathcal{L}$, the square and cube theorems [F2] with the kernel argument of step 3.1 define ${\phi }_{\mathcal{L}}:A\to B={\mathrm{Pic}}^0$ with finite scheme-theoretic kernel, hence image of dimension $g$. Since $A$ is proper and $B$ separated, the image of ${\phi }_{\mathcal{L}}$ is closed, connected and of dimension $g$; at the identity $\dim {\mathcal{O}}_{B,0}\ge g$, while its embedding dimension is bounded by $g$ by step 2.1. Thus ${\mathcal{O}}_{B,0}$ is regular of dimension $g$, and translation makes $B$ smooth. The closed image has the same local dimension, so its defining ideal in this regular local domain is zero. It is therefore open and closed in the connected group $B$, hence $B$ itself, and $B={\mathrm{Pic}}^0$ is a smooth proper connected group of dimension $g$; ${\phi }_{\mathcal{L}}$ is an isogeny. The identity component represents precisely algebraically trivial classes on all tests: a connected family of line bundles maps into one connected component of the Picard scheme, so differences of its fibres lie in $B$; conversely the universal bundle on the connected finite-type scheme $B$ connects every geometric point to the identity. Since $B$ is open, a classifying map factors through it exactly when every geometric fibre class lies there, including on nonreduced tests. Invariant differentials trivialize ${\mathrm{\Omega }}_A$ by [F3].