Statement of the main theorem

version du 2016-11-14 à 13h36TU (19c1b56)

Contents

1. Statement of the main theorem. . . 1

2. Functorial resolutions. . . 2

3. Resolution of log regular log schemes. . . 10

4. Proof of Theorem 1.1 — preliminary steps. . . 24

5. Proof of Theorem 1.1 — abelian inertia. . . 28

References. . . 34

In this exposé we state and prove Gabber’s modification theorem mentioned in the introduction (see step (C)). Its main application is to Gabber’s refined — i.e. prime to`— local uniformization theorem. This is treated in exposéIX. A relative variant of the modification theorem, also due to Gabber, has applications to prime to`refinements of theorems of de Jong on alterations of schemes of finite type over a field or a trait. This is discussed in exposéX. In §1, we state Gabber’s modification theorem in its absolute form (Theorem1.1). The proof of this theorem occupies §§4—5. A key ingredient is the existence of functorial (with respect to regular morphisms) resolutions in characteristic zero; the relevant material is collected in §2. We apply it in §3to get resolutions of log regular log schemes, using the language of Kato’s fans and Ogus’s monoschemes. The main results, on which the proof of1.1is based, are Theorems3.3.16and3.4.15. §§2and3can be read independently of §§1,4,5.

Though we basically follow the lines of Gabber’s original proof, our approach differs from it at several places, especially in our use of associated points and saturated desingularization towers, whose idea is due to the second author. In2.3.13and2.4we discuss material from Gabber’s original proof.

We wish to thank Sophie Morel for sharing with us her notes on resolution of log regular log schemes and Gabber’s magic box. They were quite useful.

1. Statement of the main theorem

Theorem 1.1. — LetXbe a noetherian, qe, separated, log regular fs log scheme (exp.VI,1.2), endowed with an action of a finite groupG. We assume thatGacts tamely (exp.VI,3.1) and generically freely onX(i.e. there exists aG-stable, dense open subset ofXwhere the inertia groupsGxare trivial). LetZbe the complement of the open subset of triviality of the log structure ofX, and letT be the complement of the largestG-stable open subset ofXon whichGacts freely.

Then there exists an fs log schemeX⁰ and aG-equivariant morphismf =f_(G,X,Z):X⁰ →Xof log schemes having the following properties:

(i) As a morphism of schemes,fis a projective modification, i.e. fis projective and induces an isomorphism of dense open subsets.

(ii)X⁰is log regular andZ⁰=f⁻¹(Z∪T)is the complement of the open subset of triviality of the log structure ofX⁰. (iii) The action ofGonX⁰is very tame (exp.VI,3.1).

When proving the theorem we will construct f_(G,X,Z) that satisfies a few more nice properties that will be listed in Theorem5.6.1. We remark that Gabber also proves the theorem, more generally, whenXis not assumed to be qe. However, the quasi-excellence assumption simplifies the proof so we impose it here. Most of the proof works for a general noetherianX, so we will assume thatXis qe only when this will be needed in

§5.

(i)The research of M.T. was partially supported by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement 268182. Part of it was done when M.T. was visiting the department of mathematics of the University of Paris-Sud and he is grateful for the hospitality.

1

1.2. — (a) Note that we do not demand thatfis log smooth. In general, it is not. Here is an example. Letkbe an algebraically closed field of characteristic6=2. LetG={±1}act on the affine planeX=A²k, endowed with the trivial log structure, byx7→ ±x. ThenXis regular and log regular, andT = {0}. The action ofGonXis tame, but not very tame, asG_{0}(=G) does not act trivially on the (only) stratumXof the stratification by the rank ofM^gp/O^∗. Letf: X⁰→Xbe the blow up ofT, with its natural action ofG. Then the pair(X⁰, Z⁰=f⁻¹(T)) is log regular,fis aG-equivariant morphism of log schemes,X⁰−Z⁰ is at the same time the open subset of triviality of the log structure and the largestG-stable open subset ofX⁰ whereGacts freely, andGacts very tamely onX⁰. However,fis not log smooth (the fiber offat{0}is the lineZ⁰with the log structure associated toN→O, 17→0, which is not log smooth over Speckwith the trivial log structure).

(b) In the above example, letD1,D2be distinct lines inXcrossing at{0}, and put the log structureM(D)on Xdefined by the divisor with normal crossingsD=D1∪D2. Then ˜X= (X, M(D))is log smooth over Speck endowed with the trivial log structure, and Gacts very tamely on (X, M(D))(exp.VI,4.6). Moreover, the modificationfconsidered above underlies the log blow up ˜f: X˜⁰ →X˜ ofXat (the ideal inM(D)of){0}. While fdepends only onX, the log étale morphism ˜fis not canonical, as it depends on the choice ofD. However, one can recoverffrom thecanonicalresolutions of toric singularities (discussed in the next section). Namely, asGacts very tamely on ˜X, the quotient ˜Y = X/G˜ is log regular (exp.VI,3.2): ˜Y = Speck[P], wherePis the submonoid ofZ²generated by(2, 0),(1, 1)and(0, 2), and the projectionp:X˜ →Y˜is a Kummer étale cover of groupG, in particular,Uis aG-étale cover ofV =p(U), whereU=X−D. Letg:Y˜⁰ →Y˜be the log blow up of{0}=p({0})in ˜Y. We then have a cartesian diagram of log schemes

X˜⁰

// ˜Y⁰

g

X˜ ^p //Y˜ ,

where the horizontal maps are Kummer étale covers of groupG. Now, as a morphism of schemes, ˜Y⁰ →Y˜is thecanonical resolutionof ˜Y, and the underlying schemeX⁰ of ˜X⁰ is thenormalizationof ˜Y⁰ in theG-étale cover p:U→V. This observation, suitably generalized, plays a key role in the proof of1.1.

2. Functorial resolutions

For simplicity, all schemes considered in this section are quasi-compact and quasi-separated. In particular, all morphisms are quasi-compact and quasi-separated.

2.1. Towers of blow ups. — In this section we review various known results on the following related top- ics: blow ups and their towers, various operations on towers, such as strict transforms and pushforwards, associated points of schemes and schematic closure.

2.1.1. Blow ups. — We start with recalling basic properties of blow ups; a good reference is [Conrad, 2007, §1].

LetXbe a scheme. By ablow upofXwe mean a triple consisting of a morphismf:Y→X, a finitely presented closed subschemeVofX(thecenter), and anX-isomorphismα:Y →^∼ Proj(L

n∈NIⁿ), whereI = I(V)is the ideal ofV. We will writeY = BlV(X). When there is no risk of confusion we will omitV andαfrom the notation. A blow up(f, V, α)is said to beemptyifV =∅. In this case,Y = Xandfis the identity. The only X-automorphism ofYis the identity. Also, it is well known thatYis the universalX-scheme such thatV×XY is a Cartier divisor (i.e. the idealI OY is invertible).

2.1.2. Total and strict transforms. — Given a blow upf:Y=BlV(X)→X, there are two natural ways to pullback closed subschemesi:Z,→X. Thetotal transformofZunderfis the scheme-theoretic preimagef^tot(Z) =Z×XY. Thestrict transformf^st(Z)is defined as the schematic closure off⁻¹(Z−V)→^∼ Z−V.

Remark 2.1.3. — (i) The strict transform depends on the centers and not only onZand the morphismY→X. For example, ifD,→Xis a Cartier divisor then the morphism BlD(X) → Xis an isomorphism but the strict transform ofDis empty.

(ii) Whilef^tot(Z)→Zis just a proper morphism, the morphismf^st(Z)→Zcan be provided with the blow up structure becausef^st(Z)→^∼ BlV×XZ(Z)(e.g., ifZ,→Vthenf^st(Z) = ∅= BlZ(Z)). Thus, the strict transform can be viewed as a genuine blow up pullback offwith respect toi.

2.1.4. Towers of blow ups. — Next we introduce blow up towers and study various operations with them (see also [Temkin, 2012, §2.2]). By atower of blow upsofXwe mean a finite sequence of lengthn≥0

X_• = (Xn

fn−1//Xn−1 //· · · ^f0//X0=X)

of blow ups. In particular, this data includes the centersV_i,→X_ifor0≤i≤n−1. Usually, we will denote the tower asX_•or(X_•, V_•). Also, we will often use notationX_n99KX₀to denote a sequence of morphisms.

Ifn = 0 then we say that the tower istrivial. Note that the morphismX_n → Xis projective, and it is a modification if and only if the centersV_iare nowhere dense. IfX_•is a tower of blow ups, we denote by(X_•)_c thecontractedtower deduced fromX_•by omitting the empty blow ups.

2.1.5. Strict transform of a tower. — Assume thatX = (X_•, V_•)is a blow up tower ofXand h:Y → Xis a morphism. We claim that there exists a unique blow up towerY = (Y_•, W_•)ofYsuch thatY_i→Y→Xfactors throughX_iandW_i=V_i×_XiY_i. Indeed, this definesY₀, W₀andY₁uniquely. SinceV₀×_XY₁=W₀×_Y0Y₁is a Cartier divisor,Y1→Xfactors uniquely throughX1. The morphismY1→X1uniquely definesW1andY2, etc.

We callY thestrict transformofX with respect tohand denote ith^st(X).

Remark 2.1.6. — The following observation motivates our terminology: ifh: Y,→Xis a closed immersion then Yi,→Xiis a closed immersion andYi+1is the strict transform ofYiunder the blow upXi+1→Xi.

2.1.7. Pullbacks. — One can also define a naive base change of X with respect to h simply as(Y_•, W_•) = X ×_XY. This produces a sequence of proper morphisms Y_n 99K Y₀ and closed subschemes W_i,→Y_i for 0 ≤i ≤ n−1. If this datum is a blow up sequence, i.e. Yi+1 →∼ BlW_i(Yi), then we say thatX ×XY is the pullbackofX and use the notationh^?(X) =X ×XY.

Remark 2.1.8. — The pullback exists if and only ifh^st(X)→^∼ X ×XY. Indeed, this is obvious for towers of length one, and the general case follows by induction on the length.

2.1.9. Flat pullbacks. — Blow ups are compatible with flat base changes h: Y → X in the sense that BlV×XY(Y) →^∼ BlV(X) ×X Y (e.g. just compute these blow ups in the terms of Proj). By induction on length of blow up towers it follows that pullbacks of blow up towers with respect to flat morphisms always exist. One can slightly strengthen this fact as follows.

Remark 2.1.10. — Assume thatX•is a blow up tower ofXandh:Y →Xis a morphism. If there exists a flat morphismf:X→Ssuch that the compositiong:Y →Sis flat and the blow up towerX_• is the pullback of a blow up towerS_•then the pullbackh^?(X_•)exists and equals tog^?(S_•).

2.1.11. Equivariant blow ups. — Assume thatXis anS-scheme acted on by a flatS-group schemeG. We will denote by p, m: X₀ = G×S X → X the projection and the action morphisms. Assume thatV,→X is aG- equivariant closed subscheme (i.e.,V×X(X₀;m)coincides withV₀=V×X(X₀;p)) then the action ofGlifts to the blow upY = BlV(X). Indeed, the blow upY₀ = BlV0(X₀)→X₀is the pullback ofY →Xwith respect to bothmandp, i.e. there is a pair of cartesian squares

Y₀

m⁰

p⁰

//X₀

m

p

Y //X

So, we obtain an isomorphismY0 →∼ G×SY (giving rise to the projectionp⁰: Y0 → Y) and a group action morphismm⁰:Y0 → Y compatible withm. Furthermore, the unit mape:X → X0satisfies the condition of Remark2.1.10(withX=S), hence we obtain the base changee⁰:Y →Y0ofe.

It is now straightforward to check that m⁰ and e⁰ satisfy the group action axioms, but let us briefly spell this out using simplicial nerves. The action of G on V defines a cartesian sub-simplicial scheme Ner(G, V),→Ner(G, X). By the flatness ofGoverSand Remark2.1.10, BlNer(G,V)(Ner(G, X))is cartesian over Ner(G), hence corresponds to an action ofGon BlV(X).

2.1.12. Flat monomorphism. — Flat monomorphisms are studied in [Raynaud, 1967]. In particular, it is proved in [Raynaud, 1967, Prop. 1.1] thati:Y,→Xis a flat monomorphism if and only ifiis injective and for anyy∈Y the homomorphismOX,y→OX,xis an isomorphism. Moreover, it is proved in [Raynaud, 1967, Prop. 1.2] that in this caseiis a topological embedding andOY = OX|Y. In addition to open immersions, the main source of flat monomorphisms for us will be morphisms of the form Spec(OX,x),→Xand their base changes.

2.1.13. Pushforwards of ideals. — Leti: Y →Xbe a flat monomorphism, e.g. Spec(OX,x),→X. By thepushfor- wardU= i?(V)of a closed subschemeV,→Ywe mean its schematic image inX, i.e. Uis the minimal closed subscheme such thatV,→Xfactors throughU. It exists by [ÉGAI9.5.1].

Lemma 2.1.14. — The pushforwardU=i?(V)extendsVin the sense thatU×XY=V. Proof. — See [Raynaud, 1967, Proof of Proposition 1.2].

2.1.15. Pushforwards of blow up towers. — Given a blow upf:BlV(Y)→Yand a flat monomorphismi:Y,→X we define the pushforwardi?(f)as the blow up alongU=i?(V), assuming thatUis finitely presented overX.

Using Lemma2.1.14and flat pullbacks we see thati^?i?(f) =fand BlV(Y),→BlU(X)is a flat monomorphism.

So, we can iterate this procedure to construct pushforward with respect toiof any blow up towerY•ofY. It will be denoted(X_•, U_•) =i_?(Y_•, V_•).

Remark 2.1.16. — (i) Clearly,i^?i_?(Y_•) =Y_•.

(ii) In the opposite direction, a blow up tower(X_•, U_•)ofXsatisfiesi_?i^?(X_•) =X_•if and only if the preimage ofYin each centerU_iof the tower is schematically dense.

2.1.17. Associated points of a scheme. — Assume thatXis a noetherian scheme. Recall that a point x ∈ Xis called associated ifmxis an associated prime ofOX,x, i.e. OX,xcontains an element whose annihilator ismx, see [ÉGAIV₂3.1.1]. The set of all such points will be denoted Ass(X). The following result is well known but difficult to find in the literature.

Lemma 2.1.18. — Leti:Y,→Xbe a flat monomorphism. Then the schematic image oficoincides withXif and only if Ass(X)⊂i(Y).

Proof. — Note that the schematic image ofican be described as Spec(F), whereF is the image of the homo- morphismφ:OX→i?(OY). Thus, the schematic image coincides withXif and only if Ker(φ) =0.

Ifi(Y)omits a pointxthen anymx-torsion elements∈OX,xis in the kernel ofφx:OX,x→i?(OY)x(we use thatOY = OX|Y by2.1.12and since the closure ofxis the support of an extension ofsto a sufficiently small neighborhood, the restrictions|Yvanishes). So, if there existsx∈Ass(X)withx /∈i(Y)then Ker(φ)6=0.

Conversely, if the kernel is non-zero then we takexto be any maximal point of its support and choose any non-zeros∈Ker(φx). In particular,s|Y∩X_x =0and hencex /∈i(Y). For any non-trivial generizationyofxthe image ofsinOX,yvanishes because Ker(OX,y→i?(OY)y) =0by maximality ofx. Thus, the closure ofxis the support ofs, and hencesis annihilated by a power ofmx. SinceXis noetherian, we can find a multiple ofs whose annihilator ismx, thereby obtaining thatx∈Ass(X).

2.1.19. Associated points of blow up towers. — If(X_•, V_•)is a blow up tower and allX_i’s are noetherian then by the set Ass(X_•)of its associated points we mean the union of the images of Ass(V_i)inX. Combining Remark 2.1.16(i) and Lemma2.1.18we obtain the following:

Lemma 2.1.20. — Leti:Y,→Xbe a flat monomorphism and letX_•be a blow up tower ofX. Theni?i^?(X_•) =X_•if and only ifAss(X_•)⊂i(Y).

2.2. Normalized blow up towers. — For reduced schemes most of the notions, constructions and results of

§2.1have normalized analogs. We develop such a "normalized" theory in this section.

2.2.1. Normalization. — The normalization of a reduced schemeXwith finitely many irreducible components, as defined in [ÉGAII6.3.8], will be denotedX^nor. Recall that normalization is compatible with open immer- sions and for an affineX=Spec(A)its normalization isX^nor=Spec(B)whereBis the integral closure ofAin its total ring of fractions (which is a finite product of fields). The normalization morphismX^nor→Xis integral but not necessarily finite.

2.2.2. Functoriality. — Recall (exp.II,1.1.2) that a morphismf:Y → Xis calledmaximally dominatingif it takes generic points of Y to generic points of X. Normalization is a functor on the category whose objects are reduced schemes having finitely many irreducible components and whose morphisms are the maximally dominating ones. Furthermore, it possesses the following universal property: any maximally dominating morphismY→Xwith normalYfactors uniquely throughX^nor. (By definition,Yis normal if its local rings are normal domains. Both claims are local onYandXand are obvious for affine schemes.)

2.2.3. Normalized blow ups. — Assume thatXis a reduced scheme with finitely many irreducible components.

By thenormalized blow upofXalong a closed subschemeVof finite presentation we mean the morphism f:BlV(X)^nor→X. The normalization is well defined since BlV(X)is reduced and has finitely many irreducible components. Note thatfis universally closed but does not have to be of finite type. As in the case of usual blow ups,Vis a part of the structure. In particular, BlV(X)^norhas noX-automorphisms and we can talk about equality of normalized blow ups (as opposed to an isomorphism).

Proposition 2.2.4. — (i) Keep the above notation. ThenBlV(X)^nor →Xis the universal maximally dominating mor- phismY →Xsuch thatYis normal andV×XYis a Cartier divisor.

(ii) For any blow up f:Y = BlV(X) → Xits normalization f^nor:Y^nor → X^nor is the normalized blow up along V×XX^nor.

Proof. — Combining the universal properties of blow ups and normalizations we obtain (i), and (ii) is its immediate corollary.

Towers of normalized blow ups and their transforms can now be defined similarly to their non-normalized analogs.

2.2.5. Towers of normalized blow ups. — Atower of normalized blow upsis a finite sequenceXn 99KX−1with n≥0of normalized blow ups with centersVi,→Xifor−1≤i≤n−1andV−1=∅. The centers are part of the datum. Note that the mapX0→X−1is just the normalization map. Thecontractionof a normalized blow up tower removes the normalized blow ups with empty centers fori≥0. It follows from [ÉGAIV38.6.3] that eachXiwithi≥0is a normalization of a reduced projectiveX−1-scheme. A tower is callednoetherianif all Xiare noetherian.

2.2.6. Normalization of a blow up tower. — Using induction on length and Proposition2.2.4(ii), we can associate to a blow up towerX = (X_•, V_•)of a reduced schemeXwith finitely many irreducible components a normal- ized blow up towerX^nor = (Y_•, W_•), whereY−1=XandYi=X^nor_i ,Wi=Vi×X_iX^nor_i fori≥0. We callX^nor thenormalizationofX.

2.2.7. Strict transforms. — If X = (X_•, V_•) is a normalized blow up tower of X = X−1 and f: Y → X is a morphism between reduced schemes with finitely many irreducible components then we define the strict transformf^st(X)as the normalized blow up tower(Y_•, W_•)such thatY−1 = Y andWi = Vi×X_iYi. Using induction on the length and the universal property of normalized blow ups, see2.2.4(i), one shows that such a tower exists and is the universal normalized blow up tower ofY such thatf =f−1extends to a compatible sequence of morphismsfi:Yi→Xi.

2.2.8. Pullbacks. — The strict transformf^st(X)as above will be called thepullbackand denotedf^?(X) if Yi →∼ Xi×XYfor any−1≤i ≤n. Recall that a morphism isregularif it is flat and has geometrically regular fibers, see [ÉGAIV₂6.8.1].

Lemma 2.2.9. — Iff: Y→Xis a regular morphism between reduced noetherian schemes then any normalized blow up towerX ofXadmits a pullbackf^?(X).

Proof. — Blow ups are compatible with flat morphisms hence we should only show that normalizations in our tower are compatible with regular morphisms: iff: Y →Xis a regular morphism of reduced noetherian schemes then the morphismh: Y^nor→X^nor×XYis an isomorphism. Sincehis an integral morphism which is generically an isomorphism, it suffices to show thatX^nor×_XYis normal.

To prove the latter we can assume thatXandY are affine. Thenfis a filtered limit of smooth morphisms h_i:Y_i → Xby Popescu’s theorem. If the claim holds forh_ithen it holds forh, so we can assume thathis smooth. We claim that, more generally, ifAis a normal domain andφ: A→Bis a smooth homomorphism thenBis normal. Indeed,Ais a filtered colimit of noetherian normal subdomainsAiand by [ÉGAIV38.8.2]

and [ÉGAIV417.7.8]φis the base change of a smooth homomorphismφ_i:A_i →B_ifor large enoughi. For eachj≥iletφ_j:A_j→B_jbe the base change ofφ_i. EachB_jis normal by [Matsumura, 1980, 21.E (iii)] andB is the colimit ofBj, henceBis normal.

2.2.10. Fpqc descent of blow up towers. — The classical fpqc descent of ideals (and modules) implies that there is also an fpqc descent for blow up towers. Namely, ifY →Xis an fpqc covering andY•is a blow up tower ofY whose both pullbacks toY×XYare equal thenY•canonically descends to a blow up tower ofXbecause the centers descend. In the same way, normalized blow up towers descend with respect to quasi-compact surjective regular morphisms.

2.2.11. Associated points. — The material of 2.1.15–2.1.19extends to noetherian normalized blow up towers almost verbatim. In particular, ifX = (X•, V•)is such a tower then Ass(X)is the union of the images of Ass(Vj) and for any flat monomorphismi:Y,→X(which is a regular morphism by 2.1.12) with a blow up towerY ofYwe always have thati^?i_?Y =Y, and we have thati_?i^?X =X if and only if Ass(X)⊂i(Y). 2.3. Functorial desingularization. — In this section we will formulate the desingularization result about toric varieties that will be used later in the proof of Theorem1.1. Then we will show how it is obtained from known desingularization results.

2.3.1. Desingularization of a scheme. — By aresolution(ordesingularization)towerof a noetherian schemeX we mean a tower of blow ups with nowhere dense centersX• such thatX= X0,Xnis regular and nofiis an empty blow up. For example, the trivial tower is a desingularization if and only ifXitself is regular.

2.3.2. Normalized desingularization. — We will also consider normalized blow up towers such that each center is non-empty and nowhere dense, X = X−1 and Xn is regular. Such a tower will be called a normalized desingularization towerofX.

Remark 2.3.3. — (i) For any desingularization towerX ofXits normalizationX^noris a normalized desingu- larization tower ofX.

(ii) Usually one works with non-normalized towers; they are subtler objects that possess more good prop- erties. All known constructions of functorial desingularization (see below) produce blow up towers by an inductive procedure, and one cannot work with normalized towers instead. However, it will be easier for us to deal with normalized towers in log geometry because in this case one may work only with fs log schemes.

2.3.4. Functoriality of desingularization. — For concreteness, we consider desingularizations in the current sec- tion, but all what we say holds for normalized desingularizations too. Assume that a classS⁰of noetherian S-schemes is provided with desingularizationsF(X) = X_•for anyX ∈ S⁰. We say that the desingulariza- tion (family)F isfunctorial with respect to a classS¹of S-morphisms between the elements ofS⁰if for anyf:Y →XfromS¹the desingularization ofXinduces that ofY in the sense thatf^?F(X)is defined and its contraction coincides withF(Y)(so, F(Y) = (Y ×XF(X))c). Note that we put the=sign instead of an isomorphism sign, which causes no ambiguity by the fact that any automorphism of a blow up is the identity as we observed above.

Remark 2.3.5. — (i) Contractions in the pulled back tower appear when some centers ofF(X)are mapped to the complement off(Y)inX. In particular, iff∈S¹is surjective then the precise equalityF(Y) =Y×XF(X) holds.

(ii) Assume thatX=Sn

i=1Xiis a Zariski covering and the morphismsXi,→Xand`n

i=1Xi→Xare inS¹. In general, one cannot reconstructF(X)from the F(Xi)’s because the latter are contracted pullbacks and it is not clear how to glue them with correct synchronization. However, all information aboutF(X)is kept in F(`n

i=1Xi). The latter is the pullback ofF(X)hence we can reconstructF(X)by gluing the restricted blow up towersF(`n

i=1Xi)|X_i. Note that F(`n

i=1Xi)|X_i can be obtained fromF(Xi)by inserting empty blow ups, and these empty blow ups make the gluing possible. This trick with synchronization of the towersF(Xi) by desingularizing disjoint unions is often used in the modern desingularization theory, and one can formally show (see [Temkin, 2012, Rem. 2.3.4(iv)]) that such approach is equivalent to the classical synchronization of the algorithm with an invariant.

(iii) Assume thatS¹contains all identities IdXwithX∈S⁰and for anyY, Z∈S⁰there existsT =Y` Z inS<sup>0</sup>such that for any pair of morphismsa:Y →X,b: Z→XinS<sup>1</sup>the morphism(a, b) :T →Xis inS<sup>1</sup>. As an illustration of the above trick, let us show that even iff, g:Y →Xare inS<sup>1</sup>but not surjective, we have an equalityF(X)×X(Y, f) = F(X)×X(Y, g)of non-contracted towers. Indeed, setY<sup>0</sup> = Y`

Xand consider the morphismsf⁰, g⁰:Y⁰ →Xthat agree withfandgand mapXby the identity. ThenF(X)×X(Y⁰, f⁰)and F(X)×X(Y⁰, g⁰)are equal becausef⁰andg⁰are surjective, hence their restrictions ontoYare also equal, but these are preciselyF(X)×X(Y, f)andF(X)×X(Y, g).

2.3.6. Gabber’s magic box. — Now we have tools to formulate the aforementioned desingularization result.

Theorem 2.3.7. — Let S⁰ denote the class of finite disjoint unions of affine toric varieties over Q, i.e. S⁰ = {`n

i=1Spec(Q[Pi])}, whereP1, . . . , Pnare fs torsion free monoids. LetS¹denote the class of smooth morphisms f:

am j=1

Spec(Q[Qj])→ an i=1

Spec(Q[Pi])

such that for each1≤j≤mthere exists1≤i=i(j)≤nand a homomorphism of monoidsφj:Pi→Qjso that the restriction offontoSpec(Q[Qj])factors through the toric morphismSpec(Q[φj]). Then there exists a desingularization F onS⁰which is functorial with respect to S¹and, in addition, satisfies the following compatibility condition: if O1, . . . ,Olare complete noetherian local rings containingQ,Z =`l

i=1Spec(Oi), andg, h:Z →Xare two regular morphisms withX∈S⁰then

(2.3.7.1) (Z, g)×XF(X) = (Z, h)×XF(X).

In the above theorem, we use the convention that different tuplesP1, . . . , Pn give rise to different schemes

`n

i=1Spec(Q[Pi]). Before showing how this theorem follows from known desingularization results, let us make a few comments.

Remark 2.3.8. — (i) Gabber’s original magic box also requires that the centers are smooth schemes. This (and much more) can also be achieved as will be explained later, but we prefer to emphasize the minimal list of properties that will be used in the proof of Theorem1.1.

(ii) It is very important to allow disjoint unions in the theorem in order to deal with synchronization issues, as explained in Remark2.3.5(ii). This theme will show up repeatedly throughout the exposé.

2.3.9. Desingularization of qe schemes over Q. — In practice, all known functorial desingularization families are constructed in an explicit algorithmic way, so one often says adesingularization algorithminstead of a desingularization family. We adopt this terminology below.

Gabber’s magic box2.3.7is a particular case of the following theorem, see [Temkin, 2012, Th. 1.2.1]. Indeed, due to Remark2.3.5(iii), functoriality with respect to regular morphisms implies (2.3.7.1).

Theorem 2.3.10. — There exists a desingularization algorithm F defined for all reduced noetherian quasi-excellent schemes overQand functorial with respect to allregularmorphisms. In addition,F blows up only regular centers.

Remark 2.3.11. — Although this is not stated in [Temkin, 2012], one can strengthen Theorem2.3.10by requir- ing thatF blows up only regular centers contained in the singular locus. An algorithmF is constructed in [Temkin, 2012] from an algorithmFVarthat desingularizes varieties of characteristic zero, and one can check that if the centers ofFVarlie in the singular loci (of the intermediate varieties) then the same is true forF. Let us explain how one can choose an appropriateFVar. In [Temkin, 2012], one uses the algorithm of Bierstone- Milman to constructF, see Theorem 6.1 and its Addendum in [Bierstone et al., 2011] for a description of this algorithm and its properties. It follows from the Addendum that the algorithm blows up centers lying in the singular loci untilXbecomes smooth, and then it performs some additional blow ups to make the exceptional divisor snc. Eliminating the latter blow ups we obtain a desingularization algorithmFVarwhich only blows up regular centers lying in the singular locus.

It will be convenient for us to use the algorithmF from Theorem2.3.10in the sequel. Also, to simplify the exposition we will freely use all properties ofF but the careful reader will notice that only the properties of Gabber’s magic box will be crucial in the end. Also, instead of working withF itself we will work with its normalizationF^norwhich assigns to a reduced qe scheme overQthe normalized blow up towerF(X)^nor. It will be convenient to use the notationFf=F^norin the sequel.

Remark 2.3.12. — (i) Since normalized blow ups are compatible with regular morphisms, it follows from The- orem2.3.10that the normalized desingularizationFfis functorial with respect to all regular morphisms.

(ii) The feature which is lost under normalization (and which is not needed for our purposes) is some control on the centers. The centersVe_iofFf(X)are preimages of the centersV_i,→X_iofF(X)under the normalization morphismsX^nor_i →Xi, so they do not have to be even reduced. It will only be important thatVei’s are equivari- ant when a smooth group acts onX. In Gabber’s original argument it was important to blow up only regular centers because they were not part of the blow up data, and one used that a regular center without codimen- sion one components intersecting the regular locus is determined already by the underlying morphism of the blow up.

2.3.13. Alternative desingularization inputs. — For the sake of completeness, we discuss how other algorithms could be used instead ofF. Some desingularization algorithms for reduced varieties overQare constructed in [Bierstone & Milman, 1997], [Włodarczyk, 2005], [Bravo et al., 2005], and [Kollár, 2007]. They all are functorial with respect to equidimensional smooth morphisms (though usually one "forgets" to mention the equidimensionality restriction). It is shown in [Bierstone et al., 2011, §6.3] how to make the algorithm of [Bierstone & Milman, 1997] fully functorial by a slight adjusting of the synchronization of its blow ups. All

these algorithms can be used to produce a desingularization of log regular schemes (see §3), so the only difficulty is in establishing the compatibility (2.3.7.1).

For the algorithm of [Bierstone et al., 2011] it was shown by Bierstone-Milman (unpublished, see [Bierstone et al., 2011, Rem. 7.1(2)]) that the induced desingularization of a formal completion at a point depends only on the formal completion as a scheme. This is precisely what we need in (2.3.7.1).

Finally, there is a much more general result by Gabber, see Theorem 2.4.1, whose proof uses Popescu’s theorem and the cotangent complex. It implies that, actually, any desingularization of reduced varieties over Qwhich is functorial with respect to smooth morphisms automatically satisfies (2.3.7.1). So, in principle, any functorial desingularization of varieties over Qcould be used for our purposes. Since Gabber’s result and its proof are powerful and novel for the desingularization theory (and were missed in [Bierstone et al., 2011], mainly due to a not so trivial involvement of the cotangent complex), we include them in §2.4.

2.3.14. Invariance of the regular locus. — Until the end of §2.3we consider only qe schemes of characteristic zero, and our aim is to establish a few useful properties ofF (andFf) that are consequences of the functoriality propertyF satisfies. First, we claim thatF does not modify the regular locus ofX, and even slightly more than that:

Corollary 2.3.15. — All centers ofF(X)andFf(X)sit over the singular locus ofX. In particular,Xis regular if and only ifF(X)is the trivial tower.

Proof. — It suffices to studyF. The claim is obvious forS =Spec(Q)becauseSdoes not contain non-dense non-empty subschemes. By functoriality,F(T)is trivial for any regularT of characteristic zero, because it is regular overS. Finally, ifT is the regular locus ofXthenF(T) = (F(X)×XT)c and hence any centerVi,→Xi

ofF(X)does not intersect the preimage ofT.

2.3.16. Equivariance of the desingularization. — It is well known that functorial desingularization is equivariant with respect to any smooth group action (and, moreover, extends to functorial desingularization of stacks, see [Temkin, 2012, Th. 5.1.1]). For the reader’s convenience we provide an elementary argument.

Corollary 2.3.17. — LetSbe a qe scheme overQ,Gbe a smoothS-group andXbe a reducedS-scheme of finite type acted on byG. Then there exists a unique action ofGonF(X)andFf(X)that agrees with the given action onX. Proof. — Again, it suffices to studyF. LetF(X)be given byXn 99KX0 =XandVi,→Xifor0≤i ≤n−1.

Byp, m:Y = G×SX → Xwe denote the projection and the action morphisms. Note thatmis smooth (e.g.

mis the composition of the automorphism(g, x)→ (g, gx)ofG×SXandp). Therefore,F(X)×X(Y;m) = F(Y) =F(X)×SGby Theorem2.3.10and Remark2.3.5(i). In particular,V₀×X(Y;p) =V₀×X(Y;m), i.e.V₀is G-equivariant. By2.1.11,X₁inherits aG-action. Then the same argument implies thatV₁isG-equivariant and X₂inherits aG-action, etc.

2.4. Complements on functorial desingularizations. — This section is devoted to Gabber’s result on a certain non-trivial compatibility property that any functorial desingularization satisfies. It will not be used in the sequel, so an uninterested reader may safely skip it.

Theorem 2.4.1. — Assume thatS is a noetherian scheme,S⁰is a class of reducedS-schemes of finite type andS¹ is a class of morphisms between elements ofS⁰ such that iff:Y → X is smooth and X ∈ S⁰thenY ∈ S⁰and f ∈S¹. LetF be a desingularization onS⁰which is functorial with respect to all morphisms ofS¹. Then any pair of regular morphismsg:Z → Xandh: Z→ Y with targets inS⁰induces the same desingularization ofZ; namely, F(X)×XZ=F(Y)×YZ.

Note that the theorem has no restrictions on the characteristic (because no such restriction appears in Popescu’s theorem). Before proving the theorem let us formulate its important corollary, whose main case is whenS=Spec(k)for a fieldkandS⁰is the class of all reducedk-schemes of finite type.

Corollary 2.4.2. — Keep the notation of Theorem2.4.1. ThenF canonically extends to the classSc⁰of all schemes that admit a regular morphism to a scheme fromS⁰and the extension is functorial with respect to all regular morphisms between schemes ofSc⁰.

The main ingredient of the proof will be the following result that we are going to establish first.

Proposition 2.4.3. — Consider a commutative diagram of noetherian schemes Z

g

~~

h

f

X

a Z⁰

g⁰

oo ^h0 //Y

 b

S

such thataandbare of finite type,gandhare regular andg⁰is smooth. Thenh⁰is smooth around the image off.

For the proof we will need the following three lemmas. In the first one we recall the Jacobian criterion of smoothness, rephrased in terms of the cotangent complex.

Lemma 2.4.4. — Letf:X→Sbe a morphism which is locally of finite presentation, and letx∈X. Then the following conditions are equivalent:

(i)fis smooth atx; (ii)H1(L_X/S⊗^L k(x)) =0.

In the lemma we use the convention Hi=H⁻ⁱ, andL_X/Sdenotes the cotangent complex ofX/S.

Proof. — (i) ⇒(ii) is trivial: asfis smooth atx, up to shrinking Xwe may assumefsmooth, then LX/S is cohomologically concentrated in degree zero and locally free [Illusie, 1972, III 3.1.2]. Let us prove (ii)⇒(i).

We may assume that we have a factorization

X ⁱ //

f

Z

 g

S

,

whereiis a closed immersion of idealIandgis smooth. Consider the standard exact sequence

(∗) I/I²→i^?Ω¹_Z/S →Ω¹_X/S→0.

By the Jacobian criterion [ÉGAIV₄17.12.1] and [ÉGA0IV19.1.12], the smoothness offatxis equivalent to the fact that the morphism

(∗∗) (I/I²)⊗k(x)→Ω¹_Z/S⊗k(x)

deduced from the left one in (∗) is injective. Now,(I/I²)⊗k(x) =H1(LX/Z

⊗L k(x))[Illusie, 1972, III 3.1.3], and (∗∗) is a morphism in the exact sequence associated with the triangle deduced from the transitivity triangle Li^?LZ/S→LX/S→LX/Z→Li^?LZ/S[1]by applying⊗k(x):^L

H1(LX/S

⊗L k(x))→H1(LX/Z

⊗L k(x))(= (I/I²)⊗k(x))→Ω¹_Z/S⊗k(x).

By (ii), H1(L_X/S⊗^L k(x)) =0, hence (∗∗) is injective, which completes the proof.

Lemma 2.4.5. — Consider morphismsf:X→Y,g:Y →S,h=gf:X→S, and letx∈X,y=f(x)∈Y. Assume that

(i)H1(L_X/S⊗^L k(x)) =0 (ii)H2(LX/Y

⊗L k(x)) =0.

ThenH1(L_Y/S⊗^L k(y)) =0. In particular, ifgis locally of finite presentation thengis smooth aty.

Proof. — It is equivalent to show that H1(L_Y/S⊗^L k(x)) =0, and this follows trivially from the exact sequence H2(LX/Y

⊗L k(x))→H1(LY/S

⊗L k(x))→H1(LX/S

⊗L k(x)).

Lemma 2.4.6. — Letf:X → S be a regular morphism between noetherian schemes. Then LX/S is cohomologically concentrated in degree zero andH0(L_X/S) =Ω¹_X/Sis flat.

Proof. — We may assumeX=SpecBandS=SpecAaffine. Then, by Popescu’s theorem [Swan, 1998, 1.1],X is a filtering projective limit of smooth affineS-schemesXα=SpecBα. By [Illusie, 1972, II (1.2.3.4)], we have

L_B/A=colimαL_Bα/A.

By [Illusie, 1972, III 3.1.2 and II 2.3.6.3], L_Bα/A is cohomologically concentrated in degree zero and H0(LB_α/A) =Ω¹_B

α/Ais projective of finite type overBα, so the conclusion follows.

Proof of Proposition2.4.3. — The compositionag⁰:Z⁰→Sis locally of finite type. SinceSis noetherian,ag⁰is locally of finite presentation, and soh⁰is locally of finite presentation too.

Next, we note that the question is local around a pointy = f(x)∈ Z⁰,x∈ Z. In view of Lemma2.4.4, by Lemma2.4.5applied toZ→Z⁰ →Yit suffices to show that H1(L_Z/Y⊗^L k(x)) =0and H2(L_Z/Z0

⊗L k(x)) =0. AsZis regular overY, the first vanishing follows from Lemma2.4.6. For the second one, consider the exact sequence

H2(L_Z/X⊗^L k(x))→H2(L_Z/Z0

⊗L k(x))→H1(L_Z0/X

⊗L k(x)).

By the regularity ofZ/Xand Lemma2.4.6, H2(L_Z/X⊗^L k(x)) =0. AsZ⁰is smooth overX, H1(L_Z⁰_/X⊗^Lk(x)) =0 by Lemma2.4.4, which proves the desired vanishing and finishes the proof.

Proof of Theorem2.4.1. — Find finite affine coveringsX=S

iXi,Y=S

iYiandZ=S

iZisuch thatg(Zi)⊂Xi

andh(Zi)⊂ Yi. SetX⁰ = `

iXi,Y⁰ = `

iYiandZ⁰ = `

iZi and letZ⁰ → X⁰ andZ⁰ → Y⁰ be the induced morphisms. It suffices to check thatF(X)×XZandF(Y)×YZbecome equal after pulling them back toZ⁰. So, we should check that(F(X)×XX⁰)×X⁰ Z⁰coincides with(F(Y)×YY⁰)×Y⁰Z⁰. The morphismsX⁰ →Xand Y⁰→Yare smooth and hence contained inS¹. So,F(X)×_XX⁰ =F(X⁰)and similarly forY. In particular, it suffices to prove thatF(X⁰)×_X⁰Z⁰ =F(Y⁰)×_Y⁰Z⁰. This reduces the problem to the case when all schemes are affine, so in the sequel we assume thatX, YandZare affine.

Next, note that it suffices to find factorizationsg=g0fandh=h0f, wheref:Z→Z0is a morphism with target inS⁰andg₀:Z₀ → X,h₀: Z₀ →Xare smooth. By Popescu’s theorem, one can writeg:Z →Xas a filtering projective limit of affine smooth morphismsg_α: Z_α →X,α∈A. AsYis of finite type overS,hwill factor through one of theZα’s ([ÉGA IV3 8.8.2.3]): there existsα ∈ A,fα: Z → Zα,hα: Zα → Y such that g = gαfα,h = hαfα. By Proposition2.4.3,hα is smooth around the image offα, so we can takeZ0to be a sufficiently small neighborhood of the image offα.

3. Resolution of log regular log schemes

All schemes considered from now on will be assumed to be noetherian. Unless said to the contrary, by log structure we mean a log structure with respect to the étale topology. We will say that a log structureM_Xon a schemeXisZariskiifε^?ε?MX→∼ MX, whereε:X_´et →Xis the morphism between the étale and Zariski sites.

In this case, we can safely view the log structure as a Zariski log structure. A similar convention will hold also for log schemes.

3.1. Fans. — Many definitions/constructions on log schemes are of "combinatorial nature". Roughly speak- ing, these constructions use only multiplication and ignore addition. Naturally, there exists a category of geometric spaces whose structure sheaves are monoids, and most of combinatorial constructions can be de- scribed as "pullbacks" of analogous "monoidal" operations. The first definition of such a category was done by Kato in [Kato, 1994]. Kato called his spacesfansto stress their relation to the classical combinatorial fans obtained by gluing polyhedral cones. For example, to any combinatorial fanCone can naturally associate a fanF(C)whose set of points is the set of faces ofC. The main motivation for the definition is that fans can be naturally associated to various log schemes.

It took some time to discover that fans are sort of "piecewise schemes" rather than a monoidal version of schemes. A more geometric version of combinatorial schemes was introduced by Deitmar in [Deitmar, 2005].

He called themF1-schemes, but we prefer the terminology of monoschemes introduced by Ogus in his book project [Ogus, 2013]. Note that when working with a log schemeX, we use the sheafMXin some constructions and we use its sharpeningMX(see3.1.1) in other constructions. Roughly speaking, monoschemes naturally arise when we work withMXwhile fans naturally arise when we work withMX.

In §3, we will show that: (a) a functorial desingularization of toric varieties overQ descends to a desin- gularization of monoschemes, (b) to give the latter is more or less equivalent to give a desingularization of fans, (c) a desingularization of fans can be used to induce a monoidal desingularization of log schemes, (d) the

latter induces a desingularization of log regular schemes, which (at least in some cases) depends only on the underlying scheme.

In principle, we could work locally, using desingularization of disjoint unions of all charts for synchroniza- tion. In this case, we could almost ignore the intermediate categories by working only with fine monoids and blow up towers of their spectra. However, we decided to emphasize the actual geometric objects beyond the constructions, and, especially, stress the difference between fans and monoschemes.

3.1.1. Sharpening. — For a monoidM, byM^∗we denote the group of its invertible elements, and itssharpen- ingMis defined asM/M^∗.

3.1.2. Localization. — By localizationof a monoid M along a subset S we mean the universal M-monoid MS such that the image ofSinMS is contained inM^∗_S. IfMis integral thenMS is simply the submonoid M[S⁻¹]⊂M^gpgenerated byMandS⁻¹. IfMis a fine then any localization is isomorphic to a localization at a single elementf, and will be denotedMf.

3.1.3. Spectra of fine monoids. — All our combinatorial objects will be glued from finitely many spectra of fine monoids. Recall that with any fine monoidPone can associate the set Spec(P)of prime ideals (with the convention that∅is also a prime ideal) equipped with the Zariski topology whose basis is formed by the sets D(f) ={p∈Spec(P)|f /∈p}forf∈P, see, for example, [Kato, 1994, §9]. The structure sheafMPis defined by MP(D(f)) =Pf, and the sharp structure sheafMP =MP/M^∗_Pis the sharpening ofMP (we will see in Remark 3.1.4(iii) that actuallyMP(D(f)) =Pf=Pf/(P^∗_f), i.e. no sheafification is needed).

Remark 3.1.4. — (i) SinceP\P^∗and∅are the maximal and the minimal prime ideals ofP, Spec(P)possesses unique closed and generic pointssandη. The latter is the only point whose stalkMP,η=P^gpis a group.

(ii) The set Spec(P)is finite and its topology is the specialization topology, i.e. Uis open if and only if it is closed under generizations. (More generally, this is true for any finite sober topological space, such as a scheme that has finitely many points.)

(iii) A subsetU⊂Spec(P)is affine (and even of the formD(f)) if and only if it is the localization of Spec(P) at a pointx(i.e. the set of all generizations ofx). Any open coveringU= S

iUiof an affine set is trivial (i.e.

Uis equal to someUi), therefore any functorF(U)on affine sets uniquely extends to a sheaf on Spec(P). In particular, this explains why no sheafification is needed when definingMP. Furthermore, we see that, roughly speaking, any notion/construction that is "defined in terms of" localizationsXxand stalksMxorMxis Zariski local. This is very different from the situation with schemes.

3.1.5. Local homomorphisms of monoids. — Any monoidMis local becauseM\M^∗is its unique maximal ideal.

A homomorphismf: M→Nof monoids islocalif it takes the maximal ideal ofMto the maximal ideal ofN.

This happens if and only iff⁻¹(N^∗) =M^∗.

3.1.6. Monoidal spaces. — Amonoidal spaceis a topological spaceXprovided with a sheaf of monoidsM_X. A morphism of monoidal spaces(f, f^#) : (Y, MY)→(X, MX)is a continuous mapf:Y →Xand a homomorphism f^#:f⁻¹(MX)→MY such that for anyy∈Ythe homomorphism of monoidsf^#_y:M_X,f(y)→MY,yis local.

Remark 3.1.7. — Strictly speaking one should have called the above category the category of locally monoidal spaces and allow non-local homomorphisms in the general category of monoidal spaces. However, we will not use the larger category, so we prefer to abuse the terminology slightly.

Spectra of monoids possess the usual universal property, namely:

Lemma 3.1.8. — Let(X, MX)be a monoidal space and letPbe a monoid.

(i) The global sections functor Γ induces a bijection between morphisms of monoidal spaces (f, f^#) : (X, MX) → (Spec(P), MP)and homomorphismsφ:P→Γ(MX).

(ii) IfMXhas sharp stalks then Γ induces a bijection between morphisms of monoidal spaces (f, f^#) : (X, MX) → (Spec(P), MP)and homomorphismsφ:P→Γ(MX).

Proof. — (i) Let us construct the opposite map. Given a homomorphismφ, for anyx ∈ Xwe obtain a ho- momorphismφx:P → MX,x. Clearly,m= P\φ⁻¹_x (M^∗_X,x)is a prime ideal and henceφxfactors through a uniquely defined local homomorphismPm →MX,x. Settingf(x) = mwe obtain a mapf:X→Spec(P), and the rest of the proof of (i) is straightforward.

If the stalks of M_X are sharp then any morphism (X, M_X) → (Spec(P), M_P) factors uniquely through (Spec(P), MP). Also, Γ(MX) is sharp, hence any homomorphism to it fromP factors uniquely through P.

Therefore, (ii) follows from (i).

3.1.9. Fine fans and monoschemes. — A finemonoscheme(resp. a finefan) is a monoidal space(X, MX)that is locally isomorphic toAP = (Spec(P), MP)(resp.AP = (Spec(P), MP)), wherePis a fine monoid. We say that (X, MX)isaffineif it is isomorphic toAP(resp.AP). A morphism of monoschemes (resp. fans) is a morphism of monoidal spaces. A monoscheme (resp. a fan) is calledtorsion freeif it is covered by spectra ofP’s with torsion freeP^gp’s. It follows from Remark3.1.4(iii) that this happens if and only if all groupsM^gp_X,xare torsion free.

Remark 3.1.10. — (i) Any fs fan is torsion free because if an fs monoid is torsion free then any of its localization is so. In particular, ifPis fs and sharp thenP^gpis torsion free. This is not true for general fine fans. For example, ifµ2={±1}thenP=N⊕µ2\ {(0,−1)}is a sharp monoid withP^gp=Z⊕µ2.

(ii) For any pointxof a fine monoscheme (resp. fan)Xthe localizationXxthat consists of all generizations ofxis affine. In particular, by Remark3.1.4(i) there exists a unique maximal point generizingx, and henceXis a disjoint union of irreducible components.

3.1.11. Comparison of monoschemes and fans. — There is an obvious sharpening functor(X, MX) 7→ (X, MX) from monoschemes to fans, and there is a natural morphism of monoidal spaces (X, MX) → (X, MX). The sharpening functor loses information, and one needs to know M^gp_X to reconstruct MX fromMX as a fibred product (see [Ogus, 2013]). Actually, there are much more fans than monoschemes. For example, the generic pointη ∈ Spec(P)is open andMP,ηis trivial hence for any pair of fine monoidsPandQwe can glue their sharpened spectra along the generic points. What one gets is sort of "piecewise scheme" and, in general, it does not correspond to standard geometric objects, such as schemes or monoschemes. We conclude that, in general, fans can be lifted to monoschemes only locally.

Remark 3.1.12. — As a side remark we note that sharpened monoids naturally appear as structure sheaves of piecewise linear spaces (a work in progress of the second author on skeletons of Berkovich spaces). In particular, PL functions can be naturally interpreted as sections of the sharpened sheaf of linear functions on polytopes.

3.1.13. Local smoothness. — A local homomorphism of fine monoidsφ:P → Qis calledsmoothif it can be extended to an isomorphismP⊕N^r⊕Z^s→^∼ Q. The following lemma checks that this property is stable under localizations.

Lemma 3.1.14. — Assume thatφ: P→Qis smooth andP⁰, Q⁰are localizations ofP, Qsuch thatφextends to a local homomorphismφ⁰:P⁰ →Q⁰. Thenφ⁰is smooth.

Proof. — Recall thatP⁰=P_afora∈P(notation of3.1.2), andφ⁰factors through the homomorphismφ_a:P⁰→ Q_a, which is obviously smooth. Therefore, replacingφwithφ_awe can assume thatP=P⁰. Letb= (p, n, z)∈ Qbe such thatQ⁰ =Q_b. Thenp∈P^∗becauseP→Q⁰is local, and henceQ⁰is isomorphic toP⊕(N^r)_n⊕Z^s. It remains to note that any localization ofN^ris of the formN^r−t⊕Z^t.

3.1.15. Smoothness. — The lemma enables us to globalize the notion of smoothness: a morphismf:Y →Xof monoschemes is calledsmoothif the homomorphisms of stalksM_X,f(y)→M_Y,yare smooth. In particular,X is smooth if its morphism to Spec(1)is smooth, that is, the stalksM_X,xare of the formN^r⊕Z^s. In particular, a smooth monoscheme is torsion free.

Analogous smoothness definitions are given for fans. Moreover, in this case we can consider only sharp monoids, and then the group componentZ^sis automatically trivial. It follows that we can rewrite the above paragraph almostverbatimbut withs=0. Obviously, a morphism of torsion free fs monoschemes is smooth if and only if its sharpening is a smooth morphism of fans.

Remark 3.1.16. — (i) Recall that any fine monoschemeXadmits an open affine coveringX=S

x∈XSpec(M_X,x). It follows that a morphism of fine monoschemesf:Y →Xis smooth if and only if it is covered by open affine charts of the form Spec(P⊕N^r⊕Z^s)→Spec(P).

(ii) Smooth morphisms of fine fans admit a similar local description, and we leave the details to the reader.

3.1.17. Saturation. — As usually, for a fine monoidPwe denote its saturation byP^sat(it consists of allx∈P^gp withxⁿ∈Pfor somen > 0). Saturation is compatible with localizations and sharpening and hence extends to a saturation functorF 7→F^sat on the categories of fine monoschemes (resp. fine fans). We also have a natural morphismF^sat→F, which is easily seen to be bijective. So, actually,(F, M_F)^sat= (F, M^sat_F ).

3.1.18. Ideals. — A subsheaf of idealsI ⊂MXon a monoscheme(X, MX)is called acoherent idealif for any pointx∈Xthe restriction ofI onXxcoincides withIxMXx. (Due to Remark3.1.4(iii) this means thatI is coherent in the usual sense, i.e. its restriction on an open affine submonoschemeUis generated by the global sections overU.) We will consider only coherent ideals, so we will omit the word "coherent" as a rule. An ideal I ⊂M_Xisinvertibleif it is locally generated by a single element.

3.1.19. Blow ups. — Similarly to schemes, for any ideal I ⊂ OX there exists a universal morphism of monoschemesh:X⁰ →Xsuch that thepullback idealh⁻¹I =IMX⁰ is invertible. We callI thecenterof the blow up. (One does not have an adequate notion of closed submonoscheme, so unlike blow ups of the scheme it would not make sense to say that "V(I)" is the center.) An explicit construction of X⁰ copies its scheme analog: it is local on the base and for an affineX=Spec(P)with an idealI⊂Pcorresponding toI one glues X⁰ from the charts Spec(P[a⁻¹I]), wherea∈IandP[a⁻¹I]is the submonoid ofP^gpgenerated by the fractions b/aforb∈I(see [Ogus, 2013] for details).

Remark 3.1.20. — Blow ups induce isomorphisms on the stalks ofM^gp; this is an analog of the fact that blow ups of schemes along nowhere dense subschemes are birational morphisms.

3.1.21. Saturated blow ups. — Analogously to normalized blow ups, one defines saturated blow up of a monoschemeXalong an idealI ⊂MXas the saturation of Bl_I(X). The same argument as for schemes shows that Bl_I(X)^satis the universal saturatedX-monoscheme such that the pullback ofI is invertible.

3.1.22. Towers and pullbacks. — Towers of (saturated) blow ups of a monoschemeXare defined analogously to towers of (normalized) blow ups. In particular, saturated towers start with the saturation morphismX₀→X₋₁. Given such a towerX_•withX=X₀(resp.X₋₁=X) and a morphismf:Y →Xwe define thepullback tower Y_• =f^?(X_•)as follows:Y₀=Y(resp.Y₋₁=Y) andY_i+1is the (saturated) blow up ofY_ialong the pullback of the centerIiofX_i+1→X_i. Due to the universal property of (saturated) blow ups this definition makes sense andY_•is the universal (saturated) blow up tower ofYthat admits a morphism toX_•extendingf.

Remark 3.1.23. — Unlike pullbacks of (normalized) blow up towers of schemes, see 2.1.7and2.2.8, we do not distinguish strict transforms and pullbacks. The above definition of pullback covers our needs, and we do not have to study the base change of monoschemes. For the sake of completeness, we note that fibred products of monoschemes exist and in the affine case are defined by amalgamated sums of monoids, see [Deitmar, 2005]. Also, it is easy to check that for a smoothf(which is the only case we will use) one indeed has thatf^?(X•) = X•×XY for any (saturated) blow up towerX•. For blow ups one checks this with charts and in the saturated case one also uses that saturation is compatible with a smooth morphismf:Y →X, i.e.

Y^sat→^∼ X^sat×XY.

3.1.24. Compatibility with sharpening. — Ideals and blow ups of fans are defined in the same way, but withMX

used instead ofMX(Kato defines their saturated version in [Kato, 1994, 9.7]). Towers of blow ups of fans are defined in the obvious way.

Lemma 3.1.25. — Let X = (X, MX) be a monoscheme, let (F, MF) = (X, MX)be the corresponding fan and let λ:MX→MFdenote the sharpening homomorphism.

(i)I 7→λ(I)induces a natural bijection between the ideals onXand onF.

(ii) Blow ups are compatible with sharpening, that is, the sharpening ofBl_I(X))is naturally isomorphic toBlλ(I)(F).

The same statement holds for saturated blow ups.

(iii) Sharpening induces a natural bijection between the (saturated) blow up towers ofXandF.

Proof. — (i) is obvious. (ii) is shown by comparing the blow up charts. Combining (i) and (ii), we obtain (iii).

3.1.26. Desingularization. — Using the above notions of smoothness and blow ups, one can copy other def- initions of the desingularization theory. By a desingularization (resp. saturated desingularization) of a fine monoscheme X we mean a blow up tower (resp. saturated blow up tower) Xn 99K X0 = X (resp.

Xn99KX−1=X) with smoothXn. By Remark3.1.20, ifXadmits a desingularization then it is torsion free, and we will later see that the converse is also true.

For concreteness, we consider below non-saturated desingularizations, but everything extends to the satu- rated case verbatim. A familyF^mono(X)of desingularizations of torsion free monoschemes is calledfunctorial (with respect to smooth morphisms) if for any smoothf:Y → Xthe desingularizationF^mono(Y)is the con- tracted pullback ofF^mono(X). The same argument as for schemes (see Remark2.3.5(ii)) shows thatF^monois already determined by its restriction to the family of finite disjoint unions of affine monoschemes.