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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 2 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 2 U. If U = fηg, 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 < 1, 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.
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