Math 256B. A Lemma on Smoothness This handout gives some lemmas useful for the Jacobian criterion of smoothness (and they also give another proof of part of Theorem 10.2). Lemma. Let f : Z ! X be an embedding of schemes of finite type over a field k . Assume that Z and X are smooth of relative dimensions d and n over k , respectively. Also assume that X is regular (we are in the process of showing that this assumption is redundant). Let z 2 Z , let x = f(z) , let I be the kernel of the map OX;x ! OZ;z induced by f , and let r = n − d . Then: (a). there are elements f1; : : : ; fr 2 I such that the images of df1; : : : ; dfr in ΩX=k ⊗ k(x) form a basis for ker(ΩX=k ⊗ k(x) ! ΩZ=k ⊗ k(x)) ; and (b). for any such choice, there is an open neighborhood U of x in X and ex- tensions of f1; : : : ; fr to sections of OX (U) (also denoted f1; : : : ; fr ) such that Z \ U is the closed subscheme of U corresponding to the ideal sheaf generated by f1; : : : ; fr . Finally, Z is regular at z . Proof. First note that the top row of the diagram (*), below, is exact (by the second exact sequence and right exactness of tensor). Since ΩX=k ⊗ k(x) and ΩZ=k ⊗ k(x) have dimensions n and d , respectively, the kernel of α0 has dimension r , so again by 0 exactness of that row, there exist f1; : : : ; fr 2 I whose images generate ker α . This proves (a). Next consider the special case in which z is a closed point. Let m be the maximal ideal of OX;x ; note that m ⊇ I . We have a commutative diagram 0 2 α (I=I ) ⊗ k(x) −−−−! ΩX=k ⊗ k(x) −−−−! ΩZ=k ⊗ k(x) −−−−! 0 ? ? ? ? ? ? (*) y y y 2 m=m −−−−! ΩX=k ⊗ k(x) −−−−! Ωk(x)=k −−−−! 0 . As noted already, the top row is exact. The bottom row is the second exact sequence for the closed immersion Spec k(x) ,! X , so it too is exact. 0 0 0 Let O = OX;x=(f1; : : : ; fr) and m = m=(f1; : : : ; fr) ; note that O is a local ring with maximal ideal m0 . Since x 2 X is a closed point and X is catenary, 2 dim OX;x = n . Since X is regular, dimk(x) m=m = n . By arrow chasing, the images 2 of f1; : : : ; fr in m=m are linearly independent (over k(x) ), so 0 0 2 dimk(x) m =(m ) = n − r = d : Thus, we have 0 0 2 0 d = dimk(x) m =(m ) ≥ dim O ≥ dim OZ;z = d : 2 Indeed, the first inequality is always true for local rings, and the second inequality holds 0 because there is a natural surjection O ! OZ;z . Thus, equality holds at every step. Equality at the first step implies that O0 is regular. Therefore it is entire (II 6.11.1A), so (0) is a prime ideal. It is the unique minimal prime, so it must occur in any chain of 0 0 prime ideals of length d in O . Since the surjection O ! OZ;z does not decrease the dimension, (0) must come from a prime ideal of OZ;z ; hence this surjection must be an isomorphism. Thus Z is regular at z , and f1; : : : ; fr generate I . Extend f1; : : : ; fr to sections of OX over some open neighborhood U of x . After shrinking U , we may assume that Z \ U is closed in U . Let I be the ideal sheaf in OU associated to r Z \ U . The cokernel of the map OU ! I determined by f1; : : : ; fr is a coherent sheaf on U not supported at x , so after shrinking U further we may assume that this cokernel is zero. Then f1; : : : ; fr generate I , proving (b). Now consider the general case. Fix f1; : : : ; fr 2 I as in part (a). Choose an open affine neighborhood Spec A of x on which f1; : : : ; fr are regular. We may also assume that Z\Spec A is closed in Spec A . Let p be the prime ideal of A corresponding to x . By smoothness of X over k , II Prop. 8.9, and II Ex. 5.7b, ΩA=k=pΩA=k e is a locally Vr free sheaf on Spec A=p . By construction, df1 ^· · ·^dfr is nonzero in ΩA=k=pΩA=k , so there is a maximal ideal m of A containing p such that df1 ^ · · · ^ dfr remains Vr nonzero in ΩA=k=mΩA=k e, and therefore df1; : : : ; dfr are linearly independent in 0 ΩA=k ⊗(A=m) . Let x 2 X be the point corresponding to m . Since it is a specialization of x in Spec A and Z \ Spec A is closed in Spec A , it follows that x0 corresponds to a closed point z0 2 Z (specializing z ). Since part (b) is stable under generization, part (b) holds for z because it holds for the closed point z0 . (Also, Z is regular at z for the same reason (by Hartshorne II, 8.14A or Mat- sumura, Commutative Ring Theory, Theorem 19.3, the localization of a regular local ring at a prime ideal is regular).) Corollary. Let Z be a scheme of finite type over a field k , and assume that it is smooth over k . Then Z is a regular scheme. Proof. The question is local on Z , so we may assume that Z is affine. Then we may n n n embed Z into Ak for some n , and the above lemma applies with X = Ak since Ak is smooth over k and is a regular scheme. Lemma 1. Let j : Z ! X be an immersion of schemes over a scheme Y , and as- sume that X , Y , and Z are of finite type over a field k . Let z 2 Z and let x = j(z) 2 X . Assume that Z and X are smooth over Y of relative dimensions d and n at z and x , respectively. Let I be the sheaf of ideals defining Z in some neighborhood of x , and let r = n − d . Then: (a). there is an open neighborhood U of x in X and sections f1; : : : ; fr 2 I (U) such that the images of df1; : : : ; dfr in ΩX=Y ⊗ k(x) are linearly independent (over k(x) ), and (b). any such f1; : : : ; fr generate I in a (possibly smaller) open neighborhood of x . 3 Proof. By shrinking Z and X , we may assume that they are smooth of relative di- mensions d and n over Y , respectively, and that j is a closed immersion. As before, part (a) holds in the stalk at x , by exactness of the sequence 2 (I =I ) ⊗ k(x) −! ΩX=Y ⊗ k(x) −! ΩZ=Y ⊗ k(x) −! 0 and the fact that the last two terms of this sequence have dimensions n and d , respec- tively. One can then extend f1; : : : ; fr to an open neighborhood U of x . For part (b), let y 2 Y be the point lying under z 2 Z and x 2 X , and let ¯ ¯ f1; : : : ; fr be functions as in part (a). Let f1;:::; fr be the corresponding functions on an appropriate neighborhood of x in the fiber Xy . Since Zy and Xy are smooth over ¯ ¯ k(y) and Zy is a closed subscheme of Xy , we can apply the earlier lemma to f1;:::; fr . ¯ ¯ Part (b) of that lemma then implies that f1;:::; fr generate the ideal corresponding to Zy in Xy in some neighborhood of x . By Nakayama's lemma, f1; : : : ; fr generate the stalk Ix , so by coherence, they generate I in an open neighborhood of x in X , as was to be shown. .