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I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:

Being a "full set of sections" of $Z/S$ is something which is sufficient to check using the constituents of a single open affine cover of $S$, and also a finite generating set of the coordinate ring of $Z$ over each such open.

Unfortunately, in my mind there are several ways in which to interpret this statement. Let's assume that finite generating set refers as a module (not as an $R$-algebra). One interpretation that I tried and gives me trouble is the following:

For simplicity, take $S$ to affine, $\newcommand{\Spec}{\text{Spec}}$ $S=\Spec(R)$, and $Z=\Spec(B)$ even with $B$ free as a module over $R$, say of rank $N$ and a basis $b_1,\ldots,b_N$; then I don't think the condition for full set of sections (say in the form (1) of definition (1.8.2) on page 33 of Katz-Mazur) for $b_1,\ldots,b_N$ implies it for every $b\in B$. When trying to prove it, roughly speaking, I ran into the problem that "additivity fails", i.e. if (1) holds for $b$ and $c$, then it doesn't hold for $b+c$ (if this were usual linear algebra over a field, I believe this translates to the fact that $b$ and $c$ viewed as linear maps on $B$ via multiplication, have different eigenvectors).

However, I didn't completely disprove it. It may be possible that the algebra structure on $B$ (or even the Hopf algebra structure) somehow guarantees that all is good? (doesn't seem so to me though, even if the matrices of $b$ and $c$ commute since $bc=cb$).

Or maybe it was meant that even though we work Zariski-locally, say on $\Spec R$ we still require the definition of full set of sections for any $\Spec R'\to\Spec R$. (i.e. I am allowed to pass to $R$-algebras). In this direction, alternatively (and maybe that's what Brian meant, but it doesn't seem exactly 'Zariski locally' anymore) one can pass to the "universal element" $f=\sum T_ib_i\in B\otimes_RR[T_1,\ldots,T_n]$ as on page 38 of K-M and then indeed, it suffices to check condition (1) (or (2)) for this universal element, yielding a big identity but in finitely many variables.

Could anybody help clarify this please? As a bonus, I would appreciate some intuition as to why the condition for full set of sections is so "linear" (in my not-very-developed understanding it seems to barely touch upon the algebra structure of $B$ and almost not at all on the Hopf algebra structure) - it seems a bit strange that it encompasses so much "geometric" information (at least the way I think of level structures).

Thanks!

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The condition that "$P_1,\dots, P_N$ is a full set of sections on $Z/S$" is local on $S$.

For the case you mention in the first interpretation, let $k$ be a field, $S = \operatorname{Spec} k[\epsilon]/(\epsilon ^2)$ and $X = \operatorname{Spec}\mathcal{O}_S(S)[\epsilon ']/(\epsilon'^2)$. Define $P_i:S\to X$ as $\epsilon'\mapsto 0$ for $P_1$ and $\epsilon'\mapsto \epsilon$ for $P_2$. Then, $b_1 = 1$ and $b_2 = \epsilon'$ are a basis of $\mathcal{O}_X(X)$ over $\mathcal{O}_S$(S) whose norms are written as $b_j(P_1)b_j(P_2)$, but the same thing does not hold for $1+\epsilon'$. This example was taken from Takeshi Saito, "Fermat's Last Theorem: The Proof."

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