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How can we understand an affine neighborhood of an $S$-valued point on a scheme, and when does it exist?

I am looking at page 111 of Haruzo Hida's Geometric Modular Forms and Elliptic Curves, and he says

"Since E/S is smooth, we have an affine open neighborhood U of 0 and ..."; here, $0 : S \rightarrow E$ is a morphism of schemes. He gives a property of smooth morphisms that was shown in section 1.9 for actual points. To me it is not obvious that this applies to general $S$-valued points - even the existence of an affine neighborhood seems unclear. Can anyone give me a reference for this?

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  • $\begingroup$ Dear NFB, There is no such general notion, and certainly in the general set-up of $E\to S$, there need not be an affine open subset of $E$ that contains the zero section. (Imagine that $S$ was a positive-dimensional projective variety, which could occur if $E$ was just a product $E_0 \times S$, so that the zero section was again a positive-dimensional projective variety.) But probably Hida wrote one thing and meant another, e.g. perhaps he is implicitly restricting to some small open subset of the base. (I haven't read the text apart from what you quoted, so there may also be addional ... $\endgroup$
    – Emerton
    May 30, 2013 at 2:30
  • $\begingroup$ ... context explaining what he means.) Regards, $\endgroup$
    – Emerton
    May 30, 2013 at 2:30
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    $\begingroup$ If the base $S$ is local then any open affine $U$ around the closed point of the identity section $e$ will contain $e$ (since the only open subscheme of a local scheme that contains the closed point is the entire space), so by passing to a local base you get such an affine open. It may be unnecessary to have such a $U$, depending on the goal. If the aim is to discussion formal completion along the identity via a power series ring then one wants the base to be local, or at least the relative tangent space along $e$ to be globally free (of rank 1). Passing to a local base is usually harmless. $\endgroup$
    – user28172
    May 30, 2013 at 2:39
  • $\begingroup$ If you can find a relatively ample line bundle $L$ on $E$ (if, as I suspect, $E$ is an elliptic curve or an abelian scheme, this exists, or is a quite harmless hypothesis), and if $S$ is affine and has trivial (torsion is enough) Picard, then you can assume that the restriction of $L$ to the $0$-section is trivial, and taking higher powers of $L$, obtain a global section $s$ of $L$ which is invertible on the $0$-section. The invertibility locus of $s$ is an affine neighborhood of this section. $\endgroup$
    – ACL
    May 30, 2013 at 7:19
  • $\begingroup$ @Emerton @ACL Yes, the setup here is that $E$ is an elliptic curve. I'm sorry that I forgot to mention this. $\endgroup$ May 30, 2013 at 7:20

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Even if there is an affine open $U$ around $0$, the map $g$ there will not exist if the relative tangent space along the identity section is not globally free over the base ring $A$ there. That is one reason you are rightly confused about transporting intuition/experience from the theory over a field to the situation over a ring.

Let's give a direct proof that after passing to the case when the base $S$ is affine, say Spec($A$), the formal completion of $E$ along the identity section is naturally a 1-parameter formal group over $A$ provided that the relative tangent space along the identity section (which is always an invertible $O_S$-module, as $E$ is $S$-smooth with pure relative dimension 1) is globally free, as can always be arranged by initial further Zariski localization on $S$. This is essentially an elaboration on the comments of Emerton and ACL.

Consider the zero scheme $E_n$ of the $(n+1)$th power of the ideal sheaf of the identity section. This is a quasi-compact separated scheme over $A$ with a quasi-coherent ideal $I$ satisfying $I^{n+1} = 0$ such that the zero scheme of $I$ is affine. It follows from Serre's cohomological criterion for affineness and devissage that each $E_n$ must be affine, call it Spec($R_n$). By design, we have $A$-algebra surjections $R_{n+1} \rightarrow R_0 = A$. For a flat scheme, the ideal sheaf of a section is flat and its formation commutes with base change (this is immediately reduced to the affine setting, where it follows from the Tor$^1$-vanishing argument). In particular, by descent to the noetherian setting and compatibility with base change we see that over general $A$ the ideal $J_n = \ker(R_{n+1} \rightarrow R_0)$ is $A$-flat and finitely presented with $n$-dimensional fibers, so it is a rank-$n$ vector bundle over $A$.

More specifically, we arranged that $J_1$ is globally free of rank 1 (it is $A$-dual to the relative tangent space along the identity section, which we arranged to be globally free of rank 1, not just invertible), so if we choose a generator $T_1$ of $J_1$ and pick compatible lifts $T_n \in J_n$ for all $n$ then the $n$ elements $T_n^i$ for $1 \le i \le n$ generate $J_n$ on geometric fibers over Spec($A$) by the theory over fields, so since $J_n$ is a rank-$n$ vector bundle it follows that these $n$ generators must be an $A$-basis of $J_n$. Thus, we have built compatible $A$-algebra isomorphisms $$A[T]/(T^{n+1}) \simeq R_n$$ for all $n \ge 0$, so this identifies the formal completion $\widehat{E}$ along the identity with ${\rm{Spf}}(A[\![T]\!])$ (using $T$-adic topology).

This pro-represents the functor $$(B,J) \mapsto \ker(E(B) \rightarrow E(B/J) = E(A))$$ on augmented $A$-algebras $(B,J)$ with some power of $J = \ker(B \twoheadrightarrow A)$ equal to 0, and since this is a group-valued functor it follows that $\widehat{E}$ acquires a structure of formal group over $A$, where the relevant fiber product is completed tensor product (as that pro-represents the fiber product of functors on the category of augmented algebras with nilpotent augmentation ideal). In other words, we acquire what is classically called a formal group law on $A[\![T]\!]$ in the usual sense.

The discussion of these matters in Hida's book is an elaboration on the early parts of Chapter 2 of the book by Katz and Mazur. The relationship between global relative 1-forms (necessarily translation-invariant) and global bases of the relative tangent space along the identity is very well-explained in section 4.2 of the book "Neron models" for smooth group schemes over any base scheme.

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