Local criterion for flatness

In algebra, the local criterion for flatness gives conditions one can check to show flatness of a module.[1]

Statement

Given a commutative ring A, an ideal I and an A-module M, suppose either

  • A is a Noetherian ring and M is idealwise separated for I: for every ideal , (for example, this is the case when A is a Noetherian local ring, I its maximal ideal and M finitely generated),

or

Then the following are equivalent:[2]

  1. M is a flat module.
  2. is flat over and .
  3. For each , is flat over .
  4. In the notations of 3., is -flat and the natural -module surjection
    is an isomorphism; i.e., each is an isomorphism.

The assumption that “A is a Noetherian ring” is used to invoke the Artin–Rees lemma and can be weakened; see (Fujiwara–Gabber–Kato, Proposition 2.2.1.)

Proof

Following SGA 1, Exposé IV, we first prove a few lemmas, which are interesting themselves. (See also this blog post by Akhil Mathew for a proof of a special case.)

Lemma 1  Given a ring homomorphism and an -module , the following are equivalent.

  1. For every -module ,
  2. is -flat and

Moreover, if , the above two are equivalent to

  1. for every -module killed by some power of .

Proof: The equivalence of the first two can be seen by studying the Tor spectral sequence. Here is a direct proof: if 1. is valid and is an injection of -modules with cokernel C, then, as A-modules,

.

Since and the same for , this proves 2. Conversely, considering where F is B-free, we get:

.

Here, the last map is injective by flatness and that gives us 1. To see the "Moreover" part, if 1. is valid, then and so

By descending induction, this implies 3. The converse is trivial.

Lemma 2  Let be a ring and a module over it. If for every , then the natural grade-preserving surjection

is an isomorphism. Moreover, when I is nilpotent,

is flat if and only if is flat over and is an isomorphism.

Proof: The assumption implies that and so, since tensor product commutes with base extension,

.

For the second part, let denote the exact sequence and . Consider the exact sequence of complexes:

Then (it is so for large and then use descending induction). 3. of Lemma 1 then implies that is flat.

Proof of the main statement.

: If is nilpotent, then, by Lemma 1, and is flat over . Thus, assume that the first assumption is valid. Let be an ideal and we shall show is injective. For an integer , consider the exact sequence

Since by Lemma 1 (note kills ), tensoring the above with , we get:

.

Tensoring with , we also have:

We combine the two to get the exact sequence:

Now, if is in the kernel of , then, a fortiori, is in . By the Artin–Rees lemma, given , we can find such that . Since , we conclude .

follows from Lemma 2.

: Since , the condition 4. is still valid with replaced by . Then Lemma 2 says that is flat over .

Tensoring with M, we see is the kernel of . Thus, the implication is established by an argument similar to that of

Application: characterization of an étale morphism

The local criterion can be used to prove the following:

Proposition  Given a morphism of finite type between Noetherian schemes, is étale (flat and unramified) if and only if for each x in X, f is an analytically local isomorphism near x; i.e., with , is an isomorphism.

Proof: Assume that is an isomorphism and we show f is étale. First, since is faithfully flat (in particular is a pure subring), we have:

.

Hence, is unramified (separability is trivial). Now, that is flat follows from (1) the assumption that the induced map on completion is flat and (2) the fact that flatness descends under faithfully flat base change (it shouldn’t be hard to make sense of (2)).

Next, we show the converse: by the local criterion, for each n, the natural map is an isomorphism. By induction and the five lemma, this implies is an isomorphism for each n. Passing to limit, we get the asserted isomorphism.

Mumford’s Red Book gives an extrinsic proof of the above fact (Ch. III, § 5, Theorem 3).

Miracle flatness theorem

B. Conrad calls the next theorem the miracle flatness theorem.[3]

Theorem  Let be a local ring homomorphism between local Noetherian rings. If S is flat over R, then

.

Conversely, if this dimension equality holds, if R is regular and if S is Cohen–Macaulay (e.g., regular), then S is flat over R.

Notes

References

  • Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461
  • Exposé IV of Grothendieck, Alexander; Raynaud, Michèle (2003) [1971], Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Paris: Société Mathématique de France, arXiv:math/0206203, Bibcode:2002math......6203G, ISBN 978-2-85629-141-2, MR 2017446
  • Fujiwara, K.; Gabber, O.; Kato, F.: “On Hausdorff completions of commutative rings in rigid geometry.” Journal of Algebra, 322 (2011), 293-321.
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