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 characteristic 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 introduced 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 resolution 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 arbitrary 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 desingularization 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 ∈ 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 < ∞. Given a modification f : X0 → X, we say 0 0 that f desingularizes X up to codimension d, if for any point x ∈ 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 \U, 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 particular, a local desingularization of formal schemes. However, we success to deduce global desingularization only in the case of rig-regular formal schemes. Can one drop the rig-regularity assumption? It is natural to expect that the answer is positive, but it seems that our method cannot manage the problem, and a more promising way is to use some kind of canonical desingularization. Finally, there is a natural question if one can prove by our method, that desingularization of arbitrary schemes (resp. of characteristic p) follows from desingularization of schemes of finite type over discrete valuation rings (resp. over fields of characteristic p). The author thinks that it is possible, but one has to strengthen Elkik’s results. In characteristic p case, one has to (locally) algebraize a rig-regular formal scheme X of finite type over a complete discrete valuation ring (perhaps, one has to blow X up along an open ideal before the algebraization). In mixed characteristic, one has to consider a more general type of formal schemes, see question 3.3.3.

1.3. Notions and notations. If X is a scheme with a closed subscheme Z, then |Z| denotes the support of Z, i.e. the underlying set of Z considered as a closed subset of X. The support of an ideal I ⊂ OX is the support of the associated closed subscheme Z = Spec(OX /I). We shall pass freely from reduced closed subschemes to closed subsets and vice versa, but we shall use different notations: we write Z ⊂ Z0, Z ∩ Z0, Z \ Z0 := Z − (Z ∩ Z0) and f −1(Z) (where f : X0 → X is a morphism) when working with subsets, and 0 0 0 we write Z,→ Z , Z ×X Z and Z ×X X when working with subschemes. By Xc, X≤c, etc., we denote the sets of points of codimension c, of codimension at most c, etc. In particular, X0 is the set of generic points of X. Recall that a noetherian ring A is called quasi-excellent if for any prime ideal p ⊂ A, the completion morphism φ : Ap → Abp is regular (i.e. φ is flat and has geometrically regular fibers) and for any finitely generated A-algebra B, the regular locus of Spec(B) is open. A universally catenary quasi- excellent ring is called excellent. A scheme X is called (quasi-) excellent if it admits an open covering by spectra of (quasi-) excellent rings. If A is a ring with an ideal P and an A-ring B, then by P -adic completion BbP of B we mean the separated completion of B in PB-adic topology. We 0 say that B is P -adic if Bf→BbP . Similarly, if X → X is a morphism of b0 schemes and I ⊂ OX is an ideal, then XI denotes the I-adic completion of 0 0 X , i.e. the completion of X along Spec(OX0 /IOX0 ). We refer to [EGA I], section 1.10, for basic properties of formal schemes. Any formal scheme in this paper is automatically assumed to be locally noetherian. Moreover, if not said to the contrary, any formal scheme considered in this paper is assumed to be noetherian. Given a locally noetherian formal scheme X, its special fiber Xs is defined as Spec(OX/P), where P is the maximal ideal of definition. Recall, that Xs is a reduced closed (formal) DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 5 subscheme of X, homeomorphic to it as a topological space. If X is a formal scheme of finite type over a complete discrete valuation ring, then one can rig an ad attach to X a generic fiber Xη (resp. Xη , resp. Xη ), which is a rigid (resp. analytic, reps. adic) space, see [BGR], ch. 9, [Ber1] or [Hub]. Only minimal familiarity with classical rigid spaces is required in this paper, however, the author expects that analytic or adic spaces may be useful in attacking question 3.3.3. Note also, that adic and generalized rigid generic fibers are defined for arbitrary noetherian formal schemes, see [Hub] or [BL1], section 5.

2. Local criterion Our main aim in this chapter is to reduce the problem of desingularization of a quasi-excellent scheme X to a local problem. We will prove in proposition 2.2.9, that it suffices to desingularize blow ups of local schemes. 2.1. Ideals and blow ups. Let us consider the following situation, which is a particular case of the situation considered in [EGA IV], 8.2.13. Assume that X is a noetherian scheme and Sα is a filtered family of open subschemes, such that the transition morphisms Sβ → Sα are affine. Then there exists an X-scheme S = proj limα Sα, the structure morphism i : S → X maps S homeomorphically onto its image, and OS is isomorphic to the restriction of OX on i(S). It follows that S is noetherian too. We will identify S with (i(S), OX |i(S)) and say that it is a pro-open subscheme of X. A typical example is obtained when Sα are affine neighborhoods of a point x, then Sf→Spec(Ox). Another example is obtained from this one by base change with respect to a morphism X0 → X. Lemma 2.1.1. Keep the above notations and assume that we are given an ideal IS ⊂ OS. Let ZS denote the support of IS and Z be its Zariski closure in X. Then there exists an ideal I ⊂ OX , such that I|S = IS and the support of I is Z. Proof. Since S is noetherian, there is a one-to-one correspondence between ideals in OS, closed subschemes and closed subschemes of finite presentation. For a scheme Y , let S(Y ) denote the set of closed subschemes of Y , then inj limα S(Sα)f→S(S) by [EGA IV], 8.6.3. Furthermore, the map S(X) → S(S) is surjective, because the map S(X) → S(Sα) is surjective for any 0 0 α by [EGA I], 6.9.7. Hence IS = I |S for some I ⊂ OX , and we the first claim of the lemma is satisfied. Let Z0 be the support of I0, then its intersection with S coincides with 0 ZS, and ZS is closed under generalizations in Z , because S is closed under generalizations in X. It follows that the set Z00 of generic points of Z0 is a 0 000 0 00 union of ZS and a finite set Z disjoint with ZS. Hence Z = Z ∪Z , where Z00 is a closed set disjoint with S. To finish the proof, we have to ”correct” I0 over Z00. Notice that U = X \Z00 is a neighborhood of S, and the support 0 00 of IU = I |U coincides with Z ∩ U. Set Y = X \ (Z ∩ Z ), then IU admits 6 M. TEMKIN a trivial extension to an ideal IY ⊂ OY (i.e. IY is trivial at all points of Y \ U), because Z ∩ U is closed in Y . Finally, let I be any extension of IY to X (it exists by [EGA I], 6.9.7), then the support of I is contained in (Z ∩ U) ∪ (Z ∩ Z00) = Z, as required. ¤

Next we discuss sheaves of ideals on a formal scheme X. Let I ⊂ OX be an ideal with the associated closed formal subscheme Z. It is not clear, how to define the reduction of Z in general, so we are forced to give the following definition. Given two ideals I, J, we say that the support of I is contained in the support of J if Jn ⊂ I for some n (if one takes into account the generic fiber of X, then it is possible to define supports set-theoretically). In general, one cannot extend to X an ideal defined on an open formal subscheme, and one cannot algebraize an ideal on X, when X = XbZ is the formal completion of a scheme X along a closed subscheme Z. The situation with open ideals is better. Any open ideal I defines a closed formal subscheme Z = Spf(OX/I), which is a scheme (i.e. its ideal of definition is nilpotent) supported on Xs. Actually, Z is a closed subscheme of some Spec(X/Pn). In particular, we may and do define the support of I as a closed subset of X. If X = XbZ , then the completion induces a bijective correspondence between ideals I ⊂ OX supported on |Z| and open ideals I ⊂ OX. In other words, open ideals are algebraizable. Lemma 2.1.2. Let X be a noetherian formal scheme with an open formal subscheme Y and a closed subset Z ⊂ X. Then any open ideal I ⊂ OY with support in Z ∩ Y extends to an open ideal J ⊂ OX with support in Z. Proof. Consider the open formal subscheme Y0 = Y ∪ (X \ Z). The ideal 0 I may be extended trivially to an open ideal I ⊂ OY0 . Now it suffices to 0 find an arbitrary extension of I to an open ideal J ⊂ OX. Choose an ideal 0 0 of definition P ⊂ OX, such that P = P|Y0 is contained in I , and consider 0 0 0 0 0 the schemes X = (Xs, OX/P), Y = (Ys, OY0 /P ) and the ideal I = I OY 0 . 0 Then I extends to an ideal J ⊂ OX by [EGA I], 6.9.7, and the preimage J ⊂ OX of J is a required extension. ¤ Next, we recall basic facts about blow ups, see [Con1], section 1, for more details. If X is a scheme with a finitely generated ideal I ⊂ OX , then the 0 2 X-scheme X = Proj(OX ⊕ I ⊕ I ⊕ ... ) is X-projective and the structure 0 morphism X → X is an isomorphism over X \ Spec(OX /I). The pair 0 (X , I) is called a blow up of X along I and is denoted BlI(X). The ideal 0 IOX0 is invertible, and X is the final object in the category of X-schemes such that the preimage of I is invertible. The construction of blow ups commutes with localizations (and, more generally, with flat base changes). 0 0 I If X = Spec(A), then X is glued from schemes Xg = Spec(A[ g ]) with g ∈ I, I 0 where A[ g ] is the subring of Ag generated by I/g. Usually the schemes Xg are called charts of the blow up. As usual, we will omit the ideal in the notation of a blow up, and say simply ”a blow up f : X0 → X”, or even ”a blow up X0 of X”, however, DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 7 we will take the ideal into account in definitions of V -admissibility and 0 strict transforms. Note that the X-scheme X = BlI(X) may be obtained by blowing up other ideals, for example BlIn (X)f→BlI (X) for any n > 0. 0 Sometimes we will say that X is a blow up of X along Y = Spec(OX /I), 0 and Y is the center of the blow up, and write X = BlY (X). For any open subscheme V , the blow up X0 is called V -admissible, if its center is disjoint with V . Sometimes it will be more convenient to express the same property in terms of closed sets, so we say that f is T -supported for a closed subscheme (or subset) T,→ X if |Y | ⊂ |T | (i.e. f is (X \ T )-admissible). More generally, given a morphism g : X → S, a closed subscheme R,→ S and open subscheme U = S \ R, we say that f is U-admissible (or R-supported) if f is g−1(U)-admissible. If X is quasi-compact and quasi-separated, then a composition of V -admissible (or T -supported) blow ups is a V -admissible (or T -supported) blow up. A simple and natural proof of this fact given in [RG], 5.1.4, is incomplete. We refer to [Con1], 1.2, where a complete and surprisingly involved version of the proof (due to Raynaud) is given. If V is an open subscheme of X such that f is an isomorphism over V , then it still may happen that f is not isomorphic to a V -admissible blow up. For example, it is the case when X = Spec(k[x, y, z, t]/(xy−zt)), V = Xreg is the complement of the origin s and I = (x, y) defines a Weil divisor which is 0 not Cartier. Then X = BlI (X) is a small resolution of X, and it cannot be obtained by blowing up an ideal supported on s, because the preimage of s is not a divisor, but a curve. Nevertheless, X0 is dominated by a V -admissible blow up. More generally, we will use the following lemma. Lemma 2.1.3. Let X be a quasi-compact quasi-separated scheme with a schematically dense open subscheme U and f : X0 → X be a U-modification, i.e. a proper morphism such that f −1(U) is schematically dense and isomorphic to U. Then there exists a U-admissible blow up X00 → X, which factors through X0. Proof. Apply the flattening theorem of Raynaud and Gruson, see [RG], 5.2.2, 0 0 to the morphism f : X → X and the sheaf OX0 (we set S = X, X = X and M = OX0 in the loc.cit.). By the theorem, there exists a U-admissible 0 blow up X → X, such that the following condition holds: let f : X → X denote the base change of f and F denote the strict transform of OX0 , then F is OX -flat. Notice that U is a schematically dense open subscheme of X,X0 and X, and let X00 be the schematic closure of the image of U under the diagonal 0 morphism U → X . Then X00 is the minimal U-modification of X which 0 00 0 dominates both X and X, and OX is isomorphic to the quotient of OX by the maximal submodule supported on the preimage of X \ U, i.e. OX00 f→F. Thus, g : X00 → X is a flat U-modification, and we will prove that it is an isomorphism. Since g is flat and an isomorphism over U, the fibers of g are discrete, i.e. g is quasi-finite. Since it is proper, g is finite. By flatness of g, g∗(OX00 ) is a locally free OX -sheaf. The rank of g∗(OX00 ) is 1 at any point 8 M. TEMKIN of X, because it is so on U. It follows that X00f→X, hence X00 is a required U-admissible blow up which dominates X0. ¤ Our last aim in this section is to introduce formal blow ups and study their basic properties. Let X be a noetherian formal scheme and I ⊂ OX be an ideal. If I is open, then a formal blow up BlI(X), which is usually called an admissible formal blow up, is defined in [BL1], section 2. We are going to extend the results of loc.cit. to the case of arbitrary ideals, because such blow ups will appear at the end of the paper. Until then, the results of [BL1], section 2, cover our needs, so the reader may consult loc.cit., instead of reading the rest of this section. Assume that X = Spf(A) is affine, and let I ⊂ A be the ideal corresponding to I and P ⊂ A be an ideal of definition. We define the formal blow up 0 0 X = Blb I (A) of X along I as the P -adic completion of X = BlI (Spec(A)). 0 0 I Since X is glued from affine charts Xg = Spec(A[ g ]) with g ∈ I, its com- 0 I I pletion is glued from affine formal schemes Xg = Spf(A{ g }), where A{ g } is I I the P -adic completion of A[ g ]. Let us give an explicit description of A{ g }. I First, we note that the homomorphism A{ g } → A{g} may be not injective (the target is empty when P ⊂ (g)), so it is of no use for us. From other side, I it is well known and easily seen that if I = (f1, . . . , fn), then A[ g ] may be de- 0 scribed as the quotient of the ring A = A[T1,...,Tn]/(gT1−f1, . . . , gTn−fn) by its g-torsion. Notice that the completion of A0 if isomorphic to Ab0 = I A{T1,...,Tn}/(gT1 − f1, . . . , gTn − fn). Since A[ g ] is noetherian, its com- I I pletion A{ g } is flat over it, in particular A{ g } has no g-torsion. It follows, I b0 that A{ g } is isomorphic to the quotient of A by its g-torsion. Lemma 2.1.4. Let A be a noetherian ring with ideals I and P , and Ab be the P -adic completion of A. Then the P -adic completion of BlI (A) is b b canonically isomorphic to BlIAb(A).

Proof. Let f1, . . . , fn be generators of I and g ∈ I be an element. We have to I b I prove that the P -adic completion of A[ g ] is canonically isomorphic to A{ g }. 0 Obviously, the P -adic completion of A = A[T1,...,Tn]/(gT1 −f1, . . . , gTn − 0 fn) is isomorphic to Ab = Ab{T1,...,Tn}/(gT1 − f1, . . . , gTn − fn). By the same flatness argument as above, the P -adic completion of A0/(g)-torsion is isomorphic to Ab0/(g)-torsion. ¤ It follows that formal blow ups of affine formal schemes are compati- b ble with formal localizations. If f ∈ A is an element, then BlIA{f} (X{f}) is isomorphic to the completion of BlIAf (Af ). Since usual blow ups are compatible with localizations, the latter is isomorphic to the completion of b (BlI (A))f , which in its turn is isomorphic to (BlI (X)){f}. It follows that for any locally noetherian formal scheme X with ideal I ⊂ OX, one can define the formal blow up Blb I(X) by gluing formal blow ups of open affine formal DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 9

0 subschemes of X. We say that a formal blow up X = Blb I(X) → X is J- supported if the support of I is contained in the support of J. Furthermore, if I is open, then its support lies in a closed subset T ⊂ Xs and we say that X0 is T -supported.

Lemma 2.1.5. Let X be a noetherian scheme with two ideals I, P ⊂ OX , X be the P-adic completion of X and I = IOX. Then the P-adic completion of BlI (X) is canonically isomorphic to Blb I(X). Proof. Both formal completion and blow up along a closed subscheme are defined locally on X, so it suffice to consider the affine case, which was established in the previous lemma. ¤

Let X be a noetherian formal scheme with a closed formal subscheme T. If T is a scheme, then Raynaud proved, that a composition of T-supported formal blow ups is isomorphic to a T-supported formal blow up, see [BL1], 2.5. The same is true for general formal blow ups. Let I ⊂ OX be a T- 0 supported ideal with formal blow up fb : X = Blb I(X) → X and J ⊂ OX0 be 0 0 00 0 0 a T ×X X -supported ideal with formal blow up fb : X = Blb J(X ) → X .

Lemma 2.1.6. Keep the above notations, then X00 is X-isomorphic to a T-supported blow up.

0 0n m Proof. Set I = IOX0 . Let m, n be natural numbers, then (fb∗(I J)) is an ideal in the OX-algebra fb∗(OX0 ) and its preimage in OX is an ideal L. We 00 will prove that for n > n0 and m > m0(n), X → X is isomorphic to the blow up of X along LI. The latter statement may be checked locally on X, because X is noetherian. So, we may assume that X = Spf(A) for a P -adic ring A. Let 0 I ⊂ A denote the ideal corresponding to I, X = Spec(A), X = BlI (X) and f : X0 → X be the blow up morphism. Since X0 is X-proper and X0 is isomorphic to the P -adic completion of X, we can apply Grothendieck’s existence theorem, see [EGA III], theorem 5.1.4 and corollary 5.1.8, to find an algebraization J ⊂ OX0 of J. Set T = Spec(A/m), where m is such that T = Spf(A/m); then I is supported on T and J is supported on the 0 00 0 preimage of T in X . By [Con1], 1.2, X = BlJ (X ) is isomorphic to a T -supported blow up of X. Since X00 is isomorphic to the P -adic completion of X00, we already obtain that X00 is isomorphic to a T-supported blow up of X. However, as we mentioned above, we have to describe the blow up explicitly. It will require a closer look on the proof of lemma 1.2 in loc.cit. 0 Set I = IOX0 . The proof in [Con1] starts with an observation that 0n ∗ for sufficiently large n and ideal M = I J , the map f f∗(M) → M is surjective. Then a sufficiently large submodule K ⊂ f∗(M) of finite type is chosen (X may be non-noetherian in loc.cit.). Since f∗(M) is coherent in our situation, one can actually choose K = f∗(M). Finally, one defines L ⊂ OX m in loc.cit. as the preimage of K under the homomorphism OX → f∗(OX0 ), 10 M. TEMKIN

00 and proves that for sufficiently large m, X f→BlIL(X). By lemma 2.1.5, 00 X f→Blb IL(X), so it remains to prove that L = L. Notice that fb∗(OX0 ) is isomorphic to the P -adic completion of f∗(OX ) by Grothendieck’s theorem on formal functions, see [EGA III], theorem 4.1.5, but f∗(OX ) coincides with its completion, because it is a finite A-algebra. By 0n 0n the same argument, fb∗(I J) and f∗(I J ) define the same ideal in f∗(OX ), and therefore L = L. ¤ 2.2. Desingularization. Definition 2.2.1. Let X be a noetherian scheme, and Z be either a closed subscheme or a closed subset. We say that Z is an snc divisor if X is regular in a neighborhood of Z and for any point x ∈ Z, there exists a regular sequence of parameters u , . . . , u ∈ O , such that in a neighborhood of 1 Q d X,x d ni x, Z is given by the equation i=1 ui = 0 with ni ∈ N. The abbreviation snc stands for strictly normal crossing. In the case of a closed subscheme, our definition differs from the standard one, because it allows non-reduced divisors. We say that Z is a Cartier divisor if it is locally given by a single regular element, i.e. for any point z ∈ Z, there exists a not zero divisor fz ∈ OX,z with OX,z/fzOX,zf→OZ,z. This condition is equivalent to requiring that the ideal I ⊂ OX defining Z is invertible. We omit the proof of the following easy lemma. Lemma 2.2.2. Assume that X is a regular scheme and Z is a closed subscheme, then Z is an snc divisor if and only if Z is a Cartier divisor and the underlying set of Z is an snc divisor in X. Definition 2.2.3. Let X and Z be as in the previous definition. The regular locus (X,Z)reg of the pair (X,Z) is the set of points x ∈ X, such that Ox is a regular ring and Z ×X Spec(Ox) is an snc divisor. The singular locus of the pair (X,Z) is defined as (X,Z)sing = X \ (X,Z)reg. We will be interested in the case when X is a noetherian integral scheme and Z is a closed subscheme not equal to X, let us call such a pair (X,Z) an integral pair. Lemma 2.2.4. Let X be a quasi-excellent scheme with a closed subscheme Z, then (X,Z)reg is open. If, furthermore, (X,Z) is an integral pair, then (X,Z)reg is non-empty. Proof. We assume that (X,Z) is an integral pair, the general case is proven 0 similarly. We may replace X by any neighborhood X of (X,Z)reg, because 0 0 (X ,Z ×X X )reg = (X,Z)reg. So, first of all replace X by Xreg. Next, removing from X all embedded components of Z and irreducible components of Z of codimension larger than 1 (these components lie in (X,Z)sing), we achieve that Z is a Cartier divisor. By the previous lemma, we may now replace Z by its reduction. Let {Zi}i∈I be the set of irreducible components of Z. For any J ⊂ I consider the scheme-theoretic intersection ZJ = ∩i∈J Zi DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 11

(we deﬁne Z∅ = X). It is well known that Z is snc at a point x ∈ X iﬀ each ZJ containing x is regular and of codimension |J| at x. Since ZJ is quasi- excellent, its singular locus is closed. Remove the closed set ∪J (ZJ )sing from X, then all subsets ZJ become regular. Finally, it remains to remove from X the irreducible components of ZJ ’s of wrong codimension. ¤

We see that (X,Z)sing is a closed subset, sometimes it will be convenient to consider it as a reduced closed subscheme. Lemma 2.2.5. Let X be a noetherian scheme with a closed subscheme Z 0 0 0 0 and f : X → X be a regular morphism. Then (X ,Z )singf→(X,Z)sing×X X , 0 0 where Z = Z ×X X . Proof. Since f is regular and the singular loci are reduced, it suffices to check −1 0 0 that f ((X,Z)sing) = (X ,Z )sing set-theoretically. By [Mat], 23.7, we have −1 0 f (Xsing) = Xsing, hence we may replace X by Xreg, and shrink other schemes accordingly. Then the lemma follows from the description of the singular locus of a pair, which was given in the previous lemma ((X,Z)sing is the union of embedded components of Z, irreducible components of Z of small dimension, etc.) ¤ Next we are going to fix our desingularization terminology. Definition 2.2.6. Given a noetherian scheme X with a closed subscheme 0 Z and an (X,Z)reg-admissible blow up f : X → X, we say that f desingu- −1 0 0 larizes the pair (X,Z) over a subset S ⊂ X if f (S) ⊂ (X ,Z ×X X )reg. If S = X (resp. S = X

We make few comments for the sake of completeness. If (X,Z)reg is empty, for example Z = X, then BlX (X) = ∅ is a desingularization of the pair (X,Z). We excluded such cases by our deﬁnition of integral pair. Note that embedded resolution of singularities allows to control the exceptional set of desingularizations. For example, a desingularization f : X0 → X of a pair (X,Xsing) provides a desingularization of X, such that the exceptional set E is an snc divisor (recall that E is the minimal closed set such that f 0 −1 is an open immersion on X \ E, hence E = f (Xsing) in our situation). Note also, that a weak desingularization f : X0 → X is usually deﬁned by 12 M. TEMKIN

0 −1 requiring only that f is a modiﬁcation and (X , f (Z))sing = ∅. Perhaps the lack of control on the exceptional set is the most important weakness of weak desingularization. The following statement will often be used implicitly. 0 Lemma 2.2.8. Let (X,Z) be as above, V = (X,Z)reg, f : X → X be a 0 0 0 00 0 V -admissible blow up, Z = Z ×X X and d be a number. If f : X → X desingularizes (X0,Z0) up to codimension d, then f 00 = f ◦ f 0 : X00 → X desingularizes (X,Z) up to codimension d. −1 0 0 0 Proof. Notice that V f→f (V ) ⊂ (X ,Z )reg, therefore f is V -admissible. 0 0 By [Con1], 1.2, the composition f ◦ f is V -admissible. Obviously Z ×X0 00 00 0−1 0

We will need the following lemma, where we do not pursue any generality. 14 M. TEMKIN

Lemma 2.2.10. Assume that there is embedded resolution of singularities over k. Then for any reduced k-scheme X of finite k-type with a closed subscheme Z, the pair (X,Z) admits a desingularization. Proof. If X is a disjoint union of integral schemes, then the lemma is trivial. Assume, this is not the case, and let X1 be an irreducible component of X and X2 be the union of all other components. Then S = X1 ×X X2 is supported on Xsing, and the blow up of X along S separates the preimages of Xi’s. Applying induction on the number of irreducible components, we 0 0 obtain that there exists an Xreg-admissible blow up X → X, such that X is a disjoint union of integral schemes. Then, it suffices to ”desingularize” 0 0 the pair (X ,Z ×X X ), and we are done. ¤ Remark 2.2.11. The lemma shows, that desingularization of pairs (X,Z) with reduced X reduces to the case of integral X (the same is true for any Z without embedded components). However, it is a stupid desingularization: it brutally separates irreducible components and kills non-reduced ones. If one wants to study not integral X’s deeper, then the regular locus (X,Z)reg should be replaced by a finer notion. For example, it seems natural to extend the notion of regularity including reduced strictly normal crossing schemes X or irreducible X’s, which are normally flat along Xred. 2.3. Embedded desingularization of varieties. In this section we deduce from Hironaka’s theorem that there is embedded resolution of singularities (in the sense of definition 2.2.7) over a field k of characteristic zero. It is done mainly for the sake of completeness, because the result is known to be true for experts. Perhaps, it is contained in one of the recent works on desingularization, but I failed to find it. Thus, our aim is to desingularize a pair (X,Z), where X is an integral k-scheme of finite type with a closed subscheme Z 6= X. Set V = (X,Z)reg. As we saw in the beginning of the proof of 2.2.9, the blow up BlZ (X) → X is dominated by a V -admissible blow up X0 → X, and replacing (X,Z) by 0 0 (X ,Z ×X X ), we achieve that Z is a Cartier divisor. By Main Theorem 0 I of [Hir], there exists an Xreg-admissible blow up X → X with regular 0 0 X. Replacing (X,Z) by (X ,Z ×X X ), we may assume that X is regular. In, particular it is now harmless to replace Z by its reduction. It follows easily from Main Theorem II of [Hir], that there is a Zsing-supported blow up 0 0 f : X → X, such that X is regular and Z ×X X is an snc divisor. However, we require slightly more in our definition, because we want f to preserve the whole snc locus of Z. We will overcome this difficulty by using proposition 4.2.1, which will be proven in sections 4.1 and 4.2. It cannot cause to a circular reasoning, because those sections use only results of sections 2.1 and 2.2. Proposition 2.3.1. There is embedded resolution of singularities over k. Proof. We have already shown that it suffices to desingularize a pair (X,Z) with regular X and a reduced divisor Z. By induction, we may assume that DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 15 there is embedded resolution of singularities over k up to dimension dim(X). We will desingularize (X,Z) by a sequence of blow ups. Let T denote the set we are allowed to modify. Clearly, we have to start with T = (X,Z)sing, but it will not be so in the sequel. Step 1. We may assume that Z = Y ∪ T , where Y is a divisor disjoint with (X,Z)sing. By proposition 4.2.1, there exists a T -supported blow up f : X0 → X with the following property: if Y 0 is the strict transform of 0 −1 0 0 0 0 0 0 Z and Z = f (Z), then Y ⊂ (X ,Z )reg. Notice that Z = Y ∪ T , where T 0 = f −1(T ), therefore X0,Y 0,Z0 and T 0 satisfy the claim of the step with the only possible exception: it may happen that X0 is singular. Let f 0 : X00 → X0 be a desingularization of X0, and set Y 00 = f 0−1(Y 0), Z00 = f 0−1(Z0) and T 00 = f 0−1(T 0). Notice that f 0 is a T -supported blow 0 0 0 up because Xsing ⊂ T . Also, f is an isomorphism over a neighborhood of 0 0 0 00 00 00 Y because Y ⊂ Xreg, hence Y ⊂ (X ,Z )reg. Therefore we may replace X0,Y 0,Z0 and T 0 by X00,Y 00,Z00 and T 00, which are as claimed. Step 2. Under assumptions of Step 1, (X,Z) admits a desingularization. Let J be the ideal defining T and d = ν(T ) = maxx∈T multx(J) be its maximal multiplicity at a point. Applying Main Theorem II of [Hir] to 0 0 E = Y , J and d, we obtain a blow up fd : X → X, such that X is regular 0 −1 0 and Z = fd (Y ∪ T ) is the union of an snc divisor E (equal to the union −1 0 −1 of the exceptional divisor and fd (E)) and a closed subset T ⊂ fd (T ) 0 with ν(T ) < d. Moreover, the center of fd is exactly the d-multiple locus of T , in particular, f is T -supported. Iterate this procedure until d = 0, then the composition f = f1 ◦ · · · ◦ fd is a T -supported blow up, such that f −1(Z) is an snc divisor. Since the source of f is regular, we actually obtain a desingularization of the pair (X,Z). ¤

3. Desingularization of special formal schemes 3.1. Quasi-excellent formal schemes and regularity conditions. If A is a quasi-excellent adic ring, then any formal localization homomorphism φf : A → A{f} is regular. Indeed, φf is a composition of the localization homomorphism A → Af and the completion homomorphism Af → A{f}, but regularity is preserved by compositions, the first homomorphism is obviously regular and the second one is regular by [EGA IV], 7.8.3 (v), because Af is quasi-excellent. It is natural to expect that the rings A{f} are quasi- excellent, but it seems to be an open question in general. Thus we are forced to give the following definition: a formal scheme X is called (quasi-) excellent if for any open affine subscheme Spf(A) the ring A is (quasi-) excellent. Many properties of quasi-excellent formal schemes may be defined and studied via their scheme analogs (compare to [Con2], section 1.2, and [Ber1], section 2.2, where one studies k-analytic spaces similarly). Consider a quasi- excellent affine formal scheme X = Spf(A) and a closed formal subscheme Z given by an ideal I, and set X = Spec(A) and Z = Spec(A/I). We define the singular locus (X, Z)sing of the pair to be the closed formal subscheme 16 M. TEMKIN defined by the ideal J ⊂ A which defines the closed subscheme (X,Z)sing of X. The following lemma shows that singular loci are compatible with formal localizations of quasi-excellent formal schemes.

Lemma 3.1.1. If f ∈ A is an element, X{f} = Spf(A{f}) and Z{f} = Z ×X X{f}, then (X{f}, Z{f})singf→(X, Z)sing ×X X{f}.

Proof. Since the homomorphism A → A{f} is regular by our assumption, the claim follows from lemma 2.2.5. ¤ The lemma implies that for any quasi-excellent formal scheme X and closed formal subscheme Z, there exists a uniquely defined closed formal 0 0 0 subscheme T = (X, Z)sing such that T ×X X f→(X , Z ×X X )sing for any open affine formal subscheme X0. We say that X is regular (resp. rig-regular) if Xsing is empty (resp. a closed subscheme of Xs). We say that Z is snc (resp. rig-snc) if the formal scheme Z ×X (X, Z)sing is empty (resp. a closed subscheme of Xs). The following lemma shows that singular loci are compatible with completions. Lemma 3.1.2. Let X be a quasi-excellent scheme with closed subschemes Z and T = (X,Z)sing, P ⊂ OX be an ideal and X, Z, T be the P-adic completions of X, Z, T . Assume that X is quasi-excellent. Then Tf→(X, Z)sing. Proof. Completions and singular loci are compatible with (formal) localizations, therefore we may assume that X = Spec(A), P corresponds to an ideal P ⊂ A, Z = Spec(A/I) and T = Spec(A/J) for ideals I,J ⊂ A. Obviously, X = Spf(Ab), where Ab is the P -adic completion of A. Since A is quasi-excellent, the homomorphism A → Ab is regular. By lemma 2.2.5, (Spec(Ab), Spec(A/Ib Ab))singf→Spec(A/Jb Ab), hence (X, Z)singf→Spf(A/Jb Ab)f→T. ¤ Corollary 3.1.3. Keep the situation of the lemma and consider the closed subscheme Y = Spec(OX /P) of X, then: (i) X is regular and Z is an snc divisor if and only if there exists a regular neighborhood U of Y such that Z ×X U is an snc divisor; (ii) X is rig-regular if and only if there exists a neighborhood U of Y such that U \ Y is regular. Similarly to regularity, one can define reducedness of quasi-excellent formal schemes and prove analogs of lemmas 3.1.1 and 3.1.2. We will need reducedness later. A further generalization (which will not be used) is given in the remark. Remark 3.1.4. (i) Similarly to regularity, if X is a quasi-excellent formal scheme and P is one of the following standard properties: Rn, CI, Gor, Sn (or their combinations like Reg, Nor, CM, Red), then one can define the non-P locus Xnon−P as a closed subscheme of X. These loci satisfy analogs of lemmas 3.1.1 and 3.1.2. If X is as in [Con2], 1.2.1, then the intersection of Xnon−P with Xs is the non-P locus in the sense of [Con2], 1.2, but the DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 17 whole Xnon−P contains more information; in particular, it remembers the non-P locus of the generic fiber. (ii) Intuitively, the rig-regularity condition means that a formal scheme X is regular outside of its special fiber Xs. If X is of finite type over a complete DVR, then one can show that X is rig-regular iff its generic fiber Xη is a regular rigid (analytic or adic) space, i.e. the local rings of Xη are regular. Probably, the same is true for an arbitrary quasi-excellent formal scheme X and its adic generic fiber. 3.2. Special formal schemes. Throughout this section k denotes a field and p = char(k). If p is positive, then a discrete valuation ring K◦ is called a p-ring if p generates the maximal ideal of K◦. It is shown in [Mat], ch. 29, that up to an isomorphism, there exists a unique complete p-ring with residue field k; it will be denoted Zp(k). Let A be a P -adically complete noetherian ring and S = Spf(A). Recall, that a formal S-scheme X is of finite type if it admits a finite covering by formal schemes of the form Spf(A{T1,...,Tn}/I). More generally, by an A-special ring we mean a topological ring A{T1,...,Tn}[[R1,...,Rm]]/I provided with the Q-adic topology, where Q is generated by the images of P and R1,...,Rm. A formal S-scheme X is called S-special if it admits a finite open covering by formal spectra of A-special rings (compare to [Ber3]). A noetherian formal scheme is called special if its special fiber is a scheme of finite type over a field. In the sequel, X denotes a special formal scheme such that Xs is of finite k-type. We say that Xs is equicharacteristic if pOX = 0. Proposition 3.2.1. Assume that X is affine. (i) If X is equicharacteristic, then it is isomorphic to a k-special formal scheme. (ii) If p > 0, then X is isomorphic to a Zp(k)-special formal scheme. Proof. Let X = Spf(B) and P be the maximal ideal of definition of B, so B/P is a finitely generated k-algebra. If B is equicharacteristic (in particular, it is automatically the case when p = 0), then we set K◦ = k and ◦ ◦ ◦ L = Fp. Otherwise we set K = Zp(k) and L = Zp. We have natural homomorphisms K◦ → k → B/P and L◦ → B, and the same argument as in the proof of Cohen’s structure theorem given in [Mat], 29.2, shows that there is a lifting K◦ → B. Indeed, it is shown in the proof of loc.cit., that K◦ is formally smooth over L◦, hence the L◦-homomorphism K◦ → B/P lifts to B/P 2, B/P 3, etc. Since B is P -adically complete, we obtain a lifting K◦ → B. Let r1, . . . , rm be generators of P and t1, . . . , tn ∈ B be such that their images in B/P generate it over k. Then there exists a continuous homo- ◦ morphism φ : C = K {T1,...,Tn}[[R1,...,Rm]] → B, which takes Ti and Rj to ti and rj. The maximal ideal of definition Q ⊂ C is generated by ◦ the maximal ideal of K and the elements Rj, hence C/Qf→k[T1,...,Tn] and the induced homomorphism C/Q → B/P is surjective. The image of 18 M. TEMKIN

Q in B generates P , hence B is of topologically finite C-type by [EGA I], 0.7.5.5 (a). Moreover, the same argument as in the proof of loc.cit. implies that the homomorphism C → B is actually surjective, because C/Q maps surjectively onto B/P . So, X is K◦-special, as required. ¤ Any K◦-special formal scheme is excellent by results of Valabrega, see [Val1], prop. 7, and [Val2], th. 9. We obtain the following corollary, which allows to apply results of the previous section to special formal schemes. Corollary 3.2.2. Any special formal scheme is excellent. The following statement is proven exactly as its analog 3.2.1. Proposition 3.2.3. Assume that X = Spf(B), B possesses a principal ideal of definition πB, and either B is equicharacteristic or π = p. Set K◦ = k if π is nilpotent, set K◦ = k[[π]] if π is not nilpotent and B is equicharacteristic, ◦ ◦ and set K = Zp(k) otherwise. Then X is isomorphic to a formal K -scheme of finite type. Remark 3.2.4. There exist special formal schemes with principal ideal of definition not covered by the proposition. For example, if B equals to Zp{T }[[R]]/(RT − p), then RB is an ideal of definition, but pB is not. 3.3. Rig-smoothness and algebraization in characteristic zero. Let O be a ring. We say that X is O-algebraizable, if it is isomorphic to the formal completion of an O-scheme of finite type along a closed subscheme. We say that X is locally O-algebraizable if it may be covered by open O- algebraizable formal subschemes. Fix the following notations: K is a complete discretely valued field with ring of integers K◦, residue field k and maximal ideal (π), S = Spf(K◦), X is a K◦-special formal scheme, L ⊂ K is a dense subfield, O = L ∩ K◦ and S = Spec(O). Proposition 3.3.1. Assume that char(K) = 0. If X is rig-regular affine and of finite S-type, then it is O-algebraizable. Proof. We have X = Spf(C) with C of topologically finite K◦-type; set X = Spec(C). Consider the K-affinoid algebra CK = C ⊗K◦ K, it is regular because Spec(CK ) = X \ Xs (Xs is the closed subset of X given by the rig condition π = 0). The regular K-affinoid space X = Sp(CK ) is Sp(K)- smooth because K is perfect, see [BLR], 2.8 (b), or an explanation in the proof of [Con2], 3.3.1. It is well known (and will be proven in proposition 3.3.2 below because of lack of an appropriate reference) that Xrig is K- smooth iff X is formally K◦-smooth outside of the open locus in the sense of [Elk], Th. 7. Applying this theorem of Elkik, we obtain that C is isomorphic to the π-adic completion of a finitely generated Oh-algebra A, where Oh is the henselization of O. Since Oh is a union of etale O-algebras, A is 0 h 0 0 isomorphic to an algebra A ⊗O0 O , where O is O-etale and A is a finitely generated O0-algebra. Then it is clear, that Spec(A0) is an O-algebraization of X. ¤ DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 19

Let A be a noetherian P -adic ring, with P -adic rings B = A{T1,...,Tn} and C = B/(f1, . . . , fl) of topologically finite A-type. Let us define a topological Jacobian ideal H similarly to its algebraic analog from [Elk], 0.2. ³ ´ C/A Set ∆ = ∂fi . For any subset L ⊂ {1, . . . , l} with |L| = r, let ∂Tj 1≤i≤l,1≤j≤n HL ⊂ B be the ideal generated by the determinants of r × r-minors of ∆ whose rows are numbered by the elements of L. Also, let JL be the ideal generated by fi’s with i ∈ L and J = (f1, . . . , fl). Then we set s X HC/A = (JL : J)HLC, L⊂{1,...,l} where (JL : J) = {x ∈ JL|xJ ⊂ JL}. Notice that a priori HC/A depends on the choices of B and fi’s. A standard argument using the Jacobian criterion of smoothness, shows that in the polynomial case (i.e. P is nilpotent) HC/A defines the not A-smooth locus of Spec(C), in particular it is independent of all choices. We refer to [Spi], 2.13, b1 b1 for details. A similar argument involving modules ΩB/Af→⊕BdTj and ΩC/A of continuous differentials shows that Spf(C) is Spf(A)-smooth iff HC/A = C (a definition of smooth morphisms of formal schemes may be found in [BL2], 1.1). In [Elk], Th 7, Spf(C) is said to be formally A-smooth outside of the open locus if the Jacobian ideal HC/A is open. Using the Jacobian criterion of rig-smoothness, see [BLR], 3.5, one can show that it happens iff Spf(C) is rig-smooth over A. For the sake of simplicity, we consider only the classical rigid case which was used in the previous proposition. Our proof is an affinoid adjustment of the proof in [Spi], 2.13.

Proposition 3.3.2. Keep the above notations, assume that A is of ﬁnite ◦ K -type and set A = A ⊗K◦ K and C = C ⊗K◦ K. Then the morphism Sp(C) → Sp(A) of rigid aﬃnoid spaces is smooth if and only if the Jacobian ideal HC/A is open.

Proof. Obviously, B/(f1, . . . , fl)f→C, where B = A{T1,...,Tn}. Use this representation of C to define the Jacobian ideal HC/A of affinoid algebras analogously to its adic analog HC/A. Notice that the definition of the Jaco- bian ideal is compatible with localization by π, hence HC/A = HC/AC, and we obtain that HC/A is open iff HC/A = C. It remains to prove that HC/A defines the not A-smooth locus of X = Sp(C). Recall that modules of differentials of rigid (analytic or adic) spaces are defined by use of modules of continuous differentials of affinoid algebras, b n see [BLR], section 1. For example, ΩB/A = ⊕i=1BdTi, while ΩB/A may be huge. Let J ⊂ B be the ideal generated by fi’s, then by [BLR], 1.2, there is a natural sequence of finite C-modules

2dA/B/C b b 0 → J/J → ΩB/A ⊗B C → ΩC/A → 0 20 M. TEMKIN

Set d = dA/B/C for shortness. Let x ∈ X be a point and m ⊂ C be its ideal. By the Jacobian criterion of smoothness, see [BLR], 2.5, X is A-smooth at a point x iff the above sequence becomes split exact after tensoring with OX,x, or, that is equivalent, the map d ⊗C OX,x has left inverse. Since OX,x b is local and ΩB/A is free, the latter may happen iff the tensored sequence is an exact sequence of free OX,x-modules. Suppose that X is A-smooth at x, then there exists a subset L ⊂ {1, . . . , l} 2 such that: (i) J/J ⊗B OX,x is freely generated by the images of the elements of fL = {fi}i∈L, and (ii) the elements of dfL are linearly independent modulo m. Identify X with the closed subspace of Y = Sp(B), then (i) implies 2 that the image of fL generates J/J ⊗B K(x)f→JOY,x/mY,xJOY,x, where K(x) = OY,x/mY,x is the residue field of x. Therefore, fL generates JOY,x by lemma of Nakayama, i.e. JLOY,x = JOY,x. Notice that the operation : is compatible with flat base changes (use that (I : J) = Ann(J/I)), in particular (JL : J)D = (JLD : JD) for any flat B-algebra D. Thus, (JL : J)OY,x = (JLOY,x : JOY,x) = OY,x, and we obtain that x is not contained in V ((J : J)) ⊂ Spec(B) (recall that set-theoretically Y coincides with the L P ∂fi set of closed points of Spec(B)). Since dfi = dTj, (ii) implies that the ³ ´ j ∂Tj ∂fi rank of (x) equals to |L|. It follows that x∈ / V (HL), hence ∂Tj i∈L,1≤j≤n x∈ / V ((JL : J)HL), and, finally, x∈ / V (HC/A). Conversely, suppose that x∈ / V (HC/A). Then there exists L ⊂ {1, . . . , l}, such that x∈ / V ((JL : J)HLC). The set fL generates JOX,x, because OX,x = (JL : J)OX,x ⊆ (JLOX,x : JOX,x). The images of the elements of fL freely 2 b n generate J/J ⊗C OX,x, because their images in ΩB/A⊗B C/mf→⊕i=1K(x)dTi are linearly independent. It follows that d ⊗C OX,x has left inverse, and X is A-smooth at x. ¤ A reduced closed subspace Z ⊂ X is uniquely defined by its points, hence it follows from the above proof, that the ideal HC/A depends only on the A-affinoid algebra C. Note also that a reduced closed formal subscheme Z ⊂ Spf(C) is uniquely defined by the sets of its formal and rigid points, hence the Jacobian ideal HC/A depends only on the homomorphism A → C. Question 3.3.3. Assume that X is special rig-regular and admits a locally principal ideal of definition, and set T = Xsing. Set O = k[π] if X is equicharacteristic, and let O be a p-ring with residue field k otherwise. Does there exist a T-supported blow up X0 → X with locally O-algebraizable X0? The positive answer to the above question would allow to reduce desingularization of arbitrary quasi-excellent schemes of characteristic p (resp. mixed characteristic) to the particular case of k(x)-schemes of finite type for points x ∈ X (resp. O-schemes of finite type, where O is a p-ring with residue field k(x)). 3.4. Desingularization and applications to schemes. Given a quasi- excellent formal scheme X with a closed formal subscheme Z and T = DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 21

0 (X, Z)sing, we say that a T-supported blow up f : X → X desingularizes −1 0 0 0 the pair (X, Z) over a subset S ⊂ X if f (S) ∩ |Z | = ∅, where Z = Z ×X X . If S = X, then we say that f is a desingularization of the pair, it happens iﬀ X0 is regular and Z0 is an snc divisor. In the following theorem we prove that certain special formal schemes of characteristic zero admit a desingularization. The theorem will be used in the proof of a much more general theorem 4.3.4.

Theorem 3.4.1. Let X be a reduced rig-regular special formal scheme of characteristic zero with a locally principal non-zero ideal of deﬁnition and Z be a closed Xs-supported subscheme. Then the pair (X, Z) admits a desingularization.

Proof. Notice that T is a reduced closed subscheme of Xs, so we can and will identify it with a closed subset of Xs. For any T-supported formal 0 0 0 0 blow up fb : X → X, the singular locus T = (X , Z )sing is T-supported (it suffices to check this claim locally on X, but if X is affine, then the statement follows from its analog for schemes). Thus, we can identify T0 0 b with a closed subset of Xs. Assume that f desingularizes the pair (X, Z) over an open formal subscheme U with X \ T ⊆ U X, for example fb = IdX and U = X\T. By noetherian induction, it suffices to prove that there exists a T-supported formal blow up X00 → X, which desingularizes X over an open formal subscheme W with U W. Let x ∈ X be a point of the closed set S = X \ U. Choose a field k of characteristic zero such that Xs is of finite k-type and ◦ set K = k[[π]] and O = k[π]. Find an open affine neighborhood X0 of 0 0 x with a principal ideal of definition. Set X0 = X0 ×X X , Z0 = Z ×X X0, b 0 U0 = U ×X X0 and S0 = S ∩ X0, and let f0 : X0 → X0 be the induced formal ◦ blow up. By lemma 3.2.3, X0 is isomorphic to a formal K -scheme of finite type, hence X0 is O-algebraizable, by proposition 3.3.1, say X0f→XbΠ, where X is an O-scheme of finite type and Π ⊂ X is the divisor defined by π. The closed subscheme Z0 ,→ X0 is supported on the special fiber of X0, therefore it is isomorphic to the completion of a closed subscheme Z,→ X supported on Π. By corollary 3.1.3, replacing X by a neighborhood of Π, we achieve that X is reduced and X \ Π is regular. Then T = (X,Z)sing lies in Π and T is isomorphic to T0 = (X0, Z0)sing by lemma 3.1.2. Let S ⊂ Π be the preimage of S0 under the homeomorphism Π → X0 and U = X \ S, then U0 is isomorphic to the formal completion of U along U ∩ Π. 0 The formal scheme X0 is obtained from X0 by blowing up an open ideal I supported on T0 = T ∩ X0, hence the corresponding ideal I ⊂ OX is supported on T . If f : X0 → X denotes the blowing up along I and 0 0 b0 0 b Π = Π ×X X , then XΠ0 f→X0 and f0 is the completion of f by lemma 2.1.5. Since fb0 desingularizes the pair (X0, Z0) over U0, f desingularizes the pair (X,Z) over U by 3.1.3. 22 M. TEMKIN

0 0 0 00 0 The pair (X ,Z ×X X ) admits a desingularization f : X → X by proposition 2.3.1 (we use lemma 2.2.10 here, because X0 may be not integral). Notice that f 0 is an S-supported blow up, hence the induced morphism X00 → X is a desingularization of (X,Z) which coincides with f over U. Passing to completions, we obtain an S0-supported formal blow up b0 00 0 00 f0 : X0 → X0. The composition X0 → X0 desingularizes (X0, Z0) by corol- b b0 lary 3.1.3 and coincides with f0 over U0. Using 2.1.2, we can extend f0 to an S-supported formal blow up fb0 : X00 → X0, in particular fb0 is T-supported. 0 0 Then fb is an isomorphism over U ×X X and induces the formal blow up 00 0 b b0 X0 → X0. Therefore, f ◦ f desingularizes X over W = U ∪ X0. ¤ Corollary 3.4.2. Let (X,Z) be an integral quasi-excellent pair, such that Xsing ⊂ Z and Z is isomorphic to a k-scheme of ﬁnite type. Then the pair (X,Z) admits a desingularization.

Proof. The blow up BlZ (X) → X is an isomorphism over V = (X,Z)reg, because Z ×X V is a Cartier divisor in V . By lemma 2.1.3, BlZ (X) is 0 0 0 dominated by a V -admissible blow up X → X, then Z = Z ×X X is a Cartier divisor in X0. Replacing X and Z by X0 and Z0, we achieve that Z is a Cartier divisor. Let X be the formal completion of X along Z, it is reduced and rig-regular by corollary 3.1.3 (i). Thus, X satisfies the assumptions of the theorem. The closed subschemes Z and T = (X,Z)sing may be identified with closed subschemes of X, because they are supported on Zredf→Xs. Then T = (X,Z)sing by lemma 3.1.2. By the previous theorem, there exists an open ideal I ⊂ OX supported 0 0 on T , such that X = BlI(X) is regular and Z ×X X is an snc divisor. Since I is open, it is the completion of an ideal I ⊂ OX supported on T . Let 0 0 0 0 X be the blow up of X along I and Z = Z ×X X . By lemma 2.1.5, X is isomorphic to the formal completion of X0 along Z0, hence Z0 is an snc divisor by 3.1.3 (ii). Since X0 \ Z0f→X \ Z is regular, X0 → X is a required desingularization. ¤ Theorem 3.4.3. Let k be a noetherian scheme of characteristic zero. Then k is quasi-excellent if and only if there is resolution of singularities over k. Proof. The converse implication is due to Grothendieck, see [EGA IV], 7.9.5. Conversely, by proposition 2.2.9, it suffices to prove that if S is an integral local k-scheme of essentially finite type, s ∈ S is a closed point, and f : S0 → 0 −1 0 S is a blow up with Ssing ⊂ f (s), then S admits a desingularization. 0 0 0 Notice that Ssing is of finite k(s)-type, hence the pair (S ,Ssing) admits a desingularization g : S00 → S0 by the previous lemma. Then it is clear that g is a required desingularization of S0. ¤

4. Strict transforms 0 Let f : X = BlS(X) → X be a blow up and Y be a closed subscheme of X. Then the strict transform of Y in X0 is deﬁned as the schematic closure DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 23 of f −1(Y \ S) in X0. It is canonically isomorphic to the blow up of Y along S ×X Y , see [Con1], section 1, for details.

4.1. Principalization of strict transform. Proposition 4.1.1. Let X be a noetherian scheme and T,→ Y be closed subschemes corresponding to OX -ideals J ⊃ I. Assume that Y is a Cartier divisor. Given a positive integer n, let Xn denote the blow up of X along n Jn = I + J and let Yn denote the strict transform of Y in Xn. Then the following statements hold. (i) Yn is canonically isomorphic to the blow up of Y along the ideal J /I ⊂ OY . (ii) Yn is a Cartier divisor for any suﬃciently large n. (iii) Assume that Y \ T is a disjoint union of its closed subschemes Ye 0 00 0 00 0 00 and Ye , and T f→Y ×X Y , where Y and Y are the schematic closures of e 0 e 00 0 00 0 00 Y and Y in X. Then the strict transforms Yn and Yn of Y and Y in Xn are Cartier divisors for any suﬃciently large n.

Proof. Recall that Yn is canonically isomorphic to the blow up of Y along Jn/I, which equals to the n-th power of J /I. Since blow ups along an ideal and its powers are canonically isomorphic, we obtain (i). Let us prove the statement of (ii). It suffices to find a finite covering U of X by open 0 0 subschemes, such that for any X ∈ U the triple (X , I|X0 , J |X0 ) satisfies (ii). So, we may assume that X = Spec(A) is affine and I corresponds to a principal ideal I = fA. Let J, Jn ⊂ A denote the ideals corresponding to J , Jn. Applying Artin-Rees lemma to the ideal J and A-modules fA ⊂ A, n+m we find a positive n0 such that for any n ≥ n0 and m ≥ 0, J ∩ I = m n m J (J ∩ I) ⊂ fJ . Fix n ≥ n0, we will prove that it is as the proposition k k−1 requires. Let us first check that Jn ∩ I = fJn for any k > 0. From the n k k−1 nk equality Jn = I + J , we obtain that Jn = fJn + J , hence it remains to use that J nk ∩ I ⊂ fJ n(k−1) (take m = n(k − 1)). Jn Consider an affine chart Ug = Spec(B) of Xn, where B = A[ g ] for some g ∈ Jn. It suffices to prove that Yn ∩ Ug coincides with the closed f subscheme Vg = V ( g ) of Ug. Notice that the intersection of any of these two schemes with Spec(Ag) coincides with Yg = Spec(Ag/fAg), and Yn∩Ug is the scheme-theoretical closure of Yg. Therefore, we have only to prove that Yg is schematically dense in Vg, or, that is equivalent, that the homomorphism f φ : B/ g B → Ag/fAg has no kernel. Suppose, conversely, that φ is not injective, then there exists an element x ∈ B \ f B such that x is divided by P g l xi i l+m f in Ag. Obviously, x = i=0 gi for some elements xi ∈ Jn, and g x ∈ fA l+m m Pl l−i for sufficiently large m. The element g x = g i=0 xig is contained l+m l+m−1 in Jn and is divided by f, therefore it is contained in fJn by the l+m l+m−1 previous paragraph, i.e. g x = fh for some h ∈ Jn . It follows that h f f x = gl+m−1 g ∈ g B, contradicting our assumptions. 24 M. TEMKIN

It remains to prove (iii). We know from (ii) that Yn is a Cartier divisor. Notice that Yn is the schematic closure of Ye = Y \ T in Xn. Since Ye = e 0 e 00 0 00 Y t Y , we obtain that Yn = Yn ∪ Yn . Therefore, it suﬃces to prove that 0 00 Yn and Yn are disjoint, but it is well known that blowing up of X along 0 00 0 00 T = Y ×X Y separates strict transforms of Y and Y , see for example [Con1], lemma 1.4 and the consequent remark. ¤ 4.2. Regularization of strict transform. In this section we assume that X is an integral noetherian scheme of dimension d and Y is a reduced closed subscheme whose generic points are regular points of X of codimension 1. Note that T = (X,Y )sing does not contain generic points of Y . Our aim is to prove the following statement. Proposition 4.2.1. Assume that there is embedded resolution of singularities over X up to dimension d. Then there exists a blow up f : X0 → X supported on T = (X,Y )sing, such that the strict transform of Y is disjoint 0 −1 with (X , f (Y ))sing. We need to trace the behavior of both strict and total transforms of Y with respect to blow ups, so it will be more convenient to consider a more general situation. In the sequel, Z will be a scheme which remembers the history of total transforms and T denotes a closed set which we are allowed to modify. So, let X and Y be as above and Z be a closed subscheme of X, 0 which contains Y ∩ (X,Y )sing and is disjoint with Y . Notice that Y \ Z is an snc divisor in X \ Z and Y ∩ (X,S)sing ⊂ Z. Let S be a closed set equal to Y ∪Z and T be a closed subset with Y ∩(X,S)sing ⊆ T ⊆ Y ∩Z. For any T -supported blow up f : X0 → X, we use the following notations: Y 0 is the strict transform of Y in X0 (notice that the morphism Y 0 → Y is birational), 0 0 0 −1 0 0 Z = Z ×X X and S = f (S) = Y ∪ Z . Then the proposition follows from the following more general lemma (take Z = T in the proposition). Lemma 4.2.2. Keep the above notations and assume that there is embedded resolution of singularities over X up to dimension d. Then there exists a 0 0 0 0 T -supported blow up f : X → X such that Y ⊂ (X ,S )reg. Proof. A required blow up will be obtained as a composition of few blow ups, which will gradually improve the strict transform of Y . Notice that while proving the lemma, we may replace X by a neighborhood of Y and shrink Z in accordance. Step 0. Given a T -supported blow up f : X0 → X, we may replace X,Y,Z 0 0 0 0 0 0 0 0 0 −1 and T by X ,Y ,Z and any T with Y ∩ (X ,S )sing ⊆ T ⊆ Y ∩ f (T ). First of all, we notice that Y 0 \Z0f→Y \Z is an snc divisor in X0 \Z0f→X \Z, hence X0,Y 0,Z0 and T 0 satisfy the assumptions of the lemma. Suppose that the proposition holds for X0,Y 0,Z0 and T 0, and let f 0 : X00 → X0 be a T - 00 00 00 00 supported blow up with Y ⊂ (X ,S )reg, where Y is the strict transform of Y 0 and S00 = f 0−1(S0). The morphism f 00 = f ◦ f 0 is a composition of T -supported blow ups, hence it is a T -supported blow up by [Con1], 1.2. DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 25

Obviously, Y 00 is the strict transform of Y in X00 and S00 is the preimage of S. Hence f 00 solves our problem for X,Y,Z and T . Step 1. We may assume that Y is irreducible. Let m be the number of irreducible components of Y . By induction, we may assume that m > 1 and the lemma holds when Y has less than m irreducible components. Let Y = Y1 ∪ Y2, where Y1 is an irreducible component of Y and Y2 is the union of the others. We will construct a required blow up X0 → X in two steps: first we will achieve that Y1 ⊂ (X,S)reg, then we will apply the induction 0 assumption to Y2. Let us check the details. Find a blow up f : X → X which solves our problem for Y1, Z1 = Y2 ∪ Z and T1 = Y1 ∩ (X,S)sing. Obviously, f is T -supported, so it suffices to solve our problem for X0, Y 0, 0 0 0 0 0 Z and T = Y ∩ (X ,S )sing. 0 0 Let Yi denote the strict transforms of Yi, then S is an snc divisor in a 0 0 0 0 0 0 0 0 0 neighborhood of Y1 and S = Y1 ∪Y2 ∪Z , in particular T = Y2 ∩(X ,S )sing. 0 0 0 0 0 0 Since Y \Z is an snc divisor in X \Z , we obtain that Y2 \Z is an snc divisor 0 0 0 0 0 as well. Thus, X ,Y2,Y1 ∪ Z and T satisfy the assumptions the lemma. 0 0 Since Y2 contains m − 1 irreducible components, there is a T -supported 0 00 0 0 0 0 0 0 blow up f : X → X , which solves our problem for X ,Y2,Y1 ∪ Z and T . Let us check, that f 0 solves our problem for X0,Y 0,Z0 and T 0 too. Indeed, 0 0 0 00 0−1 0 f does not modify Y1, because it is disjoint with T . So, S = f (S ) is an 00 0 00 snc divisors in neighborhoods of both Y1 f→Y1 and Y2 , and we obtain that 00 00 00 it is an snc divisor in a neighborhood of Y = Y1 ∪ Y2 . We finished the only stage where a play with T is required. In the sequel, we automatically set T 0 = Y 0 ∩ f −1(T ) for any T -supported blow up f : X0 → X. Step 2. We may assume in addition to Step 1, that the generic points of Z are of codimension 1 in X. Replace Z by its reduction and remove all irreducible components of Z contained in Y ; it does not change S. Notice 0 −1 that for any T -supported blow up f : BlR(X) → X, we have Z = f (R) ∪ 0 0 Z , where Z is the strict transform of Z. The generic points of f −1(R) 0 are of codimension 1, because R ×X X is a Cartier divisor. Let Ze be the union of all components of Z of codimension larger than 1, then Y ∩ Ze ⊂ T because Ze ⊂ (X,S)sing. Blowing up X along Y ×X Ze we achieve that the strict transform of Ze is disjoint with Y . Since X \ Z is a neighborhood of Y , we may replace X by X \ Z. Then Z0 contains only components of codimension 1, as required. Notice that this condition is preserved by any further substitution from Step 0. Step 3. We may assume in addition to Step 2, that there exists a Cartier divisor Y2 and a closed subscheme Y1 ,→ X, such that Y2 \T = (Y \T )t(Y1 \ T ). Notice that Y ∩ Xreg is a Cartier divisor in Xreg, hence BlY (X) → X is an isomorphism over X \ (Y ∩ Xsing) ⊃ X \ T . Use lemma 2.1.3, to find a 0 0 0 T -supported blow up f : X → X dominating BlY (X), then Y2 = Y ×X X 0 0 0 is a Cartier divisor. Define Y1 to be the schematic closure of Y2 \ Y . Since 0 −1 0 −1 0 0 −1 0 0 Y \ f (T )f→Y2 \ f (T ) and T = Y ∩ f (T ), we obtain that Y2 \ T = 26 M. TEMKIN

0 0 0 0 0 0 0 (Y \ T ) t (Y1 \ T ). Now, we may replace X,Y,Z and T by X ,Y ,Z and T 0. Step 4. We may strengthen the condition of Step 3 by achieving that Y itself is a Cartier divisor. Let J and I be the OX -ideals of Y2 and Y ×X Y1, respectively. By proposition 4.1.1 (iii), choosing sufficiently large n and blowing up the ideal I + J n, we obtain a T -supported blow up f : X0 → X such that the strict transform of Y is a Cartier divisor. Replace X,Y,Z and T by X0,Y 0,Z0 and T 0, as earlier. In the sequel I is the invertible ideal defining Y . Step 5. We may assume in addition to Step 4, that Y is regular and T and W = Z ×X Y are snc divisors in Y . Notice that (Y,W )sing ⊂ T , because S = Y ∪ Z is an snc divisor in a neighborhood Ux of each x ∈ Y \ T , and therefore W ∩Ux is an snc divisor in Y ∩Ux. Since dim(Y ) ≤ d−1 and there is embedded resolution of singularities over X in dimensions smaller than d, 0 there exists a closed T -supported subscheme R,→ Y , such that Y = BlR(Y ) 0 0 0 0 0 is regular, W = W ×Y Y is an snc divisor in Y and T = T ×Y Y is a Cartier divisor. Consider R as the closed subscheme of X, and let J be its OX -ideal. By proposition 4.1.1 (ii), there exists n such that the strict 0 0 0 0 transform of Y in X = BlI+J n (X) is a Cartier divisor. Define Y ,Z and T 0 0 0 0 0 as usual, then Y f→Y by 4.1.1 (i), in particular, T f→(T ×X X ) ×X0 Y and |T 0| = T 0. To check that X0,Y 0,Z0 and T 0 satisfy the conditions of the step, 0 0 0 0 0 0 we notice that Z ×X0 Y f→Z ×X Y f→W is an snc divisor in Y . Finally, T is a divisor supported on W 0, hence it is an snc divisor too. Step 6. We may achieve in addition to Step 5, that X is regular and Z is a reduced Cartier divisor. Notice that X is regular in a neighborhood of Y , because Y is a regular Cartier divisor. Shrinking X we make it regular. Then the reduction of Z is a Cartier divisor, so we may replace Z by Zred. Step 7. Let T1,...,Tn be the irreducible components of T , then we may achieve in addition to Step 6, that each Ti belongs to a unique irreducible component Zi of Z, and each Y ∪ Zi is an snc divisor. Set Te = T1, and let J ⊂ OX be its ideal and m be its multiplicity in W . Consider the 0 m 0 0 blow up f : X = BlI+J (X) → X, then Y f→BlTe(Y )f→Y and W = 0 0 0 0 0 Y ×X0 Z f→Y ×X Zf→Y ×Y W is isomorphic to W . Also, define T and Te0 to be the preimages of T and Te in Y 0. The following explicit calculation shows that there exists a unique irreducible component Ze0 of Z0, which contains Te0, and then Ze0 ∪ Y 0 is an snc divisor in a neighborhood of Y 0. It suffices to check the above claim locally over a point x ∈ Te. By our assumptions X is regular, Y is a regular divisor in X and Te is a regular divisor in Y , hence we can assume that X = Spec(A), Z = V (f), I = I(X) = x1A and J = J (X) = x1A + x2A for some elements f, x1, x2 ∈ A. Note, that for any point x ∈ Te, the sequence x1, x2 may be completed 0 to a regular family of parameters x1, . . . , xn ∈ OX,x. Notice that X = x1 0 Spec(A[ m ]) is a regular chart of X = BlI+Jm (X) (with regular family of x2 DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 27

x1 0 0 parameters m , x2, . . . , xn) and Y is a regular divisor in X given by the x2 0 0 x1 0 e0 condition m = 0. Set Z = Z ×X0 X and notice that Z = {x2 = 0} is x2 0 e0 −1 e 0 e0 an irreducible component of Zred, because Z ⊂ f (T ). Obviously, Y ∪ Z 0 is an snc divisor, so we have only to show that no other component of Z contains Te0. The latter may be checked locally over the generic point η = Te0. Recall m that Z is given by the condition f = 0 and notice that f = x2 +x1y, because m e the images of f and x2 in A/x1A coincide (they define T in Y ). Shrink X 0 m m x1 so that y ∈ A, then Z is given by the condition 0 = x +x1y = x (1+ m y) 2 2 x2 0 x1 and, therefore, is the union of mZe and the divisor {1 + m y = 0}, which is x2 disjoint with Y 0, as claimed. 0 0 0 0 0 Replacing X,Y,Z and T by X ,Y , Zred and T (actually, Zred is obtained 0 from Z by replacing mZe0 by Ze0), we achieve that Te satisfies the condition of the step. It remains to iterate this procedure for other Ti’s. Step 8. We may achieve in addition to Step 7, that all irreducible components of W satisfy the conditions of Step 7. We will show that it suffices to shrink X. Fix an irreducible component Wf ⊂ W not contained in T . First, there exist a unique irreducible component of Z containing Wf, because Wf \ T is reduced. Let us denote this component Ze. Second, if Wf0 is another irreducible component of Y ∩ Ze, then V = Wf ∩ Wf0 = ∅. Indeed, Y ∪ Ze is not an snc divisor at any point of V . If V is not empty, then it is of codimension 2 in Y and the only components of W containing V are Wf and Wf0 (W is an snc divisor in Y ). Since T is a union of irreducible components of W , we obtain that either Wf0 ⊂ T or V 6⊂ T . In the first case, Y ∪ Ze is an snc divisor by the previous step. In the second case, Y ∪ Ze is an snc divisor at any point of V \ T , because S is an snc divisor at such point. All in all, we obtain a contradiction with V being non-empty. Now, we are ready to check that Y ∪Ze is an snc divisor in a neighborhood of Y . It suffices to check that Y ∪ Ze is an snc divisor in a neighborhood of Wf, because the same argument works for other components of Y ∩ Ze. We can work locally at a point x ∈ Wf, so assume that X = Spec(A), f m Y = {x1 = 0} and W = {x1 = x2 = 0} for a regular sequence of parameters x1, . . . , xn ∈ OX,x. Furthermore, Ze is a Cartier divisor because it is an irreducible component of the Cartier divisor Z and X is regular, so let Ze = e f m V (f). Shrinking X we may assume that Y ×X Z = W , then f = x2 + x1y. Since Y ∪ Ze is an snc divisor at any point of Wf \ T and Wf 6⊂ T , we obtain that m = 1. Hence Y ∪ Ze is an snc divisor at any point of Wf. Step 9. If the conditions of Step 8 are satisfied, then S is an snc divisor. Let Z1,...,Zn be the irreducible components of Z, which have non-empty intersection with Y . It follows from the previous step, that Y ∪ Zi is an snc divisor for any i. Since ∪i(Y ∩ Zi) is an snc divisor in Y , it follows that 28 M. TEMKIN

Y ∪ (∪iZi) is an snc divisor in X. Then it is clear that Y ∪ Z is an snc divisor in a neighborhood of Y , as required. ¤

4.3. Main results.

Lemma 4.3.1. Let X be an integral quasi-excellent scheme with a closed subset Z, such that T = (X,Z)sing is of ﬁnite type over a ﬁeld k of characteristic zero. Assume that dim(X) = d and there is embedded resolution of singularities over X up to dimension d. Then the pair (X,Z) admits a desingularization.

0 Proof. Let f : X → X be a T -supported blow up dominating BlZ (X) → X, it exists by lemma 2.1.3. Then Z0 = f −1(Z) is the support of the Cartier 0 0 0 divisor Z ×X X . Replace X and Z by X and Z . By corollary 3.4.2, there exists a desingularization f : X0 → X of X. Replacing X and Z by X0 and f −1(Z), we achieve in addition that X0 is regular. Now, X and Y = Z satisfy assumptions of proposition 4.2.1, hence there exists a T -supported blow up f : X0 → X, such that the strict transform Y 0 of Z is disjoint with 0 0 0 0 −1 T = (X ,S )sing for S = f (Z). The Zariski closure Z0 of S0 \ Y 0 is of ﬁnite k-type, because Z0 ⊂ f −1(T ). 0 0 0 0 Notice that (X ,Z )sing ⊂ T ⊂ Z , hence we can apply corollary 3.4.2 to ﬁnd a desingularization g : X00 → X0 of the pair (X0,Z0). Let Y 00,Z00 and S00 be the preimages of Y 0,Z0 and S0. Since g is T 0-supported, T 0 ∩ Y 0 = ∅ and S0 is an snc divisor in a neighborhood of Y 0, we obtain that S00 is an snc divisor in a neighborhood of Y 00f→Y 0. Also, S00 = Y 00 ∪ Z00 and Z00 is an snc divisor, hence the whole S00 is an snc divisor. The induced morphism f 0 : X00 → X is a T -supported blow up, because T 0 ⊂ Z0 ⊂ f −1(T ). Therefore f 0 provides a desingularization of (X,Z). ¤

Now, the main result of the paper may be proven exactly as its particular case 3.4.3.

Theorem 4.3.2. Let k be a noetherian scheme of characteristic zero. Then k is quasi-excellent if and only if there is embedded resolution of singularities over k .

Proof. The converse implication is due to Grothendieck. Conversely, we will prove by induction on d that there is embedded resolution of singularities over k up to dimension d. The case d = 0 is trivial. Assume that there is embedded resolution of singularities over k up to dimension d − 1. By proposition 2.2.9 (iv), it suffices to prove that, if S is a local integral k- scheme of essentially finite k-type and dimension d, s ∈ S is a closed point, f : S0 → S is a modification and Z0 ⊂ S0 is a closed subset, such that 0 0 −1 0 0 (S ,Z )sing ⊂ f (s), then (S ,Z ) admits a desingularization. It remains to notice, that f −1(s) is a k(s)-variety, and use the above lemma. ¤ DESINGULARIZATION OF QUASI-EXCELLENT SCHEMES 29

We would like to deduce desingularization of formal schemes. So, let X be a universally quasi-excellent formal scheme (i.e. any formal X-scheme of ﬁnite type is quasi-excellent) with a closed subscheme Z. b Corollary 4.3.3. Assume that X = BlJ (X0) and Z = Z0 ×X0 X, where X0 = Spf(A) is a reduced universally quasi-excellent formal scheme of characteristic zero, I,J ⊂ A are ideals and Z0 = Spf(A/I). Then the pair (X, Z) admits a desingularization.

Proof. Set X0 = Spec(A), X = BlJ (X0) and Z = Spec(A/I) ×X0 X, then the pair (X, Z) is isomorphic to the P -adic completion of the pair (X,Z), where P is an ideal of definition of A. By the previous theorem, the pair (X,Z) admits a desingularization X0 → X (use lemma 2.2.10 if X is not integral). Using lemma 3.1.2 one checks that the P -adic completion X0 → X of the blow up X0 → X is a required desingularization of the pair (X, Z). ¤ One could expect that the corollary allows to desingularize arbitrary universally quasi-excellent formal scheme of characteristic zero by patching local desingularizations (in the case of schemes, proposition 2.2.9 does the job). Unfortunately, it cannot be done in general because ideals may not admit extension from an open formal subscheme. For this reason we are forced to consider the case, when blowing up an open ideal suffices for desingularization, i.e. the case then X is rig-regular and Z is rig-snc. Recall that a (formal) scheme X is called quasi-paracompact, if it admits an open quasi-compact covering {Xi}i∈I of finite type, i.e. each Xi intersects only finitely many Xj’s. Any integral quasi-paracompact locally noetherian scheme is noetherian, but quasi-paracompactness is a much more interesting property in the case of formal schemes. For example, Drinfeld’s upper half plane, non-Archimedean Stein spaces and analytifications of varieties over a non-Archimedean field admit quasi-paracompact formal models, which may be chosen to be irreducible if the corresponding non-Archimedean space is irreducible. (Irreducibility is understood here in the sense of [Con2], it is a rather subtle notion, because it is not preserved by localizations, unlike the scheme case.) Note, that Berkovich considered in [Ber2] and later works quasi-paracompact formal schemes of locally finite presentation over the ring of integers of a non-Archimedean field (they are simply called formal schemes of locally finite presentation in loc.cit.). One can define analytic generic fiber for such formal schemes. Recently, Bosch extended Raynaud’s theory to quasi- paracompact formal schemes and rigid spaces, see [Bo], 2.8.3. Let us say that a (formal) scheme is paranoetherian if it is quasi-paracompact and locally noetherian. Theorem 4.3.4. Let X be a reduced universally quasi-excellent paranoetherian formal scheme of characteristic zero and Z be a closed formal subscheme. Assume that X is rig-regular and Z is a rig-snc divisor, then the pair (X, Z) admits a desingularization. 30 M. TEMKIN

Proof. The proof exploits the same reasoning as was used in the proofs of proposition 2.2.9 and theorem 3.4.1. Since there are mild complications due to lack of quasi-compactness, we give the full argument. By our rig- assumptions, T = (X, Z)sing is a reduced closed subscheme of Xs. It follows 0 0 0 that (X , Z ×X X ) is an Xs-supported scheme for any formal blow up X = Blb I(X) with open I. Each connected component may be desingularized separately, so assume that X is connected. Choose a locally finite open affine covering {Xi}i∈I . By connectedness of X, I is at most countable. We assume that I = N, because the case of finite I is similar and easier. Let us say that an ideal I ⊂ OX has quasi-compact support if it is trivial outside of a quasi-compact formal open subscheme. We will use induction whose step is the following statement. Let I be a T-supported open ideal with quasi-compact support, 0 such that the blow up f : X = Blb I(X) → X desingularizes the pair (X, Z) over Ui−1 = (X \ T) ∪ (∪j

It seems natural to expect that the rig-assumptions in the above theorem may be removed, but perhaps one has to use stronger methods to prove such a generalization. Some kind of canonical desingularization should be used, because a badly chosen desingularization of an open formal subscheme may have no extension to the whole formal scheme.

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