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Desingularization of Quasi-Excellent Schemes

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Desingularization of Quasi-Excellent Schemes DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES M. TEMKIN 1. Introduction In 1964 Hironaka published a famous work [Hir], in which he in partic- ularly proved that any integral scheme of finite type over a field of charac- teristic zero may be desingularized by modifying only its singular locus. A question to what extent the theorem of Hironaka may be generalized was asked by Grothendieck in [EGA IV]. Grothendieck introduced the notions of excellent and quasi-excellent schemes (the word ”quasi-excellent” was in- troduced later) and proved that if k is a locally noetherian scheme such that any integral k-scheme of finite type admits a resolution of singularities, then k is quasi-excellent, see loc. cit., 7.9.5. Then Grothendieck remarks, see loc. cit., 7.9.6, that it may happen that the converse is also true, and, indeed, it is a common belief now that noetherian excellent schemes admit resolu- tion of singularities. In the same remark Grothendieck claims that: (i) the approach of Hironaka allows to desingularize any noetherian quasi-excellent scheme of characteristic zero, and (ii) resolution of singularities of a quasi- excellent scheme reduces to the case of Spec(A), where A is a complete local ring. Unfortunately, we can only guess what was the reasoning beyond these claims. Perhaps, Grothendieck meant in his first claim, that any algorithm for canonical desingularization of varieties should work in the case of an arbi- trary noetherian quasi-excellent scheme. Note, however, that the original algorithm of Hironaka is very subtle, mainly due to the lack of canonicity. In the last decade a large progress on resolution of singularities over a field of characteristic zero was made, nevertheless, the problem of desingularization of excellent schemes of characteristic zero remained open. Many desingu- larization algorithms were found, see [BM], [Vil], [Wl]. The new algorithms are simpler and more functorial (in particular they are canonical), though they all grew from Hironaka’s algorithm and are based on Hironaka’s ideas. Probably, some new algorithms work well for any noetherian quasi-compact scheme, but it is not so easy to prove it. For example, I heard an opinion that it is the situation with the algorithm of Bierstone and Milman. The main aim of this paper is to prove that indeed, any integral noetherian quasi-excellent scheme X of characteristic zero admits a desingularization X0, in particular, a noetherian scheme k is quasi-excellent iff any integral k-scheme of finite type admits a desingularization. Perhaps Grothendieck’s second remark and our method are based on similar ideas. It is not clear to 1 2 M. TEMKIN me how one can reduce desingularization to the case of X = Spec(A) with a complete local ring A, but we will see that it suffices to desingularize those blow ups X0 of Spec(A) which are regular outside of the preimage of the closed point. Also, we will prove a more general embedded desingularization theorem which states in addition, that if Z is a closed subset in X, then one can achieve that its preimage in X0 is a strictly normal crossing divisor, and X0 ! X is a blow up whose center S is such for any point x 2 S either X is singular at x or Z is not an snc divisor at x. An interesting feature of our proof is that it uses Hironaka’s theorem as a black box, and does not need to know how the desingularization of varieties is obtained. Also, we define quasi-excellent formal schemes and prove that in the characteristic zero case there is embedded resolution of singularities of rig-regular quasi- paracompact quasi-excellent formal schemes. 1.1. Overview. A basic idea of our method is to pass from a quasi-excellent scheme X to its formal completion X along a closed subscheme Z containing the singular locus and isomorphic to a scheme of finite type over a field k Then a desingularization of X may be obtained from a desingularization of X, and if X admits an algebraization over k, then it may be desingularized by use of Hironaka’s theorem. For example, one can desingularize isolated singularities, because any complete local ring with an isolated singularity is algebraizable due to Artin, see [Art], 3.8. The results of Artin were generalized by Elkik in [Elk], in particular, she proved that any affine rig- smooth formal scheme of finite type over a complete ring is algebraizable. Since rig-smoothness is equivalent to rig-regularity in the characteristic zero case, this method may be applied to any quasi-excellent scheme X whose singular locus Xsing is of finite type over a field k of characteristic zero. There are two difficulties one has to overcome: the singular locus may have a complicated global structure, and general rig-regular formal schemes may be not algebraizable. We propose the following approach, which solves both problems. For simplicity, we describe it in the case of a noetherian scheme X of dimension d < 1. Given a modification f : X0 ! X, we say 0 0 that f desingularizes X up to codimension d, if for any point x 2 Xsing, the codimension of f(x0) is at least d. Using induction on n = dim(X), one can reduce the general desingularization problem to the case, when X 0 0 admits a desingularization X ! X up to codimension n. But then Xsing is mapped to a finite set of closed points of X, in particular it is a union of k(xi)-schemes of finite type. Usually, desingularizing X up to codimension dim(X) does not help to desingularize X, because it does not improve the local structure of singularities (for example X0, may have singularities in any positive codimension). However, we see that the global structure of the singular locus does improve, and it suffices for our needs. Let us say few words about the induction process. In general, we use noetherian induction on an open set U, such that f ¡1(U) is regular. Let x 0 0 be a generic point of X nU, then the singular locus of S = Spec(OX;x)£X X DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 3 ¡1 0 is contained in f (x), hence Ssing is of finite k(x)-type. As we explained above, desingularization of such S0 reduces to desingularization of varieties 0 00 0 via completion and algebraization. We find a desingularization fS : S ! S , 0 0 which is a blow up with center supported on Ssing. Then fS may be extended to a blow up f 0 : X00 ! X0, such that X00 is isomorphic to X0 over U, and 00 00 the preimage of x does not intersects Xsing. Hence X ! X desingularizes X over an open subset W containing U and x. Now, let us describe briefly the content of the paper. In section 2.1, we study extensions of ideals and blow ups, and introduce the notion of formal blow up (along any ideal). In next section we fix our desingularization terminology and prove a useful criterion 2.2.9, which says when there is resolution of singularities over a scheme k. It turns out, that it suffices to desingularize any scheme S0 obtained by blowing up a local k-scheme S of essentially finite type. We study formal schemes in chapter 3. In its first section we define the notion of (rig-) regularity for a wide class of formal schemes. In next two sections we study algebraization of formal schemes. Finally, in section 3.4, we prove that certain formal schemes may be desingularized, see theorem 3.4.1, and deduce that any noetherian quasi- excellent scheme X of characteristic zero admits a desingularization, see theorem 3.4.3. The author expects that the algebraization theorem of Elkik admits a generalization, which also treats rig-snc divisors. The current version of the theorem allows to desingularize a divisor Z on a quasi-excellent scheme X only in the case when Z is of finite type over a field. To treat the case of an arbitrary Z, we have to study in the last chapter the behavior of strict transform Z0 of Z under a blow up f : X0 ! X. We prove in proposition 4.2.1, that it is possible to achieve that f ¡1(Z) is an snc divisor in a neighborhood of Z0. Combining this proposition with the results of chapter 3, we are able to prove that any noetherian quasi-excellent pair (X; Z) of characteristic zero may be desingularized, see theorem 4.3.2. We end the paper with analogous theorem for formal rig-smooth schemes, see theorem 4.3.4. 1.2. Some questions. The following natural questions are not studied in this paper, and they are worth an additional research. Hironaka proves that one can achieve desingularization by composing blow ups with regular centers. Is it possible to deduce that the same is true for an arbitrary quasi-excellent scheme of characteristic zero? Perhaps our method can do the job, but the induction process should become more involved. Hironaka considers desingularization of irreducible non-reduced schemes. Can one generalize his theorem to quasi-excellent schemes? It seems that our method should work here. Another question is how one can define desingularization of reducible schemes? See remark 2.2.11. 4 M. TEMKIN We prove in corollary 4.3.3, that blow ups of affine quasi-excellent formal schemes of characteristic zero may be desingularized, obtaining, in particu- lar, a local desingularization of formal schemes. However, we success to de- duce global desingularization only in the case of rig-regular formal schemes.
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