version du 2016-11-14 à 13h36TU (19c1b56)
Table des matières
1. The main theorem. . . 1
2. Prime to`variants of de Jong’s alteration theorems. . . 12
3. Resolvability, log smoothness, and weak semistable reduction. . . 17
References. . . 28
In this exposé we state and prove a variant of the main theorem ofVIII(see exp.VIII,1.1) for schemesX which are log smooth over a baseSwith trivialG-action. See1.1for a precise statement. The proof is given in
§1 and in the remaining part of the exposé we deduce refinements of classical theorems of de Jong, for schemes of finite type over a field or a trait, where the degree of the alteration is made prime to a prime`invertible on the base. Sections2and3are independent and contain two different proofs of such a refinement, so let us outline the methods briefly.
For concreteness, assume thatkis a field,S = Spec(k), andXis a separatedS-scheme of finite type. Two methods to construct regular`0-alterations ofXare: (1) use a pluri-nodal fibration to construct a regularG- alterationX0 → Xand then factorX0 by an`-Sylow subgroup ofG, and (2) construct a regular`0-alteration by induction on dim(S)so that one factors by an`-Sylow subgroup at each step of the induction. The first approach is presented in §2. It is close in spirit to the approach of [de Jong, 1997] and its strengthening by Gabber-Vidal, see [Vidal, 2004, §4]. The weak point of this method is that one uses inseparable Galois alter- ations. In particular, even whenkis perfect, one cannot obtain a separable alteration ofX.
The second approach is realized in §3, using [Temkin, 2010]; it outperforms the method of §2 whenkis perfect. Moreover, developing this method the second author discovered Theorem3.5that generalizes Gab- ber’s theorems 2.1and2.4to the case of a general base S satisfying a certain resolvability assumption (see
§3.3). In addition, ifSis of characteristic zero then the same method allows to use modifications instead of
`0-alterations, see Theorem3.9. As an application, in Theorem3.10we generalize Abramovich-Karu’s weak semistable reduction theorem. Finally, we minimize separatedness assumptions in §3, and for this we show in
§3.1how to weaken the separatedness assumptions in Theorems1.1and exp.VIII,1.1.
1. The main theorem
Theorem 1.1. — Letf:X→Sbe an equivariant log smooth map between fs log schemes endowed with an action of a finite groupG. Assume that:
(i)Gacts trivially onS;
(ii)XandSare noetherian, qe, separated, log regular, andfdefines a map of log regular pairs(X, Z)→(S, W)(see exp.VI,1.4:(X, Z)and(S, W)are log regular pairs andf(X−Z)⊂S−W));
(iii)Gacts tamely and generically freely onX.
LetT be the complement of the largest open subset ofXover whichGacts freely. Then there exists an equivariant projective modificationh:X0 →Xsuch that, ifZ0 =h−1(Z∪T), the pair(X0, Z0)is log regular, the action ofGonX0 is very tame, and(X0, Z0)is log smooth over(S, W)as well as the quotient(X0/G, Z0/G)whenGacts admissibly onX ([SGA 1V1.7]).
Remark 1.1.1. — (a) In the absence of the hypothesis (i) it may not be possible to find a modificationhsatis- fying the properties of1.1, as the example at the end of exp.VIII,1.2shows.
(b) By [Kato, 1994, 8.2] the log smoothness off and the log regularity ofS imply the log regularity ofX. Conversely, according to Gabber (private communication), ifXis log regular andfis log smooth and surjective, thenSis log regular.
(c) We will deduce Theorem1.1from Theorem exp.VIII,1.1. Recall that in the latter theorem we assumed thatXis qe, though Gabber has a subtler argument that works for a generalX. This forces us to assume that
(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
S(and henceX) is qe in Theorem1.1. However, our argument also shows that once one removes the quasi- excellence assumption from exp.VIII,1.1, one also obtains the analogous strengthening of Theorem1.1.
For the proof of1.1we will use the following result on the local structure of equivariant log smooth maps.
Proposition 1.2(Gabber’s preparation lemma). — Letf :X →Y be an equivariant log smooth map between fine log schemes endowed with an action of a finite groupG. Letxbe a geometric point ofX, with imageyinY. Assume that Gis the inertia group atxand is of order invertible onY. Assume furthermore thatGacts trivially onMxandMy(ii)
and we are given an equivariant charta:Y →SpecΛ[Q]aty, modeled on some pairingχ:Gab⊗Qgp→µ=µN(C) (in the sense of (exp.VI,3.3)), where Qis fine,Λ = Z[1/N, µ], withNthe exponent ofG. Then, up to replacingX by an inert equivariant étale neighborhood ofx, there is an equivariant chartb : X → SpecΛ[P]extendinga, such thatQgp → Pgpis injective, the torsion of its cokernel is annihilated by an integer invertible onX, and the resulting mapb0 : X→ X0 = Y×SpecΛ[Q]SpecΛ[P]is smooth. Moreover, up to further shrinkingXaroundx,b0 lifts to an inert equivariant étale mapc : X → X0×SpecΛSpec SymΛ(V), whereVis a finitely generated projectiveΛ-module equipped with aG-action. IfX,Y, andQare fs, withQsharp, thenPcan be chosen to be fs with its subgroup of unitsP∗ torsionfree.
Proof of1.2. — This is an adaptation of the proof of [Kato, 1988, 3.5] to the equivariant case. Consider the canonical homomorphism ofloc. cit.
(1.2.1) k(x)⊗OX,xΩ1X/Y,x→k(x)⊗ZMgpX/Y,x sending1⊗dlogtto the class of1⊗t, where
MgpX/Y,x=MgpX,x/(OX,x∗ +Imf−1(MgpY,y)).
It is surjective, and asGfixesx, it isG-equivariant. AsGis of order invertible ink(x)and acts trivially on the right hand side, (1.2.1) admits aG-equivariant decomposition
(1.2.2) k(x)⊗OX,xΩ1X/Y,x=V0⊕(k(x)⊗ZMgpX/Y,x),
whereV0is a finite dimensionalk(x)-vector space, endowed with an action ofG. Let(ti)1≤i≤rbe elements ofMgpx such that the classes of1⊗tiform a basis ofk(x)⊗ZMgpX/Y,x. By the method of (exp.VI,3.5) we can modify theti’s to make them eigenfunctions ofG. More precisely, forg∈G, we have
gti=zi(g)ti,
with zi(g) ∈ Ox∗, and g 7→ zi(g) is a 1-cocycle of G with values inOx∗. By reduction modmx, it gives a 1-cocycleψi ∈ Z1(G, µ) = Hom(G, µ), asµis naturally embedded ink(x)∗ since Xis overΛ. Liftingµ in Ox∗, g 7→ zi(g)/ψi(g)is a 1-cocycle of G with values in1+ mx, hence a coboundaryδi ∈ B1(G, 1+ mx), g7→δi(g) =gui/ui, forui∈1+ mx. Replacingtibytiu−1i , we may assume thatzi=ψi,i.e.
gti=ψi(g)ti, for characters
ψi:G→µ.
LetZbe the free abelian group with basis(ei)1≤i≤r, andh:Z→Mgpx the homomorphism sendingeitoti. As in the proof of [Kato, 1988, 3.5], consider the homomorphism
u:Z⊕Qgp→Mgpx
defined byhonZand the compositionQgp→Mgpy →Mgpx on the second factor. We have gu(a) =ψ(g⊗a)u(a)
for some homomorphism
ψ:Gab⊗(Z⊕Qgp)→µ
extendingχand such thatψ(g⊗ei)u(ei) =ψi(g)h(ei). As inloc. cit., ifudenotes the composition u:Z⊕Qgp→Mgpx →Mgpx(=Mgpx /Ox∗)
we see thatk(x)⊗uis surjective, hence the cokernelCofuis killed by an integerminvertible ink(x). Using thatOX,x∗ ism-divisible, one can choose elementsai∈Mgpx andbi∈Z⊕Qgp(1≤i≤n) such that the images
(ii)IfMis the sheaf of monoids of a log scheme,Mdenotes, as usual, the quotientM/O∗.
of theai’s generateMgpx andami =u(bi). LetEbe the free abelian group with basisei(1≤i≤n), and letFbe the abelian group defined by the push-out diagram
(1.2.3) E m //
E
Z⊕Qgp w //F
,
where the left vertical arrow sendseitobi. The lower horizontal map is injective and its cokernel is isomorphic toE/mE, in particular, killed bym. The relationami =u(bi)implies thatuextends to a homorphism
v:F→Mgpx
whose compositionv:F→Mgpx →Mgpx is surjective. Associated withvis a morphism ϕ:Gab⊗F→µ
extendingψ, such thatgv(a) = ϕ(g⊗a)v(a)fora ∈ F. LetP := v−1(Mx) ⊂ F. ThenP is a fine monoid containingQ,Pgp = F, andvsendsPtoMx. As in exp.VI,3.5, exp.VI,3.10we get aG-equivariant chart of X(x)associated withϕ, which, up to replacingXby an inert equivariant étale neighborhood atx, extends to an equivariant chart
b:X→SpecΛ[P]
extending the charta:Y →SpecΛ[Q]. The homomorphismQgp→Pgpis injective, and the torsion part of its cokernel injects into the cokernel ofw : Z⊕Qgp →Fin (1.2.3), which is killed bym. Consider the resulting map
b0:X→X0=Y×SpecΛ[Q]SpecΛ[P].
This map is strict. Showing that the underlying schematic map is smooth atxis equivalent to showing thatb0 is log smooth atx. To do this, asXandX0 are log smooth overY, by the jacobian criterion [Kato, 1988, 3.12] it suffices to show that the map
k(x)⊗Ω1X0/Y →k(x)⊗Ω1X/Y induced byb0is injective. We have
k(x)⊗Ω1X0/Y =k(x)⊗Pgp/Qgp=k(x)⊗Z
(the last equality by the fact thatF/(Z⊕Qgp)is killed bym), and by construction (cf. (1.2.2)), we have k(x)⊗Z=k(x)⊗MgpX/Y,x,
which by the map induced byb0injects intok(x)⊗Ω1X/Y.
Let us now prove the second assertion. For this, asb0 is strict, we may forget the log structures ofXand X0, and by changing notations, we may assume thatX0 = Y and the log structures ofXandYare trivial. In particular, we have
k(x)⊗Ω1X/Y =V0,
with the notation of (1.2.2). As the question is étale local onX, and closed points are very dense in the fiberXy, in particular, any point has a specialization at a closed point ofXy, we may assume thatxsits over a closed point ofXy, and even, up to base changingYby a finite radicial extension, thatxis a rational point ofXy. We then have
k(x)⊗Ω1X/Y = mx/(m2x+ myOx),
wheremdenotes a maximal ideal. By a classical result in representation theory (see1.3below) there is a finitely generated projectiveΛ[G]-moduleV such thatV0 = k(x)⊗V. The homomorphismV → mx/(m2x+ myOx) therefore lifts to a homomorphism ofΛ[G]-modules
V→mx,
inducing an isomorphismk(x)⊗V→k(x)⊗Ω1X/Y. By the jacobian criterion, it follows that the (G-equivariant) map
X→Y×SpecΛSpec SymΛ(V) is étale atx, and as in exp.VI,3.10can be made inert by shrinkingX.
Let us prove the last assertion. NowXandYare fs, andQis fs and sharp. First of all, asMxis saturated, P=ν−1(Mx)is fs. Then (cf. [Gabber & Ramero, 2013, 3.2.10]) we have a split exact sequence
0→H→P→P0→0
withP0∗torsionfree andHa finite group. AsQis fs and sharp,Qgpis torsionfree, so the compositionQgp → Pgp→Pgp0 is still injective, as well as the compositionH→Pgp→Pgp/Qgp, henceHis contained in the torsion part ofPgp/Qgp, and we have an exact sequence
0→H→(Pgp/Qgp)tors→(Pgp0 /Qgp)tors→0,
where the subscripttorsdenotes the torsion part. Thus(Pgp0 /Qgp)torsis killed by an integer invertible onX. As Mxis torsionfree, the compositionP→Mx→Mxfactors throughP0, into a mapv0:P0→Mx. Consider the diagram
Mx
//Mgpx
P0 v0 //Mx //Mgpx ,
where the square is cartesian. AsP0gp is torsionfree, the mapPgp0 → Mgpx defined by the lower row admits a lifting s : P0gp → Mgpx , sending P0 to Mx. One can adjust s to make it compatible with the morphism
˜
a : Qgp → Mgpy → Mgpx given by the chart a : Q → MY. Indeed, ifj : Qgp ,→ Pgp0 is the inclusion, the homomorphismssj/a˜ :Qgp→OX,x∗ can be extended toP0gpas the torsion part ofPgp0 /Qgpis killed by an integer invertible onX. Assume that this adjustment is done. Asvis a chart,P/v−1(Ox∗)→Mxis an isomorphism, and sinceHis contained inv−1(Ox∗),P0/s−1(Ox∗)→Mxis an isomorphism as well, hences:P0→Mxis a chart at xcompatible witha. A second adjustment is needed to make itG-equivariant. To do so, one can proceed as above, by considering the1-cocyclezofGwith values in Hom(Pgp0 ,Ox∗)given by
gs(p) =z(g, p)s(p).
The image ofzinZ1(G,Hom(Pgp0 , k(x)∗))is a homomorphism ϕ0:Gab⊗Pgp0 →µ.
The quotient g 7→ (p 7→ z(g, p)/ϕ0(g, p))belongs to B1(G,Hom(Pgp0 /Qgp, 1+ mx)), hence can be written g7→(p7→gρ(p)/ρ(p))forρ :Pgp0 /Qgp→1+ mx. So, replacingzbyg7→z(g, p)ρ(p)−1, we may assume that z=ϕ0, in other words, the map
b0:X→SpecΛ[P0]
defined by the pair(s, ϕ0)is an equivariant chart ofXatx(extendinga).
One can give an alternate, shorter proof of the last assertion which does not use the above decomposition of PintoH⊕P0. Consider again the cokernelCof the mapuintroduced a few lines above diagram (1.2.3). Write Cas a direct sum of cyclic groups of ordersmi|m. Chooseai ∈Mgpx andbi∈Z⊕Qgp(1≤i≤n) such that ami i=u(bi), and theai’s induce an isomorphism
M
i
Z/miZ→∼ C.
Replace diagram (1.2.3) by the following push-out diagram
(1.2.4) L
iZei
//L
iZ(m1
iei)
Z⊕Qgp w //F ,
where the upper horizontal map is the natural inclusion and the left vertical one sendseitobi. In this way, we haveF= Pgp ⊃Z⊕QgpandPgp/(Z⊕Qgp)→∼ C. AsXis fs,Mgpx is torsionfree, so the mapν:F →Mgpx , defined similarly as above (using (1.2.4) instead of (1.2.3)), sends(Pgp)torsto0, hence
(Pgp)tors= (Z⊕Qgp)tors= (Qgp)tors, which finishes the proof.
Lemma 1.3. — LetGbe a finite group of exponentn, letΛ=Z[µn][1/n], letkbe a field overΛ, and letLbe a finitely generatedk[G]-module. There exists a finitely generated projectiveΛ[G]-moduleVsuch thatL=k⊗ΛV.
Proof. — First, observe that sincenis invertible inΛ, anyΛ[G]-module which is finitely generated and projec- tive overΛis projective overΛ[G][Serre, 1978, §14.4, Lemme 20].
Suppose first that char(k) =0, and letkbe an algebraic closure ofk. Then,Ldescends to aQ[µn][G]-module W, ask⊗kL descends [Serre, 1978, §12.3] and the homomorphism Rk(G) → Rk(G)given by extension of scalars is injective [Serre, 1978, §14.6]. One can then take forVaG-stableΛ-lattice inW(projective overΛ), which is necessarily projective overΛ[G]by the above remark.
Suppose now that char(k) = p > 0. LetI kbe a Cohen ring fork. AsΛis étale overZ,Λ → klifts (uniquely) toΛ→I. On the other hand, asLis projective of finite type overk[G], by [Serre, 1978, §14.4, Prop.
42, Cor. 3]Llifts to a finitely generated projectiveI[G]-moduleE, free overI. LetKbe the fraction field ofI. ThenE⊗Kdescends to aQ[µn][G]-moduleE0. Choose aG-stableΛ-latticeVinE0(projective overΛ, hence, projective of finite type overΛ[G]). By [Serre, 1978, §15.2, Th. 32],k⊗ΛVhas the same class inRk(G)asL. But, ask[G]is semisimple by Maschke’s theorem,Landk⊗ΛVare isomorphic ask[G]-modules.
Proof of1.1(beginning).
The strategy is to check that, at each step of the proof of the absolute modification theorem (exp.VIII,1.1), the log smoothness ofX/Sis preserved, and, at the end, that of the quotient(X/G)/Sas well. For some of them, this is trivial, as the modifications performed are log blow ups. Others require a closer inspection.
1.4. — Preliminary reductions.We may assume that conditions (1) and (2) at the beginning of (exp.VIII,4) are satisfied, namely:
(1)Xis regular,
(2)Zis aG-strict snc divisor inX.
Indeed, these conditions are achieved by G-equivariant saturated log blow up towers (exp.VIII,4.1.1, exp.VIII,4.1.6).
We will now exploit Gabber’s preparation lemma1.2to give a local picture offdisplaying both the log stratification and the inertia stratification ofX. We work étale locally at a geometric pointxinXwith image sinS. Up to replacingXby theGx-invariant neighborhood X0 constructed at the beginning of the proof of exp.VIII,5.3.8, andGbyGx, whereGxis the inertia group atx, we may assume thatG = Gx. Indeed, the morphism(X0, Gx) → (X, G)is strict and inert, and by exp.VIII,5.4.4the tower f(G,X,Z) is functorial with respect to such morphisms.
We now apply1.2. LetNbe the exponent ofG. AssumeSstrictly local ats. We may replaceΛ=Z[1/N, µ]
by its localization at the (Zariski) image ofs, so thatΛis either the cyclotomic fieldQ(µ)or its localization at a finite place of its ring of integers, of residue characteristicp=char(k(s))not dividingn. Choose a chart
a:S→SpecΛ[Q]
withQfs and the inverse image ofOS,s∗ inQequal to{1}, so thatQis sharp andQ→∼ Ms. LetCdenotek(s) ifOS,scontains a field, and a Cohen ring ofk(s)otherwise. Let(yi)1≤i≤m be a family of elements ofmssuch that the images of theyi’s in OS,s/Is form a regular system of parameters, whereIs = I(s, Ms)is the ideal generated by the image ofMs−OS,s∗ by the canonical mapα :Ms → OS,s. By [Kato, 1994, 3.2], the charta extends to an isomorphism
(1.4.1) C[[y1,· · ·, ym]][[Q]]/(g)→∼ ObS,s,
whereg∈C[[y1,· · ·, ym]][[Q]]is0ifC=k(s), and congruent top=char(k(s))> 0modulo the ideal generated byQ −{1}and (y1,· · · , ym) otherwise. By 1.2, up to shrinking X around x, we can find aG-equivariant commutative diagram (with trivial action ofGon the bottom row)
(1.4.2) X
c //X0
b //Spec(Λ[P]⊗ΛSymΛ(V))
S a //SpecΛ[Q]
,
where:
(i) the square is cartesian;
(ii)a,b, andcare strict, where the log structure on SpecΛ[Q](resp. Spec(Λ[P]⊗ΛSymΛ(V))) is the canonical one, given byQ(resp.P);Pis an fs monoid, withP∗torsionfree;Gacts onΛ[P]byg(λp) =λχ(g, p)p, for some homomorphism
χ:Gab⊗Pgp→µ
(iii)Vis a free, finitely generatedΛ-module, equipped with aG-action;
(iv) the right vertical arrow is the composition of the projection onto the factor SpecΛ[P]and SpecΛ[h], for a homomorphismeh :Q →Psuch thathgpis injective and the torsion part of Cokerhgpis annihilated by an integer invertible onX;
(v)cis étale and inert.
(vi) Consider the map
v:P→Mx
defined by the chartX → SpecΛ[P]induced bybc. Up to localizing onX0 aroundx, we may assume thatv factors through the localizationP(p)ofPat the prime idealpcomplementary of the facev−1(OX,x∗ ). Replacing PbyP(p),Pdecomposes into
(1.4.3) P=P∗⊕P1,
with P∗ = v−1(OX,x∗ ) free finitely generated over Z, andP1 sharp, and the image ofx bybc in the factor SpecΛ[P1]is the rational point at the origin. Thenvinduces an isomorphismP1 →∼ Mx. By the assumptions (1), (2), we have Mx →∼ Nr. One can therefore choose(ei ∈ P1) (1 ≤ i ≤ r)forming a basis ofP1. Then v(ei) =ti∈Mx⊂OX,xis a local equation for a branchZiofZatx,(Zi)1≤i≤ris the set of branches ofZatx, andGacts ontithrough the characterχi=χ(−, ei) :G→µ.
Furthermore:
(vii) The square in (1.4.2) is tor-independent.
Indeed, by the log regularity ofSand the choice of the charta, we have, by [Kato, 1994, 6.1], TorZ[Q]i (OS,s,Z[P]) = 0fori > 0.
Though this will not be needed, one can describe the local structure of (1.4.2) more precisely as follows. Let (1.4.4) Y :=Spec(Λ[P]⊗ΛSymΛ(V)) =Spec(Λ[P∗]⊗ΛΛ[P1]⊗ΛSymΛ(V))
and letY0:=SpecC[[y1,· · ·, ym]][[Q]]×SpecΛY, with the notation of1.4.1. We may assume thatX=X0. Then the completion ofXatxis either isomorphic to the completion ofY0atx, or a regular divisor in it, defined by the equationg0 =0, whereg0is the image ofginObY0,x, with the notation of1.4.1.
1.5. — Step 3 and log smoothness (beginning). We will now analyze the modifications performed in the proof of Step 3 in exp.VIII,4.1.9, exp.VIII,4.2.13. The permissible towers used inloc. cit.are iterations of operations of the form: for a subgroupHofG, blow up the fixed point (regular) subschemeXH, and replaceZby the union of its strict transformZstand the exceptional divisorE. Though such a blow up is not a log blow up in general, we will see that it still preserves the log smoothness ofXoverS.
We work étale locally aroundx, so we can assumeX=X0in1.4.2. We then have a cartesian square
(1.5.1) XH b
H //
f
YH
X b //Y
,
withYas in (1.4.4). We also have cartesian squares
(1.5.2) Z b //
f
T
X b //Y ,
whereT ⊂Yis the snc divisorP
Ti,Tidefined by the equationei∈P1(1.4.3), and
(1.5.3) Z×XXH //
T×YYH
X //Y
.
Lemma 1.6. — The squares (1.5.1), (1.5.2), and (1.5.3) are tor-independent.
Proof. — For (1.5.2), this is becauseZ(resp.T) is a divisor inX(resp.Y) (cf. [SGA 6VII1.2]). For (1.5.1), as the square (1.4.2) is tor-independent (by1.4(vii)), it is enough to show that the composite (cartesian) square
(1.6.1) XH
//YH
S //SpecΛ[Q]
is tor-independent. We have a decomposition
(1.6.2) YH= (SpecΛ[P∗])H×(SpecΛ[P1])H×(Spec SymΛ(V))H,
(products taken over SpecΛ), and the map to SpecΛ[Q]is the composition of the projection onto(SpecΛ[P∗])H× (SpecΛ[P1])H and the canonical map induced by SpecΛ[Q] → SpecΛ[P], which factors through the fixed points ofH,Gacting trivially on the base. Let us examine the three factors.
(a) We have
(Spec SymΛ(V))H=Spec SymΛ(VH),
where VH is the module of coinvariants, a free module of finite type over Λ, as H is of order invertible in Λ. Therefore SpecΛ[Q]×SpecΛ(Spec SymΛ(V))H is flat over Spec Λ[Q], and its enough to check that (SpecΛ[P∗])H×(SpecΛ[P1])His tor-independent ofSover SpecΛ[Q].
(b) The restriction toP∗ = v−1(OX,x∗ )of the1-cocyclez(v)∈Z1(H, Hom(P, k(x)∗))associated withv: P→ Mx(hv(a) =z(v)(h, a)v(a)forh∈H,a∈P, see the proof of1.2and (exp.VI,3.5), is a1-coboundary, hence trivial, asB1(H, Hom(P, k(x)∗)) =0. Therefore
(SpecΛ[P∗])H=SpecΛ[P∗].
(c) Recall that
P1= M
1≤i≤r
Nei,
witheisent by vto a local equation of the branchZiofZ, and thatGacts onΛ[Nei]through the character χi:G→µ. LetA⊂{1,· · · , r}be the set of indicesisuch thatχi|His trivial. Then
(SpecΛ[P1])H=SpecΛ[M
i∈A
Nei].
LetIbe the ideal ofPgenerated by{ei}i /∈A. It follows from (b) and (c) that (SpecΛ[P])H=SpecΛ[P]/(I),
where(I)is the ideal ofΛ[P] generated byI. By [Kato, 1994, 6.1], TorΛ[Q]i (OS, Λ[P]/(I)) = 0fori > 0, and therefore (1.6.1), hence (1.5.1) is tor-independent. It remains to show the tor-independence of (1.5.3). For this, again it is enough to show the tor-independence of
(1.6.3) Z×XXH //
T×YYH
S //SpecΛ[Q]
.
By (a), (b), (c), we have
T×YYH=X
i∈A
SpecΛ[P]/(Ji)×Spec SymΛ(VH),
whereJi ⊂ Pis the ideal generated byei ∈ P1, and(Ji)the ideal generated byJi inΛ[P]. The desired tor- independence follows from the vanishing of TorΛ[Q]i (OS, Λ[P]/(JB)), where for a subsetBofA,JBdenotes the ideal generated by theei’s fori∈B.
Lemma 1.7. — Consider a cartesian square
(1.7.1) V0
//V
X0 g //X ,
where the right vertical arrow is a regular immersion. If (1.7.1) is tor-independent, then the left vertical arrow is a regular immersion, and
BlV0(X0) =X0×XBlV(X).
LetW→Xbe a second regular immersion, such thatV×XW→Wis a regular immersion, and letW0=X0×XW. If moreover the squares
(1.7.2) V0
//V
X0 g //X
,
and
(1.7.3) V0×X0W0
//V×XW
X0 g //X ,
are tor-independent, then the left vertical arrows are regular immersions, and W0st=X0×XWst,
whereWst(resp.W0st) is the strict transform ofW(resp.W0) inBlV(X)(resp.BlV0(X0)).
Proof. — Let I (resp. I0) be the ideal of V (resp. V0) in X(resp. X0). By the tor-independence of (1.7.1), if u : E → I is a local surjective regular homomorphism [SGA 6 VII 1.4], the Koszul complex g?K(u)is a resolution ofOV0, henceV0 → X0 is a regular immersion. Moreover, by [SGA 6 VII1.2], for anyn ≥ 0, the natural mapg?In→I0nis an isomorphism, and therefore BlV0(X0) =X0×XBlV(X). The tor-independence of (1.7.2) and (1.7.3) imply that of
V0×X0W0 //
V×XW
W0 //W
.
The second assertion then follows from the first one and the formulas (exp.VIII,2.1.3(ii)) Wst=BlV×XWW,
W0st=BlV0×X0W0W0.
1.8. — Step 3 and log smoothness (end). As recalled at the beginning of 1.5, we have to show that, ifHis a subgroup ofG, then the log regular pair(X1, Z1)is log smooth overS, whereX1:=BlXH(X)andZ1is the snc divisorZst∪E,Zst(resp. E) denoting the strict transform ofZ(resp. the exceptional divisor) in the blow-up h:X1→X.
The question is again étale local aboveXaround x, so we may assume thatX = X0 and we look at the cartesian square (1.4.2)
X //
Y
S //SpecΛ[Q]
,
withYas in (1.4.4), and the associated cartesian squares (1.5.1), (1.5.2), and (1.5.3).
Claim.We have
(1.8.1) BlXH(X) =X×YBlYH(Y),
(1.8.2) Zst=X×YTst.
Proof. — In view of1.6and1.7, (1.8.1) follows from the fact that the immersionYH→Yis regular. For (1.8.2) recall that
T =T0×SpecΛSpec SymΛ(V), whereT0⊂SpecΛ[P]is the snc divisor
T0= X
1≤i≤r
div(zi) withzi∈Λ[P]the image ofei∈P1as in1.4.3. Hence
(1.8.3) T = X
1≤i≤r
Ti,
whereTi=div(zi)×SpecΛSymΛ(V), andTst=P
1≤i≤rTist. We have (1.6.2) YH= (SpecΛ[P])H×Spec SymΛ(VH),
with(SpecΛ[P])Hdefined by the equations(zi=0)i /∈A, with the notations of1.6(c). In particular, the immer- sionYH×YZi→Ziis regular, hence, by1.7, we haveZsti =X×YTist, hence (1.8.2), which finishes the proof of the claim.
Since the mapS→SpecΛ[Q]is strict, in order to prove the desired log smoothness, we may, by this claim, replace the triple(X, XH, Z)overSby(Y, YH, T)over SpecΛ[Q]. We choose coordinates onP∗,P1=Nr,V:
P∗= M
1≤i≤t
Zfi, P1= M
1≤i≤r
Nei, V = M
1≤i≤s
Λyi
Λ[P] =Λ[u±11 ,· · ·, u±1t , z1,· · · , zr], SymΛ(V) =Λ[y1,· · · , ys], withui(resp.zi) the image offi(resp.ei) inΛ[P], in such a way that
Λ[P]H=Λ[u±11 ,· · ·, u±1t , zm+1,· · ·, zr],
i.e. is defined inΛ[P]by the equations(z1=· · ·=zm =0), for somem,1≤m≤r, and Λ[VH] =Λ[yn+1,· · ·, ys],
i.e. is defined inΛ[V]by the equationsy1=· · ·=yn=0for somen,1≤n≤s. Then YH⊂Y=SpecΛ[u±11 ,· · ·, u±1t , z1,· · ·, zr, y1,· · ·, ys] is defined by the equations
z1=· · ·=zm =y1=· · ·=yn=0.
Then
Y0 :=BlYH(Y) is covered by affine open pieces:
Ui=SpecΛ[(u±1j )1≤j≤t, z10,· · · , zi−10 , zi, zi+10 ,· · ·, zm0 , zm+1,· · · , zr, y10,· · · , yn0, yn+1,· · · , ys]
(1 ≤ i ≤ m), with Ui → Y given byzj → zizj0 for1 ≤ j ≤ m,j 6= i,yj → ziyj0,1 ≤ j ≤ n, and the other coordinates unchanged, and
Vi=SpecΛ[(u±1j )1≤j≤t, z10,· · · , zm0 , zm+1,· · ·, zr, y10,· · ·, yi−10 , yi, yi+10 ,· · ·, yn0, yn+1,· · · , ys] (1 ≤ i ≤ n), withVi → Y given byzj 7→ yizj0 for 1 ≤ j ≤ m,yj 7→ yiyj0,1 ≤ j ≤ n,j 6= i, and the other coordinates unchanged. Recall thatYhas the log structure defined by the log regular pair(Y, T), whereT is the snc divisor
T = (z1· · ·zr=0),
andY0is given the log structure defined by the log regular pair(Y0, T0), whereT0is the snc divisor T0=F∪Tst,
whereFis the exceptional divisor of the blow up ofYHandTstthe strict transform ofT. Consider the canonical morphisms
Y0 b //Y g //Σ:=SpecΛ[Q].
They are both morphisms of log schemes. The morphismgis given by the homomorphism of monoidsγ : Q→P, i.e.
q∈Q7→(γ1(q),· · ·, γt(q), γt+1(q),· · ·, γt+r(q), 0,· · ·, 0)∈Λ[u±11 ,· · ·, u±1t , z1,· · ·, zr, y1,· · ·, ys].
The blow upbhas been described above in the various charts. Note thatbis not log étale, or even log smooth, in general. However, the compositiongb:Y0→Σis log smooth. We will check this on the charts(Ui),(Vi).
(a)Chart of typeUi. We haveF= (zi =0),Tst= (Q
1≤j≤r,j6=izi0 =0). Hence the log strucure ofUiis given by the canonical log structure ofΛ[Nr]in the decomposition
Ui=SpecΛ[Zt]×SpecΛ[Nr]×SpecΛ[y10,· · ·, yn0, yn+1,· · ·, ys]
with the basis elementekofNrsent to thek-th place in(z10,· · · , zi−10 , zi, zi+10 ,· · ·, zm0 , zm+1,· · · , zr)(and the basis elementfkofZtsent touk), the third factor having the trivial log structure. Checking the log smoothness ofgb : Ui → Σamounts to checking the log smoothness of its factor Spec Λ[P] → Σ= SpecΛ(Q), which is defined by the composition of homomorphisms of monoids
Q γ//Zt⊕Nr Id⊕β//Zt⊕Nr,
whereβis the homomorphismNr →Nrsendingejtoej+eifor1 ≤j≤ m, j6= i,eitoei, andejtoejfor m+1≤j≤r. Recall ((1.4.2), (iv)) thatγgpis injective and the torsion part of its cokernel is invertible inΛ. As βgpis an isomorphism, the same holds for the composition(Id⊕β)γ, hencegb:Ui→Σis log smooth.
(b)Chart of typeVi. We haveF = (yi = 0),Tst = Q
1≤j≤mzj0Q
j≥m+1zi. Hence the log structure ofViis given by the canonical log structure ofΛ[Nr+1]in the decomposition
Vi=SpecΛ[Zt]×SpecΛ[Nr+1]×SpecΛ[(yj0)1≤j≤n,j6=i, yn+1,· · ·, ys]
with the basis elementek ofNr+1sent to thek-th place in(z10,· · ·, zm0 , zm+1,· · · , zr)ifk ≤ r, ander+1sent to yi (and the basis element fk of Zt sent to ui), the third factor having the trivial log structure. Again, checking the log smoothness ofgb:Vi→Σamounts to checking the log smoothness of its factor SpecΛ[Zt]× SpecΛ[Nr+1]→SpecΛ(Q). This factor is defined by the composition of homomorphisms of monoids
Q γ//Zt⊕Nr Id⊕β//Zt⊕Nr+1
whereβ:Nr→Nr+1sendsejtoej+er+1for1≤j≤m, and toejform+1 ≤j≤r. Thenβgpis injective, and its cokernel is isomorphic toZ, hence(βγ)gpis injective, and we have an exact sequence
0→Cokerγgp→Coker(βγ)gp→Z→0.
In particular, the torsion part of Coker(βγ)gpis isomorphic to that of Cokerγgp, hence of order invertible inΛ, which implies thatgb:Vi→Σis log smooth.
This finishes the proof that Step 3 preserves log smoothness.
1.9. — End of proof of1.1.We may now assume that in addition to conditions (1) and (2) of1.4, condition (3) is satisfied as well, namely
(3)Gacts freely onX−Z(i.e. Z = Z∪T in the notation of1.1or (exp.VIII,1.1)), and, for any geometric point x→X, the inertia groupGxis abelian.
We have to check:
Claim. Iff(G,X,Z) : (X0, Z0)→(X, Z)is the modification of (exp.VIII,5.4.4), then(X0, Z0)and(X0/G, Z0/G)are log smooth overS.
Working étale locally around a geometric pointxofX, we will first choose a strict rigidification(X, Z)of (X, Z)such that(X, Z)is log smooth overS. We will define(X, Z)as the pull-back byS→ Σ= SpecΛ[Q]of a rigidification(Y, T)of(Y, T)which is log smooth overΣ, with the notation of(1.4.4). Using thatG(=Gx)is abelian, one can decomposeVinto a sum ofG-stable lines, according to the characters ofG:
V= M
1≤i≤s
Λyi
with Gacting on Λyi through a characterχi : G → µN, i.e. gyi = χi(g)yi. We defineT to be the divisor z1· · ·zry1· · ·ys = 0 inY = SpecΛ[u±11 ,· · · , u±1t , z1,· · · , zr, y1,· · · , ys]. The action of G on (Y, T) is very tame at x because the log stratum at x is Spec Λ[u±11 ,· · · , u±1t ], hence very tame in a neighborhood of x by (exp.VIII, 5.3.2) (actually on the whole ofY, cf. (exp. VIII,4.6, exp. VIII,4.7(a)). On the other hand, (SpecΛ[y1,· · ·, ys], y1· · ·ys =0)is log smooth over SpecΛ, and as SpecΛ[P]is log smooth overΣ,(Y, T)is log smooth overΣ. Sincef(G,X,Z) is compatible with base change by strict inert morphisms, it is enough to
check that iff(G,Y,T) = f(G,Y,T,T) : (Y0, T0)→(Y, T)is the modification of (exp.VIII,5.4.4) then(Y0, T0)is log smooth overΣ. Recall (exp.VIII,5.3.9) that we have a cartesianG-equivariant diagram
(1.9.1) (Y0, T0) h0 //
α0
(Y, T)
α
(Y10, T10) h1 //(Y1, T1) ,
where the horizontal maps are the compositions of saturated log blow up towers, and the vertical ones Kum- mer étaleG-covers. From (1.9.1) is extracted the relevant diagram involvingh:=f(G,Y,T,T),
(Y0, T0)
β
h //(Y, T)
(Y10, T10)
,
whereT10 =h−11 (T1), withT1=T/G,T0=α0−1(T10), andh(resp.β) is the restriction ofh0(resp.α0) over(Y, T) (resp.(Y10, T10)). In particular,βis a Kummer étaleG-cover (as Kummer étaleG-covers are stable under any fs base change). AsGacts trivially onS, this diagram can be uniquely completed into a commutative diagram
(1.9.2) (Y0, T0)
β
h //(Y, T)
f
(Y01, T01) g //Σ
.
Herefis log smooth andβis a Kummer étaleG-cover. Thoughh0andh1are log smooth,handh1are not, in general. However, it turns out that:
(∗)g: (Y10, T10)→Σ, hencegβ=fh: (Y0, T0)→Σ, are log smooth, which will finish the proof of the claim, hence of1.1. We first prove
(∗∗)With the notation of (1.9.2),(Y1, T1)is log smooth overΣ.
Let us writeY =SpecΛ[P], with
(1.9.3) P=P×Ns=Zt×Nr×Ns.
AsG acts very tamely on(Y, T), the quotient pair(Y1 = Y/G, T1 = T /G)is log regular. More precisely, by the calculation in (exp.VI,3.4(b)), this pair consists of the log schemeY1 = SpecΛ[R]with its canonical log structure, where
R=Ker(Pgp→Hom(G, µN))∩P,
Pgp →Hom(G, µN)being the homomorphism defined by the pairingχ:G⊗Pgp →µN. The inclusionR⊂P is a Kummer morphism, andPgp/Rgp is annihilated by an integer invertible inΛ. AsQgp →Pgp is injective, with the torsion part of its cokernel annihilated by an integer invertible inΛ, the same is true forQgp→Pgp, hence also forQgp→Rgp. Thus(Y1, T1) =SpecΛ[R]is log smooth overΣ.
Finally, let us prove (∗). It is enough to work locally onY10 so we can replace the log blow up sequence (Y10, T10) → (Y1, T1)with an affine chart (i.e. we replace the first log blow up with a chart, then do the same for the second one, etc.). ThenY10 =SpecΛ[R0], andRgp →∼ R0gpby exp.VIII,3.1.19. Note thatR0 →∼ Za×Nb whereD1, . . . , Dbare the components ofT10. We can assume thatD1, . . . , Dc ⊂T10andDc+1, . . . , Dbare not contained inT10. LetR0 →∼ Za×Nc denote the submonoidR0 that defines the log structure of(Y10, T10). Note thatR0consists of all elementsg0∈R0such that(g0=0)⊂T10(as a set). Also, byν:R→R0we will denote the homomorphism that defines(Y10, T10)→(Y1, T1).
We showed in exp.VIII,5.3.9thatT1=T/Gis aQ-Cartier divisor inY1and observed that thereforeT10 is a Cartier divisor inY10. Note that the inclusionR⊂R, where
R=Ker(Pgp→Hom(G, µN))∩P
defines a log structure onY1. Denote the corresponding log scheme(Y1, T1). We obtain the following diagram of log schemes (on the left). The corresponding diagram of groups is placed on the right; we will use it to establish log smoothness ofg. Existence of dashed arrows requires an argument; we will construct them later.
(1.9.4) (Y10, T10)
h1 //(Y1, T1)
""
R0gp ∼ Rgp
νgp
oo
(Y10, T10) //(Y1, T1) //Σ R?0gpOO
Rgp
? _
oo ?OO
Qgp
oo``
Part (ii) of the following remark clarifies the notation (Y1, T1). It will not be used so we only sketch the argument.
Remark 1.10. — (i) Note that(Y1, T1)may be not log smooth overΣ. For example, even whenΣis log regular, e.g. Speckwith trivial log structure,(Y1, T1)does not have to be log regular, asT1may even be non-Cartier.
Nevertheless, ash1is log smooth (even log étale),(Y10, T10)is log smooth overΣ. Moreover,Y10 is regular, and T10an snc divisor in it.
(ii) AlthoughT1may be bad, one does have thatROY∗1=OY1∩i?OY∗1\T1for the embeddingi:Y1\T1,→Y1. This can be deduced from the formulas forRandRand the fact thatROY∗1 =OY1∩j?OY∗
1\T1by log regularity of(Y1, T1).
Note thatQ → Pfactors throughP, henceQ → Rfactors through R = R∩P. It follows from (1.9.3) that Pconsists of all elementsf ∈Pwhose divisor(f = 0)is contained inT (as a set). Thereforeg∈ Rlies inRif and only if(g= 0) ⊂T1(as a set). This fact and the analogous description ofR0 observed earlier imply that ν:R→R0takesRtoR0. Thus, we have established the dashed arrows in (1.9.4).
Letϕ :Q → R0 be the homomorphism defining the composition(Y10, T10) →(Y1, T1)→ Σ. Since the latter is log smooth, ϕis injective, and the torsion part of Coker(ϕgp)is annihilated by an integerminvertible in Λ. Note thatRgp,→R0gp,→R0gp, and therefore we also have thatQgp,→R0gp and the torsion of its cokernel is annihilated bym. Therefore,(Y10, T10)is log smooth overΣ, which finishes the proof of (∗), hence of1.1. Remark 1.11. — In the proof of (∗) above, we first proved thatgis log smooth, and deduced thatgβis, too. In fact, asβis a Kummer étaleG-cover, the log smoothness ofgβimplies that ofg. More generally, we have the following descent result, due to Kato-Nakayama ([Nakayama, 2009, 3.4]):
Theorem 1.12. — Let X0 g //X f //Y be morphisms of fs log schemes. Ifgis surjective, log étale and exact, and fgis log smooth, thenfis log smooth.
The assumption ongis equivalent to saying thatgis a Kummer étale cover (cf. [Illusie, 2002, 1.6]).
2. Prime to`variants of de Jong’s alteration theorems
LetXbe a noetherian scheme, and`be a prime number. Recall that a morphismh : X0 → Xis called an
`0-alterationifhis proper, surjective, generically finite, maximally dominating (i.e., (exp.II,1.1.2) sends each maximal point to a maximal point) and the degrees of the residual extensionsk(x0)/k(x)over each maximal pointxofXare prime to`. The next theorem was stated inIntroduction,3(1):
Theorem 2.1. — Letkbe a field,`a prime number different from the characteristic ofk,Xa separated and finite type k-scheme,Z⊂Xa nowhere dense closed subset. Then there exists a finite extensionk0 ofk, of degree prime to`, and a projective`0-alterationh:X˜ →XaboveSpeck0→Speck, withX˜ smooth and quasi-projective overk0, andh−1(Z)is the support of a relative strict normal crossings divisor.
Recall that a relative strict normal crossings divisor in a smooth schemeT/S is a divisor D = P
i∈IDi, whereIis finite, Di ⊂T is anS-smooth closed subscheme of codimension1, and for every subsetJofIthe scheme-theoretic intersectionT
i∈JDiis smooth overSof codimension|J|inT.
We will need the following variant, due to Gabber-Vidal (proof of [Vidal, 2004, 4.4.1]), of de Jong’s alteration theorems [de Jong, 1997, 5.7, 5.9, 5.11], cf. [Zheng, 2009, 3.8]:
Lemma 2.2. — LetXbe a proper scheme overS=Speck, normal and geometrically reduced and irreducible,Z⊂Xa nowhere dense closed subset. We assume that a finite groupHacts onX→S, faithfully onX, and thatZisH-stable. Then there exists a finite extensionk1ofk, a finite groupH1, a surjective homomorphismH1 →H, and anH1-equivariant diagram with a cartesian square (whereS=Speck,S1=Speck1)
(2.2.1) X
X1
oo b X2
~~oo a
Soo S1
satisfying the following properties:
(i)S1/Ker(H1→H)→Sis a radicial extension;
(ii)X2is projective and smooth overS1;
(iii) a : X2 → X1 is projective and surjective, maximally dominating and generically finite and flat, and there exists anH1-admissible dense open subsetW ⊂X2over a dense open subsetUofX, such that ifU1 = S1×SUand K=Ker(H1→Aut(U1)),W→W/Kis a Galois étale cover of groupKand the morphismW/K→U1induced bya is a universal homeomorphism;
(iv)(ba)−1(Z)is the support of a strict normal crossings divisor inX2.
Proof. — We may assumeXof dimensiond≥1. We apply [Vidal, 2004, 4.4.3] toX/S,Z, andG =H. We get the data ofloc. cit., namely an equivariant finite extension of fields(S1, H1)→(S, H)such thatS1/Ker(H1→ H) → Sis radicial, an H1-equivariant pluri-nodal fibration(Yd → · · · → Y1 → S1,{σij}, Z0 = ∅), and an H1-equivariant alterationa1 : Yd → XoverS, satisfying the conditions (i), (ii), (iii) ofloc. cit. (in particular a−11 (Z) ⊂Zd). Then, as in the proof of [Vidal, 2004, 4.4.1], successively applying [Vidal, 2004, 4.4.4] to each nodal curvefi:Yi→Yi−1, one can replaceYiby anH1-equivariant projective modificationYi0of it such that Yi0is regular, and the inverse imageZi0 ofZi :=S
jσij(Yi−1)∪f−1i (Zi−1)inYi0 is anH1-equivariant strict snc divisor. Then,X2:=Yd0 is smooth overS1andZd0 is a relative snc divisor overS1. This follows from the analog of the remark following [Vidal, 2004, 4.4.4] with “semistable pair over a trait" replaced by “pair consisting of a smooth scheme and a relative snc divisor over a field". In particular,(ba)−1(Z)redis a subdivisor ofZd0, hence an snc divisor. After replacingH1byH1/Ker(H1→Aut(X2))the open subsetsUandVof (iii) are obtained as at the end of the proof of [Vidal, 2004, 4.4.1].
2.3. — Proof of2.1. There are three steps.
Step 1. Preliminary reductions. By Nagata’s compactification theorem [Conrad, 2007], there exists a dense open immersionX⊂XwithXproper overS. Up to replacingXbyXandZby its closureZ, we may assume Xproper overS. By replacingXby the disjoint sum of its irreducible components, we may further assumeX irreducible, and geometrically reduced (up to base changing by a finite radicial extension ofk). Up to blowing upZinXme may further assume thatZis a (Cartier) divisor inX. Finally, replacingXby its normalizationX0, which is finite overX, andZby its inverse image inX0, we may assumeXnormal.
Step 2. Use of2.2. Choose a finite Galois extension k0ofksuch that the irreducible components ofX0 = X×S S0 (S0 = Speck0) are geometrically irreducible. Let G = Gal(k0/k)and H ⊂ G the decomposition subgroup of a componentY0ofX0. We apply2.2to (Y0/S0,Z0∩Y0), whereZ0=S0×SZ. We find a surjection H1→Hand anH1-equivariant diagram of type2.2.1:
(2.3.1) Y0
Y1
b0
oo Y2
a0
oo~~
S0oo S1
,
satisfying conditions (i), (ii), (iii), (iv) withSreplaced by S0, andX2 → X1 → XbyY2 → Y1 → Y0. AsG transitively permutes the components ofX0,X0is, as aG-scheme overS0, the contracted product
X0=Y0×HG,
i.e. the quotient ofY0×GbyHacting onY0on the right and onGon the left (cf. proof of exp.VIII,5.3.8), and similarlyZ=Z0×HG. Choose an extension of the diagram H1
u //H i //G into a commutative diagram
