On Smoothability

On smoothability

Jarosław Buczyński, Joachim Jelisiejew

June 6, 2014

Abstract In this note we present some results on smoothability of zero-dimensional schemes. In particular we compare embedded and abstract smoothings and show that a disjoint union is smoothable if and only if each of its components is smoothable. This was a preliminary version. The current version is [BJ17].

Let k be a ﬁeld. We do not require k to be algebraically closed or characteristic zero. Throughout this paper all considered schemes are Spec k-schemes.

Deﬁnition 1 (abstract smoothing). Let R a ﬁnite scheme. We say that R is (abstractly) smoothable if there exist an irreducible, locally Noetherian scheme T and a T -scheme Z → T such that

1. Z → T is ﬂat and ﬁnite,

2. T has a k-point t, such that Zt ' R. We call t the special point of T .

3. Zη is a smooth scheme over η, where η is the generic point of T . The scheme Z is called an abstract smoothing of R. We will sometimes denote it by (Z,R) → (T, t), which means that t is the k-point of T , such that Zt ' R. Deﬁnition 2 (embedded smoothing). Let Y be a scheme and R be a ﬁnite closed subscheme of Y . We say that R is smoothable in Y if there exist an irreducible, locally Noetherian scheme T and a closed subscheme Z ⊆ Y × T such that Z → T is an abstract smoothing of R. The scheme Z is called an embedded smoothing of R ⊆ Y .

First we mention a useful base-change property for smoothings.

Lemma 3 (Base change for smoothings). Let T be an irreducible, locally Noetherian scheme with the generic point η and a k-point t. Let (Z,R) → (T, t) be an abstract smoothing of a scheme R over T . Let T 0 be an irreducible, locally Noetherian scheme with a morphism f : T 0 → T such that η is, topologically, in the image of f and there exists a k-point of T 0 mapping to t. Then the 0 0 0 pullback Z = Z ×T T is an abstract smoothing of R over T . Moreover, if R was embedded into some Y and Z ⊆ Y × T was an embedded smoothing, then Z0 ⊆ Y × T 0 is also an embedded smoothing of R ⊆ Y .

0 0 0 0 0 0 Proof. Z → T is ﬁnite and ﬂat. The generic point η of T maps to η under f so that Zη0 → η 0 0 is a pullback of a smooth morphism Zη → η. In particular it is smooth, so that Z → T is a smoothing of R. If Z ⊆ Y × T was a closed subscheme, then Z0 ⊆ Y × T 0 is also a closed subscheme.

1 Corollary 4. Let Z → T be an abstract or embedded smoothing of some scheme. Then, after a base change, we may assume that T ' Spec A, where A is a one-dimensional Noetherian complete local domain with quotient ﬁeld k. If k is algebraically closed, we may furthermore assume that A = k[[t]].

Proof. Let A1 = OT,t, then A1 is Noetherian by the assertions on T . Using Lemma 3 we may assume that T = Spec A1. Let A be the completion of A1 at the maximal ideal m of A1. Then A is Noetherian and faithfully flat over A1, so that Spec A → Spec A1 is surjective. Let p ∈ Spec A be any point mapping to the generic point of T . Then T 0 = Spec A/p is integral and satisfies all the assumptions of Lemma 3. We make a base change to T 0, which satisfies the assertions on T except, perhaps, one-dimensionality. Since Z → T 0 is finite, the relative differentials sheaf is coherent over T 0, so that there exists 0 an open neighbourhood U of η such that Zu is smooth for any u ∈ U. If U = {η}, then T is at most one-dimensional by the Theorem of Artin-Tate, see [GW10, Cor B.62]. If not, then we 0 may take an irreducible closed subset V ( T such that the fiber over the generic point of V is smooth and make a base change to V . Since dim V < dim T 0 < ∞, after a finite number of such base changes we obtain that dim V ≤ 1. Then dim V = 1, since V is irreducible with at least two points, corresponding to smooth and non-smooth fiber. Since V is a closed subset of T , it is a spectrum of a Noetherian complete local ring with quotient field k. Moreover V is also a one-dimensional domain. This finishes the proof of the first part. Suppose now that k is algebraically closed. We may assume that T := V = Spec A, where V is as above. Let m be the maximal ideal of A. The normalisation A˜ of A is a finite A-module, see e.g. [Nag58, Appendix 1, Cor 2]. Then T → T˜ = Spec A˜ is finite and dominating, thus it is onto. Since k is algebraically closed, any point in the preimage of the special point is a k-point, thus Spec T˜ → Spec T satisfies assumptions of Lemma 3. Now A˜ is a one-dimensional normal Noetherian domain which is complete with respect to the m-adic filtration, see [Eis95, Thm 7.2a]. From the first part of the proof it follows that we may replace A˜ by a local ring (B, n, k) of A˜. Since B may be regarded as a quotient of A˜, see [Eis95, Cor 7.6], it is also m-adically complete. But mB is n-primary ideal, so that mB-adic and n-adic topologies on B agree. Then B is also n-adically complete. Summing up, the k-algebra B is a one-dimensional Noetherian normal local complete domain with quotient field k. Thus B is regular, so from the Cohen Structure Theorem, see [Eis95, Thm 7.7], it follows that B is isomorphic to k[[t]].

Now we recall the correspondence between smoothing of R and its connected components. This is a known folklore result, however we were unable to ﬁnd a reference for it.

Proposition 5. Let R = R1 t R2 t ... t Rk be a finite scheme. If (Zi,Ri) → (T, t) are abstract F smoothings of Ri over some base T , then Z = Zi → T is an abstract smoothing of R. Conversely, let (Z,R) → (T, t) be an abstract smoothing of R over T = Spec A, where A = (A, m, k) is a local complete k-algebra. Then Z = Z1 t ... t Zk, where (Zi,Ri) → (T, t) is an abstract smoothing of Ri. F Proof. The first claim is clear, since we may check that Z = Zi is flat and finite locally on F connected components of Z. Let η be the generic point of T , then Zη = (Zi)η is smooth over η since (Zi)η are all smooth. For the second part, note that Z is affine. Let Z = Spec B, then B is a finite A-module. Since A is complete Noetherian k-algebra, by [Eis95, Thm 7.2a, Cor 7.6] we get that B =

Bn1 × ... × Bnn , where n1,..., nk are precisely the maximal ideals of B. In particular we get that

B/ni ' Bni /niBni . Note that Bni is a ﬂat A-module, as a localisation of B, and also a ﬁnite A- module, since it may be regarded as a quotient of B. Moreover Bni /mBni = Bni /niBni = B/ni.

2 Now, maximal ideals ni correspond bijectively to connected components Ri of R. If Spec B/ni =

Ri, then Zi := Spec Bni → Spec A = T is a smoothing of Ri.

Corollary 6. Let R = R1 t R2 t ... t Rk be a ﬁnite scheme. Then R is abstractly smoothable if and only if each Ri is abstractly smoothable.

Proof. If each Ri is abstractly smoothable, then we may choose smoothings over the same base T , then the claim follows from Proposition 5. Conversely, if R is smoothable, then we may choose a smoothing over a one-dimensional Noetherian complete local domain by Corollary 4. Now, the results follows from Proposition 5.

Now we will compare the notion of abstract smoothability and embedded smoothability of a scheme R. First, we need a technical lemma.

Lemma 7. Let (A, m, k) be a local k-algebra and T = Spec A with a k-point t = m. Let Z → T be a ﬁnite ﬂat T -scheme with a unique closed point. Let Y → k be a separated k-scheme and f : Z → Y × T be a morphism of T -schemes. If ft : Zt → Y is a closed immersion, then f is also a closed immersion.

Proof. Since Y ×T → T is separated and Z → T is finite, from the cancellation property ([Har77, Ex II.4.8]) it follows that f : Z → Y × T is finite, in particular proper, thus the image of Z in Y × T is closed. Then it is enough to prove that Z → Y × T is a locally closed immersion. Let U ⊆ Y be an open affine neighbourhood of ft(p), where p is the unique closed point of Z. Since the preimage of U × T in Z is open and contains p, the morphism Z → Y × T factors through U × T . We claim that Z → U × T is a closed immersion. Note that it is a morphism of affine schemes. Let B, C denote the global sections of Z and U × T , then the above morphism corresponds to a morphism of A-algebras C → B. Since the base change A → A/m induces an isomorphism C/mC → B/mB, we have B = mB + C, thus C → B is onto by Nakayama Lemma and the fact that B is a finite A-module. Thus f : Z → U × T → Y × T is a locally closed immersion.

The following Theorem 8 together with it immediate Corollary 9 is a generalisation of [CN09, Lem 2.2] and [BB14, Prop 2.1]. Similar ideas are mentioned in [CEVV09, Lem 4.1]. The theorem uses the notion of formal smoothness, see [Gro67, Def 17.1.1], which we recall briefly. A k-scheme X is formally smooth if and only if for every affine k-scheme Y and every closed subscheme Y0 defined by a nilpotent ideal of Y , every morphism Y0 → X extends to a morphism Y → X.A k-scheme is smooth (in the sense of e.g. [Har77, III.10]) if and only if it is locally of finite type and formally smooth, see [Gro67, 17.3].

Theorem 8 (Abstract smoothing vs embedded smoothing). Let R be a ﬁnite scheme over k which is embedded into a formally smooth, separated scheme X. Then R is smoothable in X if and only if it is abstractly smoothable.

Proof. Clearly from definition, if R is smoothable in X, then it is abstractly smoothable. It remains to prove the other implication. Let us consider first the case when R is irreducible. Let (Z,R) → (T, t) be an abstract smoothing of R. Using Corollary 4 we may assume that T is a spectrum of a complete local ring (A, m, k). Since Z → T is finite, Z ' Spec B, where B is a finite A-algebra. In particular, B is complete in the m-adic topology, see e.g. [Eis95, Thm 7.2a], thus B is the inverse limit of n Artinian k-algebras B/m , where n ∈ N. By definition of X being formally smooth, the morphism R = Spec B/m → X lifts to n Spec B/m → X for every n ∈ N. Together these morphisms give a morphism Z = Spec B → X,

3 which in turn gives rise to a morphism of T -schemes Z → X × T . This morphism is a closed immersion by Lemma 7. This finishes the proof in the case of irreducible R. Now consider a not necessarily irreducible R. Let R = R1 t ... t Rk be the decomposition into irreducible (or connected) components. By Proposition 5, the smoothing Z decomposes as Z = Z1t...tZk, where (Zi,Ri) → (T, t) are smoothings of Ri. The schemes Ri are irreducible, so by the previous case, these smoothings give rise to embedded smoothings Zi ⊆ X ×T . Moreover, the images of closed points of Zi are pairwise different in X × T , thus we get an embedding of Z = Z1 t ... t Zk ⊆ X × T , which is the required embedded smoothing. Corollary 9. Suppose that R is a finite scheme and X and Y are two smooth, separated k- schemes. If R can be embedded in X and in Y , then R is smoothable in X if and only if R is smoothable in Y .

If we are given a morphism between schemes, we may drop the “formally smooth” hypothesis of Theorem 8, as the following Corollary 10 shows.

Corollary 10. Let R be a ﬁnite scheme embedded in X and smoothable in X. Let Y be a separated scheme with a morphism X → Y which induces an isomorphism of R which its scheme- theoretic image S ⊆ Y . Then R ' S is smoothable in Y .

Proof. Let Z ⊆ X × T → T be an embedded smoothing of X over a base (T, t). The morphism X → Y induces a morphism Z → Y ×T which, over t, induces a closed embedding R ⊆ Y . Then we need to prove that Z → Y × T is a closed immersion. By Corollary 4 and Proposition 5 we may reduce to the case when R is irreducible. Then the claim follows from Lemma 7.

Using Corollary 10 we may strengthen Corollary 9 a bit, obtaining a direct generalisation of [BB14, Prop 2.1].

Corollary 11. Let X be a ﬁnite type scheme and R ⊆ X be a ﬁnite closed subscheme of X, supported in the smooth locus of X. If R is abstractly smoothable, then R is smoothable in X.

Proof. Let Xsm be the smooth locus of X. By Theorem 8 R is smoothable in Xsm and by Corollary 10 is it also smoothable in X.

Acknowledgements

The authors thank Dustin Cartwright for pointing them to [Eis95, Thm 7.2a, Cor 7.6].

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