A L
E S
E D L ’IN IT ST T U
F O U R
ANNALES
DE
L’INSTITUT FOURIER
Roland HUBER
A finiteness result for the compactly supported cohomology of rigid analytic varieties, II
Tome 57, no3 (2007), p. 973-1017.
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A FINITENESS RESULT FOR THE COMPACTLY SUPPORTED COHOMOLOGY OF RIGID ANALYTIC
VARIETIES, II
by Roland HUBER
Abstract. — Leth:X→Y be a separated morphism of adic spaces of finite type over a non-archimedean fieldkwithY affinoid and of dimension61, letLbe a locally closed constructible subset ofXand letg: (X, L)→Y be the morphism of pseudo-adic spaces induced byh. LetAbe a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring ofkand letF be a constantA-module of finite type on(X, L)´et. There is a natural classC(Y) of A-modules onY´et generated by the constructibleA-modules and the Zariski- constructibleA-modules. We show that, for everyn∈N0, the higher direct image sheaf with proper supportRng!F is generically constructible, and ifh is locally algebraic,Rng!F is an element ofC(Y). As an application we obtain a comparison isomorphism for the`-adic cohomology of a separated scheme of finite type overk and its associated adic space.
Résumé. — Soith:X →Y un morphisme séparé d’espaces adiques de type fini sur un corps non archimédienkavecY affinoïde et de dimension61. SoitL un sous-ensemble constructible localement fermé dans X et soitg: (X, L)→Y le morphisme d’espaces pseudo-adiques induit deh. SoitAun anneau noethérien de torsion première à la caractéristique résiduelle de ket soitF un faisceau de A-modules localement constant de type fini sur(X, L)´et. Il y a une classe natu- relleC(Y)des faisceaux deA-modules surY´etengendrée par des faisceaux deA- modules constructibles et des faisceaux deA-modules Zariski-constructibles. Nous montrons que le faisceau image directe à support propreRng!F est génériquement constructible, et sihest localement algébrique,Rng!F est un élément deC(Y). En conséquence, on obtient un théorème de comparaison entre cohomologie`-adique d’un schéma séparé de type fini surket de l’espace adique associé.
1. Introduction
Letkbe a non-archimedean field, leth:X →Y be a separated morphism of adic spaces of finite type overSpa (k, k◦)withY affinoid anddimY 61,
Keywords:Rigid analytic spaces, adic spaces, compactly supported cohomology.
Math. classification:14G22, 14F20.
letLbe a locally closed constructible subset ofX and letg: (X, L)→Y be the morphism of pseudo-adic spaces induced byh. Let A be a noetherian torsion ring with torsion prime tochar(k◦/k◦◦) and let F be a constant A-module of finite type on (X, L)´et. There is a natural class C(Y) of A- modules onY´etgenerated by the constructibleA-modules as defined in [9], 2.7 and the Zariski-constructibleA-modules. We are interested to know if Rmg!F ∈ C(Y). In [11] is proved that this is fulfilled if char(k) = 0 and
|A| <∞. In this paper we will show that without any restriction on the characteristic of k and the cardinality of A the following two statements hold
(I) For every m ∈N0, Rmg!F is generically constructible onY, i.e., there exists an open subset U of Y such that the restriction Rmg!F|U is constructible on U and every x ∈ Y whose support supp(x) = {c ∈ OY(Y) | c(x) = 0} ∈ SpecOY(Y) is a generic point ofSpecOY(Y)is contained inU.
(II) Ifhis locally algebraic then, for everym∈N0, Rmg!F∈C(Y).
As a consequence of (II) we will obtain a comparison isomorphism for `- adic cohomology,
Hcq(X,(Fn)n∈N)−→∼ Hcq(Xad,(Fnad)n∈N),
where X is a separated scheme of finite type over Speck (here k is as- sumed to be algbraically closed) andXadis its associated adic space over Spa (k, k◦). (Forchar(k) = 0 this comparison theorem is already proved in [10]).
The main new ingredient of the proof of (I) is a result on algebraization of finite morphisms of adic spaces (Lemma 7.3).
(II) can be deduced from (I). In the following we sketch this. By virtue of (I) it suffices to show thatRmg!F is constructible around each y ∈ Y(k) (withk the algebraic closure ofk). For simplicity let us assume that
Y :=B1k= Spa (khTi, k◦hTi), {y}:={0}=V(T)⊆Y.
The topological spaceSpa (khTi, k◦hTi)is a subspace of the valuation spec- trumSpvkhTiofkhTi,
0∈Y = Spa (khTi, k◦hTi)⊆SpvkhTi.
The element 0 has no proper generalization inSpa (khTi, k◦hTi)but there is a unique valuationv∈SpvkhTiwhich is a proper generalization of 0 in SpvkhTi and extends the valuation| | ofk. One may expect thatv can be helpful to study, for a sheafE onSpa (khTi, k◦hTi)´et, the behavior ofE around 0. Some evidence for this appears in the paper [18]. In the present
paper we usev in order to prove (II).
If (Un)n∈N is a fundamental system of neighbourhoods of zero in k then (UnhTi)n∈N is a fundamental system of neighbourhoods of zero in khTi whereUnhTi={P
a`T` ∈khTi |a`∈Un for all`}. Let khTiT be the lo- calization ofkhTiwith respect to the multiplicative system{1, T, T2, . . .}.
We endow khTiT with the ring topology such that (Tn·UnhTi)n∈N is a fundamental system of neighbourhoods of zero. We will show that the topo- logical ring khTiT is strictly noetherian. Hence we have the affinoid adic space
Y0:= Spa (khTiT, k◦hTi).
Sincev(T)6= 0, the valuationv ofkhTiextends uniquely to a valuationw of khTiT. The valuation w is continuous with respect to the topology of khTiT andw(b)61for allb∈k◦hTi, i.e.,
w∈Y0. For an elementa∈k∗ with|a|<1put
Yε0:={x∈Y0| |a(x)|x<1} ⊆Y0. This set is independent ofa. We have
(1) There is a natural morphism of adic spaces π:Y − {0} −→Y0.
πis an open embedding and it extends to a homeomorphism τ:Y −→Yε0
withτ(0) =w.
(Remark. The setY0−Yε0 consists of exactly one element).
Sinceh:X →Y is locally algebraic, we may assume that the diagram of pseudo-adic spaces
(X−h−1(0), L−h−1(0))
u
Y − {0} π //Y0
whereu is the restriction ofg can be extended to a cartesian diagram of pseudo-adic spaces
(X−h−1(0), L−h−1(0)) π
0 //
u
(X0, L0)
g0
Y − {0} π //Y0
where the morphism of adic spacesX0→Y0 underlyingg0 is of finite type and separated and L0 is a locally closed constructible subset of X0. Let F0 be the constant A-module on (X0, L0)´et such that π0∗(F0) = F|(X − h−1(0), L−h−1(0)). Obviously,
(2) Rmg!F|Y − {0}=π∗Rmg!0F0.
We will show that (I) holds analogously forg0 andF0 instead ofg andF. Hence as the support of w is the generic point of SpeckhTiT, we obtain thatRmg!0F0 is constructible atw. Then (1) and (2) imply thatRmg!F is constructible around 0.
Throughout the paper is assumed that, for an affinoid analytic adic X, OX(X)is a stictly noetherian Tate ring.
I thank S. Bosch for his reference to [19] in Remark 2.9.
2. Some strictly noetherian Tate rings
For a topological ring A, we call a subring of A which is open inA and whose subspace topology is adic a ring of definition ofA, and we call Aa Tate ring if it has a ring of definition and a topologically nilpotent unit ([7]).
Let A be a topological ring and let f be an element of A. Let Af be the localization of the ring A with respect to the multiplicative system {1, f, f2, . . .}and let ρ:A→Af be the natural mapping. There is a ring topology on Af such that {ρ(fnU) | n ∈ N and U a neighbourhood of 0 in A} is a fundamental system of neighbourhoods of zero. The ring Af equipped with this topology is denoted byAf. The mappingρ:A→Af is a universal ring homomorphism fromA to a topological ring which maps f to a unit and is open.
If f0 is an element of A such that we have an equality V(f) = V(f0) of subsets ofSpecA then we have an equality of topological ringsAf =Af0.
IfAis hausdorff (complete, resp.) and the idealker(ρ) ={a∈A|fna= 0 for somen∈N}is annihilated by somefmthenAf is hausdorff (complete, resp.). If there exists a units of A such that sf is topolgically nilpotent inAthenAf has a topologically nilpotent unit (e.g.,ρ(sf)) and, for every ring of definitionA0 ofA,ρ(A0)is a ring of definition ofAf. Therefore, if Ais a Tate ring thenAf is a Tate ring.
The aim of this section is to show that if A is a strictly noetherian Tate ring thenAf is strictly noetherian, too.
2.1. — Let A be a topological ring which has a fundamental system of neighbourhoods of 0 consisting of additively closed subsets ofA. We put, for every subsetU ofA,
AhX1, . . . , Xni = {X
aνXν∈A[[X1, . . . , Xn]]|(aν)ν∈Nn
0 is a zero sequence inA}
UhX1, . . . , Xni = {X
aνXν∈AhX1, . . . , Xni |aν ∈U for allν∈Nn0} U[X1, . . . , Xn] = {X
aνXν∈A[X1, . . . , Xn]|aν ∈U for allν∈Nn0}.
In the following we endowA[X1, . . . , Xn]andAhX1, . . . , Xniwith the ring topology such that if U is the set of all neighbourhoods of 0 in A then {U[X1, . . . , Xn]| U ∈U} and{UhX1, . . . , Xni |U ∈U}are fundamental systems of neighbourhoods of0 inA[X1, . . . , Xn]andAhX1, . . . , Xni.
A complete Tate ringAis called strictly noetherian if, for everyn∈N, the ringAhX1, . . . , Xniis noetherian.
Remark 2.2. — (i) LetAbe as in (2.1). Then(A[T]T)∧=AhTiT. (ii) For a ring B and an element s of B let (B, s) indicate that the
ringB is endowed with thesB-adic topology. Then(B[T], sT)∧= ((B, s)hTi, sT).
We say that a topological ringAsatisfies (N) if every idealIofAis finitely generated and the naturalA-module topology ofI([8], §2) agrees with the subspace topology ofI in A.
Remark 2.3. — (i) LetAbe a topological ring and letIbe a finitely generated ideal ofA. The naturalA-module topology ofI agrees with the subspace topology ofI inA if and only if for some (and then for any) finite system of generators a1, . . . , an ofI the map- ping
(∗) An→I, (x1, . . . , xn)7→x1a1+. . .+xnan
is open whereAn is equipped with the product topology andI is equipped with the subspace topology ofA.
(ii) LetAbe a Tate ring that is hausdorff and unequal{0}. Let k k be a norm ofA, i.e.k kis a mappingA→R>0such that
(a) k0k= 0, k1k= 1
(b) kx+yk6max(kxk,kyk)for allx, y∈A (c) kxyk6kxk · kyk for allx, y∈A
(d) ({x∈ A | kxk 6r} |r ∈ R>0) is a fundamental system of neighbourhoods of 0inA.
Assume that there is a topologically nilpotent unit s of A such thatks−1k=ksk−1or, equivalently,ksak=ksk · kakfor alla∈A.
(Such a norm ofA always exists).
LetIbe an ideal ofAwith a finite system of generatorsa1, . . . , an. Then the mapping (∗) in (i) is open if and only if
(∗∗) there exists some K ∈ R>0 such that for everyx ∈ I there exist x1, . . . , xn ∈Awith x=x1a1+. . .+xnan and kxik6 K· kxkfori= 1, . . . , n.
(iii) A complete Tate ring satisfies (N) if and only if it is noetherian ([8], 2.4.ii).
Lemma 2.4. — LetAbe a Tate ring that is hausdorff.
(i) IfAhTisatisfies (N) thenA[T]satisfies (N).
(ii) IfAandAhTˆ isatisfy (N) thenAhTiandA[T]satisfy (N).
(iii) Let n ∈ N such that A and AhTˆ 1, . . . , Tni satisfy (N). For i ∈ {1, . . . , n} let {i ,i} be [ , ] or h ,i. Then the Tate ring A{1T11}. . .{nTnn}satisfies (N).
Proof. We fix a norm k k :A →R>0 ofA such that ks−1k =ksk−1 for some topologically nilpotent unitsofA. We equip all rings occuring in (i) and (ii) with the Gauss norm with respect tok k.
i) We will show that every idealI ofA[T]has a finite system of generators a1, . . . , an for which(∗∗)in Remark 2.3(ii) holds.
First we reduce the situation to the case thatIisT-saturated. So letIbe an ideal ofA[T]and letI0 be theT-saturation ofI. Assume that the assertion holds for I0, i.e., there exist a finite system of generators e01, . . . , e0n of I0 and someK∈R>0such that for everyx∈I0 there existx1, . . . , xn∈A[T] withx=x1e01+. . .+xne0n andkxik6K· kxk fori= 1, . . . , n. Letm∈N such thatTme0i∈Ifori= 1, . . . , n. For everyr∈N0letπrbe the mapping
A[T]→A, X
apTp7→ar.
By hypothesis AhTi satisfies (N) and hence A satisfies (N), too. Let er,1, . . . , er,n(r)be elements ofI∩(Tr·A[T])such thatπr(er,1), . . . , πr(er,n(r)) generate the idealπr(I∩(Tr·A[T]))ofA. Applying(∗∗)in Remark 2.3(ii) to these generators, we obtain by induction onr= 0,1, . . . that for every r∈N0 there exists someKr∈R>0 such that for everyx∈I there exists a familyxij, i= 0, . . . , r, j= 1, . . . , n(i)in Asuch that
x− X
i=0,...,r j=1,...,n(i)
xijeij ∈ I∩(Tr+1·A[T])
and kxijk 6 Ki· kxk for i = 0, . . . , r, j = 1, . . . , n(i). Therefore, in order to prove the assertion for I it is enough to consider the elements x ∈ I∩(Tm·A[T]). Then T−m·x∈ I0 and so there exist x1, . . . , xn ∈A[T] with T−m·x= x1e01+. . .+xne0n and kxik 6 K· kT−m·xk = K· kxk for i = 1, . . . , n. Then for ei := Tme0i ∈ I (i = 1, . . . , n) we get x = x1e1+. . .+xnen. Thus we see that it is enough to prove the assertion for I0, i.e., we may assume thatI isT-saturated.
Foru, v∈N0 withu6vput
A[T]u,v:={p∈Tu·A[T]|deg(p)6v}.
Forx=P
apTp∈AhTiand u∈N0 put
ux:=X
p6u
apTp ∈ A[T].
We consider the ideal ofAhTigenerated byI. SinceAhTisatisfies (N), there existe1, . . . , en ∈Isuch that for everyx∈I there existx1, . . . , xn∈AhTi with x = x1e1 +. . .+xnen and kxik 6 kxk for i = 1, . . . , n. Then for L:= max(1,ke1k, . . . ,kenk), m:= max(deg(e1), . . . ,deg(en)), u:= deg(x) andy:=x−u(x1)e1−. . .−u(xn)en hold
y∈I∩A[T]u,u+m and kyk6L· kxk.
SinceIisT-saturated, we obtain
y=Tu·z for somez∈I∩A[T]0,m.
Therefore it suffices to consider the elementsz∈I∩A[T]0,m. Similarly as in the reduction above we get that there existf1, . . . , fs∈I∩A[T]0,m and K ∈ R>0 such that for every z ∈ I∩A[T]0,m there exist z1, . . . , zs ∈ A withz=z1f1+. . .+zsfs andkzik6K· kzk fori= 1, . . . , s.
ii) By virtue of (i) it suffices to show thatAhTisatisfies (N). Let I be an ideal ofAhTi. We will show that there exists a finite system of generators a1, . . . , an of I for which (∗∗) in Remark 2.3(ii) holds. As in the proof of (i) we may assume thatI is T-saturated. We consider the ideal of AhTˆ i
generated byI. SinceAhTˆ isatisfies (N), there exist elementsg1, . . . , gmofI such that for everyx∈I and everyε∈R>0 there existx1, . . . , xm∈AhTi withkx−Pm
i=1xigik < εand kxik 6kxk for i= 1, . . . , m. Letπ denote the ring homomorphism
AhTi →A, X
aiTi 7→a0.
SinceA satisfies (N), there exist elementsgm+1, . . . , gn ofI such that for everyx∈π(I)there existxm+1, . . . , xn∈Awithx=xm+1π(gm+1) +. . .+ xnπ(gn)andkxik6kxk fori=m+ 1, . . . , n. We will show that for every x∈I there exist x1, . . . , xn ∈AhTisuch that x=x1g1+. . .+xngn and kxik6kxkfori= 1, . . . , n.
Letxbe an element ofIwithx6= 0. We choose a decreasing zero sequence (σp)p∈N0 in R>0 with σ0 = kxk. We will construct, for every p ∈ N0, elementsyp,1, . . . , yp,n ofAhTisuch that
a) kyp,ik6σp fori= 1, . . . , n b) If xp,i := Pp
q=0Tqyq,i ∈ AhTi (i = 1, . . . , n) and zp := x− Pn
i=1xp,igi thenkzpk6σp+1andzp∈Tp+1·AhTi.
Then for
xi:= X
q∈N0
Tqyq,i∈AhTi (i= 1, . . . , n) we havex=x1g1+. . .+xngn andkxik6kxkfori= 1, . . . , n.
Let p ∈ N0 such that yq,1, . . . , yq,n for q = 0, . . . , p−1 are already con- structed. Sincezp−1 ∈I∩(Tp·AhTi)(for p= 0 put zp−1 =x) andI is T-saturated, we haveT−p·zp−1∈I. Chooseyp,1, . . . , yp,m∈AhTisuch that ifz:=T−p·zp−1−(yp,1g1+. . .+yp,mgm)andλ:= max(1,kgm+1k, . . . ,kgnk) thenkzk ·λ6σp+1 and kyp,ik 6kT−p·zp−1k =kzp−1k fori= 1, . . . , m.
Then chooseyp,m+1, . . . , yp,n∈Asuch thatπ(z) =yp,m+1π(gm+1) +. . .+ yp,nπ(gn)andkyp,ik6kπ(z)k fori=m+ 1, . . . , n.
iii) For i = 1, . . . , n the Tate ring (A{1 T11}. . .{i−1 Ti−1i−1})∧hTii = AhTˆ 1, . . . , Tiisatisfies (N), since AhTˆ 1, . . . , Tni satisfies (N). Then the as-
sertion follows from (ii) by induction oni.
Proposition 2.5. — Let A be a Tate ring that is complete and strictly noetherian. Then, for everyf ∈A, the Tate ringAfis complete and strictly noetherian.
Proof. We may assume that f is power bounded in A. Then we have the continuous ring homomorphism σ : AhTi → A with σ(a) = a for all a ∈ A and σ(T) = f. Consider the induced continuous ring homomor- phism τ : AhTiT → Af. The mapping τ is surjective and open, as σ is
surjective and open . Hence it suffices to show thatAhTiT is strictly noe- therian.
Remark 2.2(i) immediately implies that, for every n ∈ N, AhTiThX1, . . . , Xni=A[X1, . . . , Xn]hTiT. By Lemma 2.4(iii) and Remark 2.3(iii) the ringA[X1, . . . , Xn]hTisatisfies (N), and hence it is noetherian.
Thus we obtain thatAhTiThX1, . . . , Xniis noetherian.
Lemma 2.6. — Let A be a strictly noetherian complete Tate ring, let f be an element ofA, letρ:A→Af be the natural mapping and let B be a complete Tate ring of topologically finite type overAf,
A−→ρ Af −→η B.
Let A0 be a ring of definition of A, and so ρ(A0) is a ring of definition ofAf. LetB0 be a ring of definition ofB of topologically finite type over ρ(A0). LetC0 be the ringB0 equipped with the adic topology such that the ring homomorphismη◦ρ:A0 →C0 is adic. LetC be the subring of B generated byC0 and(η◦ρ)(A), endowed with the group topology such thatC0is an open topological subgroup ofC. (ThenCis a Tate ring, the mappingη◦ρ:A→Cis continuous andB =C(η◦ρ)(f)). Then
(i) The topological ringC satisfies (N).
(ii) The ring homomorphismη◦ρ:A→C∧ is of topologically finite type, more precisely, the ring homomorphism η◦ρ:A0 → C0∧ is of topologically finite type.
Proof. We may assume thatf ∈A0. By virtue of the mappingτ:AhTiT → Afin the proof of Proposition 2.5 we may replaceA, f, B, A0, B0byAhTi, T, AhTiThX1, . . . , Xni, A0hTi, ρAhTi(A0hTi)hX1, . . . , Xni. Then with Remark 2.2(ii) we obtainC0=A0[X1, . . . , Xn]hTi, and henceC=A[X1, . . . , Xn]hTi which satisfies (N) by Lemma 2.4(iii) and Remark 2.3(iii). Furthermore, C0∧=A0hX1, . . . , XnihTiis of topologically finite type overA0hTi For a ringAand a subsetRofA, let us call anA-moduleM R-noetherian if for every sub-A-moduleP ofM there exists somen(P)∈Nsuch that for everyr∈Rn(P):={sn(P)|s∈R} there exists a finitely generated sub-A- moduleP0 of M with rP ⊆P0 ⊆P. The ringAis called R-noetherian if theA-moduleAisR-noetherian. (Remark. In our application (Proposition 2.8, the valuation ofknot discrete) holds that for every r∈R and every n∈N there exists some t ∈R withr ∈tnA. Then if M is R-noetherian then we can putn(P) = 1for every sub-A-moduleP ofM).
Lemma 2.7. — LetAbe a ring andRa subset of A.
(i) IfAisR-noetherian then every finitely generatedA-module isR- noetherian.
(ii) Assume that there exists a ring topology on A such that the fol- lowing two conditions are satisfied
(a) A is an open topological subring of some topological ring B which satisfies (N).
(b) If λ:A→Aˆis the completion ofAthen the ringAˆ isλ(R)- noetherian.
ThenAisR-noetherian.
Proof. We proof (ii). LetIbe an ideal ofA. By hypothesis (b) there exists a n(I·A)ˆ ∈Nsuch that for everyr∈Rn(I·A)ˆ there exists a finitely generated idealJ ofAˆwithr·I·Aˆ⊆J ⊆I·A. We will show that for everyˆ r∈Rn(I·A)ˆ there exists a finitely generated idealKofAwith rI⊆K⊆I.
By hypothesis (a) the idealI·BofBis finitely generated. Hence there exists a finite subsetSofIwithI·B=S·B. Furthermore by (a) there exists an open subsetU ofB withU∩(I·B) =S·A. Letr∈Rn(I·A)ˆ be given and letJ be a finitely generated ideal ofAˆwithr·I·Aˆ⊆J ⊆I·A. We chooseˆ a finite subsetT ofIwithJ ⊆T·A. Thenˆ rI⊆(S∪T)·A⊆I. Indeed, for everyx∈rIthere exists a family(at)t∈T inAwithx−P
t∈Tat·t∈U∩A.
SinceU∩(I·B) =S·A, we obtainx−P
t∈Tat·t∈S·A.
Proposition 2.8. — We consider ring homomorphisms k−→τ A−→ρ Af −→η B
wherek is a complete non-archimedean field,Aand B are complete Tate rings, f is an element of A, ρ is the natural ring homomorphism and τ andη are continuous ring homomorphisms of topologically finite type. Let A0 be a ring of definition ofA of topologically finite type overk◦ and let B0 be a ring of definition ofB of topologically finite type over the ring of definitionρ(A0)ofAf.
ThenB0is(η◦ρ◦τ)(k◦◦)-noetherian. (If the valuation ofkis discrete then B0 is noetherian).
Proof. LetC0 andCbe as in Lemma 2.6. Letλ:C0→C0∧be the comple- tion ofC0. By Lemma 2.6(ii) the ring homomorphismσ:=λ◦η◦ρ◦τ:k◦→ C0∧ is of topologically finite type. Then according to [14], Satz 5.1 the ring C0∧ is σ(k◦◦)-noetherian. By Lemma 2.6(i) the topological ringC satifies (N). Then we can conclude from Lemma 2.7(ii) thatB0is(η◦ρ◦τ)(k◦◦)-
noetherian.
Remark 2.9. — Let k be a non-archimedean field (not necessarily com- plete) and letn∈N. Then
(i) The topological ringk[X1, . . . , Xn]satisfies (N).
(ii) The ringk◦[X1, . . . , Xn] isk◦◦-noetherian
The first assertion follows from Lemma 2.4(iii). Since the completion (k◦[X1, . . . , Xn])∧ = (k∧)◦hX1, . . . , Xni is (k∧)◦◦- noetherian ([14], Satz 5.1), the second assertion follows from the first one and Lemma 2.7(ii).
I learned from S. Bosch that a better result than (i) holds. Namely, ifV is a valuation ring then every(V − {0})-saturated idealI of V[X1, . . . , Xn] is finitely generated. Indeed, the ringV[X1, . . . , Xn]/I is flat and of finite type over the integral domainV and hence of finite presentation overV by [19], 3.4.7.
3. Some analytic adic spaces
We fix an affinoid analytic adic spaceX and an elementf ofOX(X). Put A:=OX(X)andA+:=OX+(X).
According to Section 2 we have the Tate ring Af. The integral closure (A+)c of A+ in Af is a ring of integral elements of Af. By Proposition 2.5 the Tate ringAf is strictly noetherian. Hence we have an adic space associated with the affinoid ring(Af,(A+)c). We denote this space byXf,
Xf := Spa (Af,(A+)c).
If Y is a further affinoid adic space and h : Y → X is a morphism of adic spaces then the continuous morphism of affinoid ringsh∗: (A, A+)→ (B, B+)withB:=OY(Y)induces a continuous morphism of affinoid rings (Af,(A+)c)→(Bh∗(f),(B+)c)and so we get a morphism of adic spaces
hf :Yh∗(f)→Xf.
(Remark. Ifhis of finite type, this does not imply in general thathf is of finite type. This is the reason why in statement (II) of the introduction the morphismhis required to be locally algebraic.)
The aim of this section is to compareX and Xf. Similar considerations are contained in [21], 4.2.
Put Xf := X−V(f). The A-algebra homomorphism Af → OX(Xf) is continuous and maps(A+)ctoOX+(Xf). (Indeed, ifA→Bis a continuous ring homomorphism from A to a topological ring B which maps f to a
unit then theA-algebra homomorphismσ : Af → B is continuous, since the mappingA→Af is open. Furthermore, ifB+is a subring ofB which is integrally closed inB and contains the image ofA+ inB then trivially σ((A+)c)⊆B+). So we get by [8], 2.1(ii) a morphism of adic spaces
πX,f :Xf →Xf.
Proposition 3.1. — πX,f is an open embedding of adic spaces.
Proof. Let A0 be a ring of definition of A and let s be a topologically nilpotent unit ofAwiths∈A0. We may assume thatf ∈sA0. ThenA0 is a ring of definition ofAf andf is a topologically nilpotent unit ofAf(more precisely, the images ofA0 and f in Af). We consider rational subsets of X andXf,
Un := RX(sn
f ) ={x∈X| |sn(x)|6|f(x)|}
Un0 := RXf(sn
f ) ={x∈Xf | |sn(x)|6|f(x)|}.
ThenUn⊆Un+1, Xf =S
n∈NUn, Un0⊆Un+10 andπX,f−1 (Un0) =Un. The mor- phism of affinoid rings πX,f∗ : (OXf(Un0),OX+f(Un0))→ (OX(Un),OX+(Un)) is an isomorphism, since both affinoid rings are completions of the affinoid ring(Af, A+[sfn]c)where A+[sfn]c is the integral closure of A+[sfn]in Af andAf is equipped with the topology such thatA0[sfn]is a ring of defini- tion withs·A0[sfn](or, equivalently,f ·A0[sfn]) an ideal of definition ([8],
Proposition 1.3).
We put, forsa topologically nilpotent unit ofA, (Xf)ε := {x∈Xf| |s(x)|<1}
(Xf)ι := {x∈Xf| |s(x)|>1}={x∈Xf | |s(x)|= 1}.
These subsets of Xf are independent of the choice of s. The set Xιf is rational inXf and the setXεf is closed and constructible inXf. Let(Xεf)◦ denote the interior ofXεf inXf.
Proposition 3.2. — (i) (Xεf)◦= im(πX,f :Xf →Xf)
(ii) LetB be the completion ofA in thef A-adic topology and letB0 be the image ofB in Bf. Then
OXf(Xιf) = Bf,equipped with the topology such thatB0 is a ring of definition andf B0 is an ideal of definition OX+f(Xιf) = integral closure of B inBf.
Hence Xιf is the analytic adic space associated with the formal completion of the schemeSpecAalong its closed subsetV(f).
Proof. i) We use the notations of the proof of Proposition 3.1 (in particular, f is a topologically nilpotent unit ofAf). By the proof of Proposition 3.1 we have
im(πX,f) = [
n∈N
{x∈Xf | |sn(x)|6|f(x)|}
which is the interior of{x∈Xf | |s(x)|<1}in Xf ([11], Lemma 1.3.ii).
ii) is obvious.
For an analytic adic spaceY and an elementtofOY(Y)and finite subsets D, E ofOY(Y)with(D∪E∪ {t})·OY(Y) =OY(Y)we put
SY(D|E
t ) := {y∈Y | |d(y)|6|t(y)|for everyd∈D and|e(y)|<|t(y)|for everye∈E}.
ThenSY(D|Et )is a locally closed constructible subset ofY. IfE=∅(resp.
D=∅ ) thenSY(D|Et )is open (resp. closed) ([9], 3.1).
Proposition 3.3. — There is a mapping σX,f :Xεf →X such that
(i) For any element t of A and finite subsets D, E of A with (D∪E∪ {t})·A=A,
σ−1X,f(SX(D|E
t )) =SXf(D|E t )∩Xεf.
(ii) The mappingσX,f is spectral, i.e.,σX,f is continuous and for any quasi-compact open subsetU ofX the preimageσX,f−1 (U)is quasi- compact.
(iii) The composite mapσX,f ◦πX,f :Xf →X is the inclusion ofXf intoX.
(iv) σX,f is functorial inX, i.e., for any morphismr:S →T of affinoid analytic adic spaces and for anyg∈OT(T)the diagram
(Sr∗(g))ε
σS,r∗(g) //
rg
S
r
(Tg)ε σT ,g //T
commutes.
The mappingσX,f is uniquely determined by (i). IfAhas a noetherian ring of definition that is contained inA+ thenσX,f is uniquely determined by (ii) and (iii). The family (σX,f)X,f of all σX,f is uniquely determined by (ii),(iii),(iv).
Proof. Letsbe a topologically nilpotent unit ofA. We consider the valua- tion spectrum SpvAofA and the subset
W :={v∈SpvA|v(a)61for everya∈A+andv(s)<1}.
According to [7] we have
X = Spa (A, A+) ={v∈W |Γv=cΓv} and we have the retraction
r:W →X, v7→v|cΓv.
The ring homomorphismρ:A→Af induces the mapping p:= Spv (ρ) : SpvAf →SpvA.
SinceXεf ⊆p−1(W), we get the mapping
σX,f :=r◦p:Xεf →X.
For any element t of A and finite subsets D, E of A with (D∪E∪ {t})·A=A, the subset of SpvA
{v∈SpvA|v(d)6v(t)for everyd∈D andv(e)< v(t)for everye∈E}
is closed under primary specializations and primary generalizations in SpvA.
Hence
σX,f−1 (SX(D|E
t )) =SXf(D|E
t )∩Xεf ,
i.e., (i) holds. (ii) follows from (i). (iii) and (iv) are easily checked.
For anyx∈X, the set{x}is the intersection of allSX(D|Et )which contain x. HenceσX,f is uniquely determined by (i).
The constructible topology of a spectral space is hausdorff. So the con- structible topology ofX is hausdorff. IfA has a noetherian ring of defini- tion that is contained inA+ (and soA+=A◦,[13], 2.4.16) then the set of all maximal points ofXf is dense in the constructible topology ofXf ([7], Lemma 3.4), in particular(Xεf)◦ is dense in the constructible topology of Xεf. By Proposition 3.2(i) we have (Xεf)◦ = im(πX,f). Therefore σX,f is uniquely determined by (ii) and (iii) ifAhas a noetherian ring of definition that is contained inA+.
LetS be as in (iv) and lethbe an element ofOS(S). Let(ri:S→Ti)i∈I
be the family of all morphisms fromS to an affinoid analytic adic spaceTi such thatOTi(Ti)has a noetherian ring of definition contained in OT+i(Ti) and there is some ti ∈ OTi(Ti) with h = r∗i(ti). Any x ∈ S is uniquely determined by the family (ri(x))i∈I (i.e., if x, y are elements of S with ri(x) = ri(y) for all i ∈ I then x = y). Hence the family of all σX,f is
uniquely determined by (ii),(iii),(iv).
Lemma 3.4. — Put U :={x∈SpecA | f(x) 6= 0} and V := {x∈ X | f(x) = 0}.
(i) The mappingσ=σX,f :Xεf →X is generalizing, specializing and closed.
(ii) Ifi:Y ,→X is the closed adic subspace ofX corresponding to the scheme-theoretic closure of U in SpecA thenif :Yi∗(f) →Xf is an isomorphism.
(iii) σ−1(V) = Xεf −(Xεf)◦. If U is dense in SpecA then σ(Xεf − (Xεf)◦) =V (and hence σ:Xεf →X is surjective).
Proof. We use the notations of the proofs of Proposition 3.1 and Proposi- tion 3.3.
i) Letxbe an element ofXεf and letv (resp.w) be a specialization (resp.
generalization) ofσ(x). We considerσ(x), v, was elements of SpvA. Then v(resp.w) is a secondary specialization (resp. secondary generalization) of σ(x)in SpvA, andp(x)∈SpvAis a primary generalization ofσ(x). Hence there existv0, w0∈SpvAsuch thatv0 (resp.w0) is a primary generalization ofv (resp.w) and a secondary specialization (resp. secondary generaliza- tion) ofp(x) ([12], Lemma 1.2.5.ii,iv). Letv00 and w00 be the elements of SpvAf with p(v00) =v0 andp(w00) =w0. Thenv00, w00 ∈Xεf andv00 (resp.
w00) is a specialization (resp. generalization) of x in Xεf and σ(v00) = v andσ(w00) =w. This shows thatσis specializing and generalizing. Asσis spectral (Proposition 3.3(ii)), we obtain thatσis closed.
ii) is obvious
iii) Ifx∈Xεf−(Xεf)◦ then|f(x)|<|s(x)|n for everyn∈N(by the proof of Proposition 3.2) and hence|f(σ(x))|<|s(σ(x))|n for everyn∈Nwhich impliesf(σ(x)) = 0, i.e. σ(x)∈V.
Assume that U is dense in SpecA. Let x be an element of V. We show that there exists somey∈Xεf withx=σ(y). SinceU is dense in SpecA, there exists a prime idealpofAsuch thatp⊆supp(x)andf 6∈p. By [12], Lemma 1.2.6 there exists a primary generalization z of x in SpvA with p= supp(z). Then forz(f)∈Γz∪ {0} we havez(f)6= 0and z(f)< cΓz. LetH be the smallest convex subgroup ofΓz which contains z(f). Then cΓz ⊆ H and so we have the primary specialization y := z|H ∈ SpvA.
Sincey(f)6= 0, we can consideryas an element of SpvAf. It is easily seen
thaty∈Xεf andx=σ(y).
Example 3.5. — Assume that Ais a Dedekind ring and f 6= 0. We want to describe the topological spaceXf.
For this we use the following natural inclusions of topological spaces Xf ⊆ SpvAf ⊆ SpvA ⊇ X.
So we considerXf andX as topological subspaces of SpvA. The elements of SpvAare written as pairs(p, B)wherepis a prime ideal of AandB is a valuation ring of the quotient fieldqf(A/p). Put
W := {x∈SpecA|f(x) = 0}={m∈MaxA|f ∈m}
V := {x∈X |f(x) = 0}={(m, B)∈X|m∈W}.
If m is a maximal ideal of A and B is a valuation ring of A/m then (m, B)∈SpvA is an element ofX if and only if the image ofA+ in A/m is contained inB and, for some (and then for any) topologically nilpotent unitsofA, the image ofsinA/mis contained in the maximal ideal ofB. We divide Xf into the three subsets (Xεf)◦, Xιf, R := Xεf −(Xεf)◦ = (Xιf)−−Xιf where(Xιf)− denotes the closure ofXιf in Xf.
a) By Proposition 3.1 and 3.2(i) the adic spaces(Xεf)◦ andXf are isomor- phic. IdentifyingXf andX as topological subspaces of SpvA, we get the equality
(Xεf)◦=X−V.
b) By Proposition 3.2(ii) the mapping
ψ:W →Xιf, m7→({0}, Am)
is a homeomorphism. SoXιf is a finite discrete topological space. For every m∈W, the subset{ψ(m)} ofXf is rational andOXf({ψ(m)}) =qf( ˆAm) and OX+f({ψ(m)}) = ˆAm where Aˆm denotes the completion of A in the m-adic topology.
c) For a maximal idealm ofAand a valuation ringB ofA/m, letP(m, B) denote the preimage ofB under the mappingAm →A/m. ThenP(m, B) is a valuation ring ofqf(A). The mapping
V →R, (m, B)7→({0}, P(m, B)) is a homeomorphism.
d) The restriction of the mappingσX,f :Xεf →XtoRis a homeomorphism from R ontoV, namely it is the inverse mapping of the bijection in (c).
ThenσX,f :Xεf →X is bijective. As σX,f is closed, we obtain thatσX,f : Xεf →X is a homeomorphism.
Remark 3.6. — LetB be a Tate ring (not necessarily complete), letf be an element ofB and letB+ be a ring of integral elements ofB. According to Section 2 we have the Tate ringBf. The integral closure(B+)c of B+ in Bf is a ring of integral elements of Bf. Assume that the completions B∧and(Bf)∧ofB andBf are strictly noetherian. Then we have the adic spaces
Y := Spa (B, B+) Z := Spa (Bf,(B+)c).
(The natural morphism(Bf)∧ →(B∧)f is not an isomorphism in general (see Remark 2.2(i)), and hence the natural morphism of adic spacesYf → Z is not an isomorphism in general.)
Letsbe a topologically nilpotent unit of B and put Zε := {z∈Z| |s(z)|<1}
Zι := {z∈Z| |s(z)|>1}={z∈Z| |s(z)|= 1}.
The setsZεandZι are independent ofs. As above (i.e., as in the case that B is complete) one can define a morphism of adic spaces
π:Y −V(f)→Z and a mapping
σ:Zε→Y for which (3.1)-(3.5) hold analogously.
4. Some constructible sheaves
LetA be a noetherian ring.
Definition 4.1. — For an analytic pseudo-adic space(X, L)withX affi- noid, Z(X, L) denotes the class of all A-modules F on (X, L)´et which satisfy the following equivalent conditions(Y := SpecOX(X))
(i) For every y ∈ Y there exist a morphism of schemes f : S → Y and a constructible A-module G on S´et such that f is of finite type,y∈f(S)and the restrictions ofF andGto the étale site of (X, L)×Y S are isomorphic.
(ii) For everyy ∈Y there exists a morphism of schemes f : S → Y such thatf is of finite type, y ∈f(S)and the restriction ofF to the étale site of(X, L)×Y Sis a constantA-module of finite type.
(iii) For every y ∈ Y there exist a locally closed subscheme R of Y and a surjective finite étale morphism of schemesS →Rsuch that y∈Rand the restriction ofF to the étale site of(X, L)×Y Sis a constantA-module of finite type.
(iv) There exist a decreasing sequence of closed adic subspaces ofX X =X0⊇X1⊇. . .⊇Xn =∅
and, for everyi∈ {0, . . . , n−1}, a finite morphism of adic spaces fi :Ti→Xi such thatfi−1(Xi−Xi+1)→Xi−Xi+1 is surjective and étale and the restriction ofF to the étale site offi−1(L∩(Xi− Xi+1))is a constantA-module of finite type.
(The equivalence of (ii) and (iii) follows from [5], 17.16.4).
Definition 4.2. — For an analytic pseudo-adic space(X, L), letC(X, L) denote the class of allA-modulesF on(X, L)´et such that, for everyx∈L, there exist a locally closed locally constructible subsetP ofLwith x∈P and a surjective étale morphism of pseudo-adic spaces (Y, M) → (X, P) withY affinoid such that the restriction ofF to the étale site of(Y, M)is an element ofZ(Y, M).
If(X, L)is a pseudo-adic space then an A-moduleF on(X, L)´et is called constructible if, for every x∈L, there exists a locally closed locally con- structible subset P of L such that x∈P and the restriction ofF to the étale site of (X, P) is a locally constant A-module finite type ([9], 2.7) Every constructibleA-module on(X, L)´et is an element ofC(X, L).
Example 4.3. — Letkbe a non-archimedean field, letTbe a1-dimensional normal affinoid adic space of finite type over Spa (k, k◦) and let f be an element ofOT(T). According to Section 3 we have the analytic adic space X := Tf (cf. Example 3.5). Let L be a convex locally pro-constructible subset ofX. AnA-moduleF on(X, L)´et is an element ofC(X, L)if and only if the following two conditions are satisfied
(i) For every x∈ Lsuch that the support supp(x) :={s∈OX(X)| s(x) = 0} ∈ SpecOX(X)is a generic point of SpecOX(X)there exists a locally closed constructible subsetP ofLsuch thatx∈P andF|P is a locally constant A-module of finite type onP´et. (ii) For every x∈ Lsuch that the support supp(x)∈ SpecOX(X) is
not a generic point ofSpecOX(X), the restriction ofF to{x}´et is a locally constantA-module of finite type and there exist an open neighbourhood Y of x in X and a finite surjective morphism of
adic spacesg :Z →Y such that g is étale overY − {x} and the restriction of F to g−1(Y ∩L− {x})´et is a constantA-module of finite type.
In order to check property (ii) of Example 4.3 we will use in the proof of Theorem 5.2 the following criterion. This criterion is the reason why we introduced in Section 3 the adic spacesXf.
Lemma 4.4. — LetX be an affinoid analytic adic space such that B :=
OX(X)is normal, letLbe a convex pro-constructible subset ofX and let F be an A-module on (X, L)´et. Let f be a non zero divisor of B and let xbe an element ofL such that the supportsupp(x)∈SpecB is a generic point of{p∈SpecB|f ∈p}.
According to Section 3 we have the mappingσ:Xεf →X and we consider X −V(f) as an open subspace both of X and Xf. Let u be the ele- ment ofXεf withσ(u) =x. Thensupp(u)∈SpecBf is a generic point of SpecBf.
Assume that there exist a locally closed constructible subsetQof σ−1(L) and a locally constantA-moduleGof finite type onQ´et= (Xf, Q)´et such thatu∈Qand the restrictions ofFandGto the étale site ofQ∩(X−V(f)) are isomophic.
Then there exist a locally closed constructible subsetP of L with x∈P and an étale morphismh:Y →X withP ⊆im(h)and a surjective finite morphismg:Z→Y which is étale overY−V(f)such that the restriction ofF to the étale site of(g◦h)−1(P−V(f))is a constantA-module of finite type. Ifxis a maximal point ofX (i.e., the valuation corresponding to x has rank 1) then one can chooseh : Y →X as an open embedding and P=h(Y)∩L.
Proof. We may assume that there exist an affinoid adic space V and an étale morphism r : V → Xf such that Q ⊆ r(V) and the restriction of Gto r−1(Q) is a constant A-module of finite type. Since there exists an affine schemeS étale overT := SpecBf such thatV is an open subspace ofS×TXf ([9], 1.7.3), there exists a finite morphisms:W →Xsuch that V is an open subspace of Ws∗(f). We may assume thatOW(W)is normal.
Letvbe an element ofV withr(v) =u. Sinceu∈Xεf, we havev∈Ws
∗(f)
ε .
The mappingσW :Ws
∗(f)
ε →W is closed (Lemma 3.4(i)). Furthermore, for z:=σW(v)∈W we haveσW−1(z) ={v}. Hence there exists an open subset ZofW such thatz∈Zandσ−1W(Z)⊆V. The mappings:W →X is open (even universally open, [13], 3.4.7 and [12], 2.1.6). HenceY :=s(Z)is an open neighbourhood ofs(z) =xinX. Letg:Z →Y be the restriction ofs
and letPbe a locally closed constructible subset ofLsuch thatx∈P ⊆Y andσ−1(P)⊆Q. SinceG|r−1(Q)is a constantA-module of finite type and sinceG|Q∩(X−V(f))∼=F|Q∩(X−V(f)), we obtain thatF|g−1(P−V(f)) is a constantA-module of finite type. If xis a maximal point of X then there exists an open neighbourhoodU ofxinY such thatg−1(U)→U is finite. For an arbitraryx∈X, replacing (Y, x)by an étale neighbourhood of(Y, x)we may assume thatg:Z →Y is finite ([11], 3.2).
Proposition 4.5. — Let f :X →Y be a finite morphism between affi- noid analytic adic spaces. LetM be a convex locally pro-constructible sub- set ofY and putL:=f−1(M). Letg: (X, L)→(Y, M)be the morphism of pseudo-adic spaces induced byf.
(i) IfF ∈Z(X, L)theng∗F ∈Z(Y, M).
(ii) Assume that M ⊆ f(X). If F is an A-module on (Y, M)´et with g∗F ∈Z(X, L)thenF ∈Z(Y, M).
Proof. (i) can be proved as the corresponding statement for constructible sheaves on schemes, cf. [4], Lemma 4.11.
(ii) follows immediately from Definition 4.1.
Proposition 4.6. — Letf :X →Y be a separated quasi-finite morphism of finite type between analytic adic spaces. LetM be a convex locally pro- constructible subset ofY and letLbe a locally closed constructible subset off−1(M). Letg: (X, L)→(Y, M)be the morphism of pseudo-adic spaces induced byf.
(i) IfF ∈C(X, L)theng!F ∈C(Y, M)andRng!F = 0forn >0.
(ii) Assume that f(L) = M. If F is an A-module on (Y, M)´et with g∗F ∈C(X, L)thenF ∈C(Y, M).
Proof. i) By [9], 5.5.6,Rng!F = 0forn >0. We fix somey∈M and show that there exist a locally closed locally constructible subsetP of M with y ∈ P and a surjective étale morphism of pseude-adic spaces (Y0, P0) → (Y, P) with Y0 affinoid such that the restriction of g!F to (Y0, P0) is an element ofZ(Y0, P0).
We may assume thatY is affinoid andM is convex and pro-constructible inY. First we show
(∗) We may assume thatL =f−1(M) and that there exists an étale morphism h:Z → X such that Z is affinoid and L⊆h(Z) and F|(Z, h−1(L))∈Z(Z, h−1(L)).
Proof of(∗). The topological spacef−1(y)is finite and discrete. For every x∈f−1(y)∩L we fix a quasi-compact open subsetUx ofX and a locally
