I'm attempting to track down the original EGA proof of Deligne's formula that for a noetherian ring $A$, ideal $\mathfrak{a}\subset A$, and $A$-module $M$ we have $$\varinjlim \operatorname{Hom}_A(\mathfrak{a}^n,M) \cong \widetilde{M}\left(\operatorname{Spec} A \setminus V(\mathfrak{a})\right)$$ which I've heard cited as EGA I prop 6.9.17. Unfortunately, the version of EGA hosted by Numdam here (warning, large file) does not even contain a section 6.9. What's going on?

  • $\begingroup$ I know I’m not answering at all the question, but there’s probably a pretty good proof somewhere in stacks project (I remember seeing this somewhere but absolutely not what for). In general, I’d first suggest trying to find the statement in the other EGA at the same number. $\endgroup$
    – Mindlack
    Mar 3 '21 at 0:08
  • $\begingroup$ I've checked the proof in Stacks (it's 01PM), but I want to read the EGA presentation too. As far as other volumes of EGA, none of those volumes available on Numdam have what I'm looking for as 6.9.17. $\endgroup$ Mar 3 '21 at 1:08

I think they are citing the 1971 edition of EGA I published by Springer.

Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.

In this edition, Proposition 6.9.17 is the following (my translation).

Proposition (6.9.17) Let $X$ be a noetherian scheme, $\newcommand{\I}{\mathscr{I}} \I$ a quasicoherent $\newcommand{\O}{\mathscr{O}} \O_X$-module, $Y$ the closed subscheme of $X$ defined by $\I$, and $U$ the open set $X \setminus Y$. Let $\newcommand{\F}{\mathscr{F}} \F$ be a coherent $\O_U$-module, $\newcommand{\G}{\mathscr{G}} \G$ a quasicoherent $\O_X$-module, $\F'$ a coherent $\O_X$-module such that $\F'|_U = \F$ (6.9.8). Then we have a canonical bijection \begin{equation} \DeclareMathOperator{\Hom}{Hom} \varinjlim_n \Hom_{\O_X}(\I^n \F', \G) \overset{\sim}{\to} \Hom_{\O_U}(\F, \G|_U) \, . \tag{} \end{equation}

  • $\begingroup$ Thanks - I guess I'll have to pick up a copy of that. It's kind of a shame that Numdam doesn't have the expanded version. $\endgroup$ Mar 3 '21 at 3:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.