A pointed genus-one curve has a regular group law over its ground field

The hypotheses needed after field extension follow from the available suppliers, rather than an assumption that the new field is algebraically closed. By [F7], $H^0(E,{\mathcal{O}}_E)=F$. Apply [F2] to ${\mathcal{O}}_E$ over $\mathrm{Spec}F$. Its computing complex and flatness of field extension imply $H^0(E_{F^{\mathrm{\prime }}},\mathcal{O})=F^{\mathrm{\prime }}$ and $h^1(E_{F^{\mathrm{\prime }}},\mathcal{O})=1$ for every field $F^{\mathrm{\prime }}\mathrm{/}F$. By [F5] these extensions remain smooth. Over an algebraic closure $A$ of $F^{\mathrm{\prime }}$, their local rings are regular domains by [F7], so different irreducible components cannot meet. There are finitely many such components, since these schemes are Noetherian, so they are open and closed. But $H^0(E_A,\mathcal{O})=A$ has no nontrivial idempotent, which excludes a nontrivial open-and-closed decomposition. Thus $E_A$ is integral. Its dimension is one by [F7], and properness is preserved by [F4]. Hence $E_{F^{\mathrm{\prime }}}$ is again a smooth proper geometrically integral genus-one curve, for every extension $F^{\mathrm{\prime }}\mathrm{/}F$.

Over any such field $F^{\mathrm{\prime }}$, a degree-one invertible sheaf $L$ on $E_{F^{\mathrm{\prime }}}$ has a one-dimensional section space and no first cohomology by [F1]. By [F9], a nonzero section determines an effective divisor of degree one representing $L$. In the degree formula ${\sum }_x{n}_x[\kappa (x):F^{\mathrm{\prime }}]=1$, all terms are nonnegative integers and each residue degree is a positive integer, so exactly one coefficient is $1$ at a point with residue field $F^{\mathrm{\prime }}$. This is one reduced $F^{\mathrm{\prime }}$-rational point. Multiplying the section by a nonzero scalar does not change its divisor; since $h^0(L)=1$, every nonzero section is such a scalar multiple. Hence each degree-one class has exactly one effective representative.

We need a family version to obtain regularity, not only an operation on points. Let $T$ be a Noetherian $F$-scheme and $p:E_T\to T$ the projection. Let $L$ be invertible of degree one on every geometric fiber. It is coherent and flat over $T$, since $p$ is smooth. On an affine neighbourhood of any point of $T$, apply [F2] to $L$ and make its computing complex finite free. Over the local residue field its cohomology has dimension one in degree zero and vanishes in all other degrees by step 2.1. If any differential matrix has a unit entry, change bases and split off the two-term identity summand; the complex relation makes all entries to and from that summand zero after these operations. Repeat finitely until all remaining differential entries lie in the maximal ideal. The residual fiber complex has zero differentials, so its terms outside degree zero have rank zero and its degree-zero term has rank one. The same elementary cancellations hold on a neighbourhood of the chosen point. Thus $p_{\ast }L$ is invertible, higher cohomology vanishes, and $p_{\ast }L$ commutes with every base change, because the local computing complex is now a single free rank-one term. These identifications use the natural cohomology maps in [F2], so agree on overlaps.

Consider evaluation $p^{\ast }{p}_{\ast }L\to L$. On any geometric fiber it is the nonzero section spanning the one-dimensional space from step 2.1, and hence multiplication by a nonzerodivisor on that integral fiber. Faithful scalar extension to an algebraic closure detects injectivity over the residue-field fiber. Apply [F3] to evaluation at every source stalk, with the base local ring at its image: the source is a rank-one finite module over the Noetherian target stalk and the target line bundle is base-flat. Evaluation is injective with base-flat cokernel. Tensor it with $L^{-1}$; its image is a locally principal ideal generated by a nonzerodivisor, so defines an effective Cartier divisor $D\subset E_T$. Its structure sheaf is flat over $T$, since it is locally a line-bundle twist of that cokernel. Every geometric fiber of $D$ is the degree-one zero divisor, hence one reduced rational point. The projection $D\to T$ is proper by [F4], is finite type, and has finite geometric fibers, so is finite by [F4]. Its finite flat algebra is locally free of rank one. Its unit map from ${\mathcal{O}}_T$ is an isomorphism on every residue field, and Nakayama makes it surjective. In rank-one local frames it is multiplication by a unit, hence an isomorphism. Therefore $D\to T$ is an isomorphism and $D$ is the graph of a unique morphism $T\to E$. This construction commutes with base change: the evaluation map does by step 3.1, and its flat cokernel makes its zero divisor commute with base change as well.

Take $T=E{\times }_{F}E$ and denote its two projection points by $P,Q$. The projection $E_T\to T$ is proper by base change from $E\to \mathrm{Spec}F$ [F4], hence separated by [F8]. Each of $P,Q,O$ defines a section of this projection. For any section $s:T\to E_T$, its image is the base change of the diagonal of $E_T\mathrm{/}T$ along $(s\circ p,{\mathrm{id}}_{E_T})$, so it is a closed immersion by [F8]. These graphs are closed immersions of smooth $F$-schemes, with the same residue field in source and target at corresponding points. Their dimensions differ by one; [F5] gives local regular rings whose dimensions differ by one even at nonclosed points, and then gives a single regular parameter generating each graph ideal. Thus $L={\mathcal{O}}_{E_T}({\mathrm{\Gamma }}_P+{\mathrm{\Gamma }}_Q-{\mathrm{\Gamma }}_O)$ is an actual invertible sheaf, of degree one on each geometric fiber. Steps 3.1–4.1 produce an $F$-morphism $m:T\to E$ with the required class rule. On $T=E$, use ${\mathcal{O}}_{E_T}(2{\mathrm{\Gamma }}_O-{\mathrm{\Gamma }}_P)$ to obtain an $F$-morphism $i:E\to E$ with the inverse class rule. The base-change property in step 4.1 shows that these are exactly the stated operations on points over every field extension. In particular, their construction takes place over $F$ itself.

For every field extension, the map $P\mapsto [P]-[O]$ into the group of divisor classes of degree zero is injective. If $[P]-[Q]$ were principal for $P\mathrm{\ne }Q$, a defining nonconstant rational function would belong to $H^0(E,\mathcal{O}(Q))$, which has dimension one by [F1] and contains the constants, a contradiction. The class formulas of step 5.1 give $[m(P,Q)]-[O]=([P]-[O])+([Q]-[O])$ and $[i(P)]-[O]=-([P]-[O])$. Associativity and commutativity of divisor-class addition, with this injection, prove associativity, commutativity, identity $O$, and both inverse identities on all points valued in all field extensions. No prior group law on $E$ is used.

The preceding identities are identities of scheme morphisms. To see this without any assumption about density of $F$-rational points, two morphisms from a reduced scheme $X$ to a separated scheme have a closed equalizer. If they agree on every residue-field point $\mathrm{Spec}\kappa (x)\to X$, the ideal of that equalizer maps to zero in every residue field. On each affine open its sections lie in every prime ideal, and hence are nilpotent; reducedness makes them zero. Thus the equalizer is all of $X$. Apply this argument to the products of $E$ occurring in the group identities. They are smooth and therefore geometrically regular and reduced by [F5]. Step 6.1 proves agreement at every field-valued point, including each residue-field point. The class rules also characterize $m$ and $i$ uniquely by the same argument. A pointed isomorphism preserves divisor classes, so intertwines the laws; field extension does too by step 4.1. Consequently [F6] makes $E$ a group variety, proper and geometrically connected, and hence an abelian variety. Its $F$-points form the asserted abelian group.