Flat and ´etalemorphisms Maxim Mornev All rings are commutative. Contents 1 Flat morphisms 1 1.1 Preliminaries on tensor product . 1 1.2 Flat modules . 2 1.3 Artin-Rees lemma and Krull intersection theorem . 4 1.4 Modules of finite length . 5 1.5 Criteria of flatness . 6 1.6 Flatness in the context of schemes . 8 2 Etale´ morphisms 9 2.1 The module of K¨ahlerdifferentials . 9 2.2 Etale´ algebras over fields . 12 2.3 Unramified morphisms . 14 2.4 Etale´ morphisms . 16 1 Flat morphisms 1.1 Preliminaries on tensor product Let A be a ring, M and A-module. For all A-modules N1;N2 we have a natural isomorphism ∼ HomA(N1 ⊗A M; N2) = HomA(N1; HomA(M; N2)): In other words ⊗AM is left adjoint to HomA(M; −). Hence ⊗AM is right exact and commutes with colimits. 1 i A Left derived functors L (⊗AM)(−) are denoted Tori (−;M). A morphism of 0 A A 0 modules M ! M induces natural morphisms Tori (−;M) ! Tori (−;M ), so Tori is a bifunctor. The most important property of Tor is its commutativity: Theorem 1.1.1. Let A be a ring, and let M; N be A-modules. For every i > 0 A A 1 there exists a natural isomorphism Tori (N; M) ! Tori (M; N). We will not need the full force of this theorem and so omit its proof. Proposition 1.1.2. Let A be a ring, I ⊂ A an ideal, and M an A-module. A Tor1 (A=I; M) = ker(I ⊗A M ! M). Proof. The short exact sequence 0 ! I ! A ! A=I ! 0 induces an exact A A sequence 0 = Tor1 (A; M) ! Tor1 (A=I; M) ! I ⊗A M ! M. A Corollary 1.1.3. Let a 2 A be a nonzero element. Tor1 (A=(a);M) is the a-torsion of M. Let A; B be rings, N1 an A-module, N2 an A; B-bimodule, and N3 a B-module. There is an isomorphism of A; B-bimodules (N1 ⊗A N2) ⊗B N3 ! N1 ⊗A (N2 ⊗B N3); which is natural in N1;N2;N3. Also recall that if A is a ring and S ⊂ A a multiplicative system, then the functor ⊗AAS is isomorphic to the functor of localization at S. 1.2 Flat modules Definition 1.2.1. Let A be a ring. A module M over A is called flat if ⊗AM is exact. Proposition 1.2.2. Let A ! B be a morphism of rings, and M a B-module. If M is flat over B and B is flat over A then M is flat over A. Proof. The functor − ⊗A M is isomorphic to the composition (− ⊗A B) ⊗B M of exact functors. Proposition 1.2.3. Let A ! B be a morphism of rings. If M is a flat A- module, then B ⊗A M is a flat B-module. Proof. The functor −⊗B (B ⊗A M) is isomorphic to the functor −⊗A M, which is exact. 1See [3], chapter 2, section 2.7 2 Proposition 1.2.4. Let ': A ! B be a morphism of rings, and M a B-module. M is flat over A if and only if for every q 2 Specmax B the module Mq is flat −1 over Ap, where p = ' q. Proof. Notice that ⊗AM sends A-modules to B-modules, with the structure of B-module inherited from M. Let q 2 Specmax B, and p = '−1q. We have an isomorphism of functors from the category of A-modules to the category of Bq-modules: (− ⊗A M)q = (− ⊗A M) ⊗B Bq = − ⊗A (M ⊗B Bq) = − ⊗AMq = − ⊗A (Ap ⊗Ap Mq) = (−)p ⊗Ap Mq: Localization is exact. Hence, if ⊗AM is flat then ⊗Ap Mq is exact. Conversely, −1 if ⊗Ap Mq is exact for every q 2 Specmax B; p = ' q, then tensoring a short exact sequence N1 ! N2 ! N3 with M we obtain a sequence of B-modules which is exact at every maximal ideal q. Therefore it is exact. Proposition 1.2.5. Let A be a ring. An A-module is flat if and only if A I ⊗A M ! M is injective (equivalently, Tor1 (A=I; M) = 0) for every finitely generated ideal I ⊂ A. Proof. The \only if" part is trivial. We want to show that for arbitrary inclusion 0 0 of A-modules N ⊂ N the induced morphism N ⊗A M ! N ⊗A M is injective. We first show that I ⊗A M ! M is injective for every ideal I. Let x 2 I ⊗A M be an element which vanishes in M. The element x is a finite linear combination of elementary tensors y ⊗ m where y 2 I; m 2 M. Thus there exists a finitely 0 0 0 0 generated ideal I ⊂ I and x 2 I ⊗A M such that the image of x in I ⊗A M 0 0 is equal to x. The map I ⊗A M ! M is injective, so x = 0 and hence x = 0, A i.e. I ⊗A M ! M is injective. As a corollary, Tor1 (N; M) = 0 if N is a cyclic module, that is, N = A=I for some ideal I ⊂ A. Let N be an arbitrary module and N 0 its submodule. Consider an index set J 0 whose elements are finite subsets of NnN . For j 2 J let Nj be the submodule of 0 0 N generated by N and j. Jf j ⊂ j then there is a natural injection Nj ! Nj0 . The inclusion order on J makes it a directed poset. Clearly, colimj2J Nj = N. Let j ⊂ j0 be an inclusion. Assume that j0nj consists of a single element. In this case Nj0 =Nj is a cyclic module. The short exact sequence 0 ! Nj ! A Nj0 ! Nj0 =Nj ! 0 induces an exact sequence Tor1 (Nj0 =Nj;M) ! Nj ⊗A A M ! Nj0 ⊗A M. Since Nj0 =Nj is cyclic, Tor1 (Nj0 =Nj;M) vanishes, and so Nj ⊗A M ! Nj0 ⊗A M is injective. A general inclusion j ⊂ j0 can be factored into a sequence of inclusions such that at each step only one new element appears. Hence Nj ⊗A M ! Nj0 ⊗A M 0 is injective, which implies that the morphism N ⊗A M ! colimj2J Nj ⊗A M is injective too. It remains to recall that ⊗AM commutes with colimits. 3 Corollary 1.2.6. Let A be a PID. An A-module M is flat if and only if it is torsion-free. Proposition 1.2.7. Let A be a ring, let 0 ! M 0 ! M 00 ! M ! 0 be a short exact sequence of A-modules, and let N be an A-module. If M is flat then 0 00 M ⊗A N ! M ⊗A N is injective. Proof. One can either refer to commutativity of Tor or do a direct proof as follows. Let 0 ! K ! F ! N ! 0 be a short exact sequences with F a free module. Consider a commutative diagram with exact rows and columns: 0 ? ? y 0 00 M ⊗A K −−−−! M ⊗A K −−−−! M ⊗A K −−−−! 0 ? ? ? ? ? ? y y y 0 00 0 −−−−! M ⊗A F −−−−! M ⊗A F −−−−! M ⊗A F −−−−! 0 ? ? ? ? ? ? y y y 0 00 M ⊗A N −−−−! M ⊗A N −−−−! M ⊗A N −−−−! 0: ? ? ? ? ? ? y y y 0 0 0: A simple diagram chase finishes the proof. Theorem 1.2.8. Let A be a local noetherian ring, and M an A-module of finite type. If M is flat then it is free. Proof. Let k be the residue field of A. Take a k-basis of M ⊗A k. Lifting it to M we obtain a morphism from a free A-module F of finite type to M. By Nakayama lemma this morphism is surjective. Let K be its kernel. Tensoring the short exact sequence 0 ! K ! F ! M ! 0 by k we obtain exact sequence K ⊗A k ! F ⊗A k ! M ⊗A k ! 0. The morphism K ⊗A k ! F ⊗A k is injective by proposition 1.2.7. The morphism F ⊗A k ! M ⊗A k is an isomorphism by construction. Hence K ⊗A k is zero. On the other hand, K is of finite type since A is noetherian. So, Nakayama lemma shows that K = 0. 1.3 Artin-Rees lemma and Krull intersection theorem Let A be a ring, I ⊂ A an ideal. Definition 1.3.1. Let M be an A-module. An I-filtration on M is a descending chain of submodules FiM ⊂ M, i 2 Z>0, such that F0M = M and IFiM ⊂ Fi+1M for every i. 4 Definition 1.3.2. Let M be an A-module. An I-filtration FiM is called stable if IFiM = Fi+1M for sufficiently large i. Proposition 1.3.3. Let A be a ring, I ⊂ A an ideal, and let N; M be A- modules. If FiN is a stable I-filtration of N then the filtration of N ⊗A M by images of FiN ⊗A M is stable. Proof. Omitted. Proposition 1.3.4. Let A ! B be a morphism of rings, I ⊂ A an ideal, M a B-module, and FiM a stable I-filtration of M as an A-module. If each FiM is a B-submodule, then FiM is a stable IB-filtration of M as a B-module. Proof. Omitted. Let M be an A-module endowed with an I-filtration FiM. Consider a graded L1 i L1 ring BI A = i=0 I and a BI A-module BF M = i=0 FiM.