Grothendieck’s lemma

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Googling for Grothendieck’s lemma turns up a whole slew of different lemmas. For some reason I started thinking of Grothedieck’s lemma as the following result, of which there are two versions:

  • If A –> B is a flat local ring map of Noetherian local rings and f in B is a nonzero divisor in the fiber ring B/m_AB, then B/fB is flat over A.
  • If A –> B is a flat local ring map of local rings, B is essentially of finite presentation over A and f in B is a nonzero divisor in the fiber ring B/m_AB, then B/fB is flat over A.

Leave a comment if you have an opinion about how to refer to this lemma. This result is (very) related to the local criterion for flatness which says instead

  • If A –> B is a local ring map of Noetherian local rings and I is an ideal of A such that B/IB is flat over A/I and moreover Tor_1^A(B, A/I) is zero, then B is flat over A.
  • If A –> B is a local ring map of local rings, B is essentially if finite presentation over A and I is an ideal of A such that B/IB is flat over A/I and moreover Tor_1^A(B, A/I) is zero, then B is flat over A.

This is particularly useful when I = m_A because B/m_A B is automatically flat over the field A/m_A. In the Algebra chapter of the stacks project we prove both of these independently although it might have been better/quicker to deduce the first from the second. Finally, there is another very related result which I think is usually called the critère de platitude par fibre which says roughly

  • If X –> Y is a morphism of locally Noetherian schemes flat over a locally Noetherian base S and if f induces flat morphisms between fibers, then f is flat.
  • If X –> Y is a morphism of schemes of flat and locally of finite presentation over a base S and if f induces flat morphisms between fibers, then f is flat.

You can in fact weaken the assumptions a bit. Of course this is a completely algebraic fact which can be reformulated in terms of maps of local rings as above. There are also versions for modules which are potentially much more useful; for these and some other results see Algebra, Section Tag 00MD, Algebra, Section Tag 00R3, and More on Morphisms, Section Tag 039A.