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I'm studying recently faithfully flat modules and I'd like to know the following:

Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of nilpotent elements of $R$?

user26857
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user93721
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1 Answers1

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If $I\subset R$ is an arbitrary ideal, then $R/I$ can only be flat if $I=I^2$ : this follows from tensoring the exact sequence $0\to I\to R$ by $R/I$ and getting the exact sequence $0\to I\otimes _R R/I=I/I^2\stackrel {0}{\to} R/I$.

This is a tool for proving that many quotients of a ring are not flat over the ring:
For example if $k$ is a field, then the ring $R=k[\epsilon]=k[T]/(T^2)$ has as nilpotents $N=(\epsilon)$, and since $(\epsilon)^2=(0)\neq (\epsilon)$, the quotient $R/N$ is not flat over $R$, which answers your question negatively (we don't have flatness of $R/N$ over $R$, let alone faithful flatness).

Georges Elencwajg
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  • To a friend: everybody makes mistakes! Grothendieck's erroneous statement about the Brauer group was the basis of at least one masters thesis and several research articles! – Georges Elencwajg Sep 08 '13 at 11:18
  • Faithfully flat = flat and surjective on spectra, but $\mathrm{Spec}(R/N) \to \mathrm{Spec}(R)$ is a homeomorphism. So actually here all comes down to the failure of flatness. – Martin Brandenburg Sep 08 '13 at 18:38
  • Dear Georges, I am curious about Grothendieck's mistake. – Cantlog Sep 08 '13 at 22:16
  • I wonder whether the condition $I=I^2$ (with $I=N(R)$) is sufficient for the flatness ? Should ask separately the question ? – Cantlog Sep 08 '13 at 22:21
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    Dear Cantlog, the formula (7) page 74 in Grothendieck's Le groupe de Brauer II is incorrect if the scheme has more than one singularity. The quoted article is in the book Dix exposés sur la cohomologie des schémas. And you could certainly ask a separate question about $I=I^2$. – Georges Elencwajg Sep 09 '13 at 07:08
  • Thanks for the reference! Sorry for boring you with, but what is wrong in this formula ? There are many definitions, so it is little hard for me to understand quickly. – Cantlog Sep 09 '13 at 09:02
  • Dear @Cantlog, have a look at the article Nontrivial, Locally Trivial Azumaya Algebras by F. R. DEMEYER and T. J. FORD, in Contemp. Math. 124 (1992) – Georges Elencwajg Sep 09 '13 at 09:38
  • Thanks Georges! I will get the article and have a look. – Cantlog Sep 10 '13 at 07:19
  • Dear @Cantlog: another notorious mistake by Grothendieck (and many others) is his definition of rational function, which doesn't make sense: here is Kleiman's celebrated article explaining the error. – Georges Elencwajg Sep 10 '13 at 09:09
  • I knew this one and the mistake is easy to understand (the definition doesn't even give a presheaf). But I don't know other mistakes in EGA. – Cantlog Sep 10 '13 at 11:08
  • @GeorgesElencwajg: The mistake in Brauer II seems to be a confusion of Pic(X) with Cl(X), right ? – Cantlog Sep 17 '13 at 11:09