Download RELATIVE RIEMANN-ZARISKI SPACES 1. Introduction Let k be an

RELATIVE RIEMANN-ZARISKI SPACES
M. TEMKIN
Max Planck Institut für Matematik, Vivatgasse 7, Bonn 53111, Germany
(e-mail: [email protected])
1. Introduction
Let k be an algebraically closed field and K be a finitely generated k-field.
In the first half of the 20-th century, Zariski defined a Riemann variety RZK
associated to K as the projective limit of all projective k-models of K.
Zariski showed that this topological space, which is now called a RiemannZariski (or Zariski-Riemann) space, possesses the following set-theoretic description: to give a point x ∈ RZK is the same as to give a valuation ring
Ox with fraction field K, such that k ⊂ Ox . The Riemann-Zariski space
possesses a sheaf of rings O whose stalks are valuation rings of K as above.
Zariski made extensive use of these spaces in his desingularization works.
Let S be a scheme and U be a subset closed under generalizations, for
example U = Sreg is the regular locus of S, or U = η is a generic point of
S. In many birational problems, one wants to consider only U -modifications
S 0 → S, i.e. modifications which do not modify U . Then it is natural to consider the projective limit S = RZU (S) of all U -modifications of S. It was
remarked in [Tem2], that working with such generalized Riemann-Zariski
spaces, one can extend the P -extension results of [Tem2] (see loc.cit. 1.3,
1.7.5 and 1.8.4) to the case of general U and S. This plan is realized in
chapter 2. In section 2.1, we prove that any point x ∈ S is uniquely determined by a point x ∈ U and a valuation ring R of k(x), which is centered on
the Zariski closure of x in S and satisfies some minimality condition. In the
next section, we use this description of S to prove the stable modification
theorem, which generalizes the main result of [Tem2].
The definition of relative Riemann-Zariski spaces may be generalized
without efforts to the case of an arbitrary separated morphism f : Y → X.
Let us say that f is decomposable if it factors into a composition of an affine
morphism Y → Z and a proper morphism Z → X. Actually, in section 2.1,
we study the case of a general decomposable morphism. We define a set
ValY (X) whose points are certain X-valuations of Y , and construct a surjection ψ : ValY (X) → RZY (X) and a section RZY (X) → ValY (X). Later,
we prove that these maps are actually bijections, see 3.2.5. It is natural
to ask if the decomposition assumption is essential. The answer is negative
because the assumption is actually empty. The main result of chapter 3 is
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2
M. TEMKIN
a decomposition theorem which states that a morphism of quasi-compact
quasi-separated schemes is decomposable if and only if it is separated.
Let us describe briefly other parts of the paper. In section 1.1 below, we
recall Nagata compactification and Thomason’s noetherian approximation
theorems, see [Nag] and [TT], C.9, and prove a slight generalization of the
second theorem. Then we notice that the decomposition theorem is essentially equivalent to the union of Nagata’s and Thomason’s theorems. Thus,
one could use Nagata’s theorem to prove that ψ : ValY (X)f
→RZY (X) in
general, but it is also possible to work in opposite direction. In section 3.1,
we establish an interesting connection between Riemann-Zariski spaces and
adic spaces of R. Huber; in particular, we obtain an intrinsic topology on
ValY (X). Then we study Y -blow ups of X, which are analogs of formal blow
ups from Raynaud’s theory, see [BL], and prove that in the decomposable
case ψ is a homeomorphism. We give a local description of spaces ValY (X)
in the next section; surprisingly, it is finer than the local structure of related adic spaces SpaY (X). Finally, we prove the decomposition theorem in
section 3.4. In particular, we prove that ψ is a homeomorphism in general
and obtain a new proof of Nagata’s theorem, see [Con] and [Lüt] for other
proofs. Nevertheless, it seems that our proof has certain similarities with
Deligne’s proof presented in [Con].
I want to mention that I was motivated by Raynaud’s theory in studying
Riemann-Zariski spaces in the decomposable case, and the basic ideas are
taken from [BL]. I give a simple illustration of those ideas in the proof of
the generalized Thomason’s theorem. I want to express my gratitude to R.
Huber for a useful discussion and to B. Conrad for pointing out various gaps
and mistakes in an earlier version of the article. This article was written
during my stay at the Max Planck Institute for Mathematics at Bonn, and
I want to thank it for hospitality.
1.1. On noetherian approximation and Nagata compactification.
For shortness, a filtered projective family of schemes with affine transition
morphisms will be called affine filtered family. Also, we abbreviate the words
”quasi-compact and quasi-separated” by the single ”word” qcqs. In [TT],
C.9, Thomason proved a very useful noetherian approximation theorem,
which states that any qcqs scheme Y over a ring Λ is isomorphic to a scheme
proj lim Yα , where {Yα }α is an affine filtered family of Λ-schemes of finite
presentation. Due to the following lemma, this theorem may be reformulated
in a more laconic way as follows: Y is affine over a Λ-scheme Y0 of finite
presentation.
Lemma 1.1.1. A morphism of qcqs schemes f : Y → X is affine if and
only if Y f
→ proj lim Yα , where {Yα }α is a filtered family of X-affine finitely
presented X-schemes.
→ proj lim Yα is as above, then Yα = Spec(Eα ) and Y = Spec(E),
Proof. If Y f
where E = inj lim Eα . Conversely, suppose that f is affine. By [EGA I], 6.9.16
RELATIVE RIEMANN-ZARISKI SPACES
3
(iii), f∗ (OY )f
→ inj lim Eα , where {Eα } is a direct filtered family of finitely presented OX -algebras. Hence Y = proj lim Spec(Eα ).
¤
We generalize Thomason’s theorem below. As a by-product, we obtain a
simplified proof of the original theorem.
Theorem 1.1.2. Let f : Y → X be a (separated) morphism of qcqs schemes.
Then f may be factored into a composition of an affine morphism Y → Z
and a (separated) morphism Z → X of finite presentation.
Proof. If Y is X-separated, then Z may be chosen X-separated by [TT], ....
Step 1. Preliminary work. If Y is affine and f (Y ) is contained in an open
affine subscheme X 0 ⊂ X, then the claim is obvious. Hence Y admits a
finite covering by open qcqs subschemes Y1 , . . . , Yn , such that the induced
morphisms Yi → X satisfy the conclusion of the theorem. By induction, we
may assume that n = 2. Then the schemes U = Y1 and V = Y2 , may be
represented as U = proj lim Uβ and V = proj lim Vγ , where the limits are
taken over affine filtered families of X-schemes of finite presentation. If f is
separated, then we also require that Vγ ’s and Uβ ’s are X-separated.
Step 2. Affine domination. By [EGA IV], 8.2.11, for β ≥ β0 and γ ≥ γ0 ,
the schemes Uβ and Vγ contain open subschemes Uβ0 and Vγ0 , whose preimages
in U and V coincide with W = U ∩ V . By [EGA IV], 8.13.1, the morphism
W → Uβ0 0 factors through Vγ0 for sufficiently large γ. Replace γ0 by γ.
By the same reason, the morphism W → Vγ00 factors through some Uβ0
and the morphism W → Uβ0 factors through some Vγ0 . Let us denote the
corresponding morphisms as fγ,β : Vγ0 → Uβ0 , fβ,γ0 and fγ0 ,β0 . Now comes an
obvious but critical argument: fβ,γ0 is separated, because the composition
fγ0 ,β0 ◦ fβ,γ0 : Uβ0 → Uβ0 0 is separated; fγ,β is affine because its composition
with a separated morphism fβ,γ0 is affine. We gather the already defined
objects in the left diagram below. Notice that everything is defined over
X, the horizontal arrows are open immersions, the vertical arrows are affine
morphisms and the indexed schemes are of finite X-presentation.
V o
? _W 
²
²
Vγ o
/U
φ0
? _ Vγ0
² 
Uβ0 h
²
/ Uβ
V o
? _W 
²
²
? _V 0 
Vγ o
φ0
γ
² 
Uβ0 /U
²
φ
/ Uγ
²
/ Uβ
Step 3. Affine extension. The main task of this step is in producing
the right diagram from the left one. It follows from the previous stage
that Vγ0 = Spec(E 0 ), where E 0 is a finitely presented OUβ0 -algebra. The
morphism φ0 : W → Vγ0 to a Uβ0 -affine scheme corresponds to a homomorphism ϕ0 : E 0 → h0∗ (OW ), where h0 : W → Uβ0 is the projection. Obviously
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M. TEMKIN
h∗ (OU )|Uβ0 f
→h0∗ (OW ), where h : U → Uβ is the projection. Hence we can
apply [EGA I], 6.9.10.1, to find a finitely presented OUβ -algebra E and a
homomorphism ϕ : E → h∗ (OU ), such that E|Uβ0 f
→E 0 and the restriction of
ϕ to Uβ0 is ϕ0 . Set Uγ = Spec(E), then ϕ induces a morphism φ : U → Uγ .
Finally, we glue Uγ and Vγ along Vγ0 obtaining a finitely presented X-scheme
Z, and notice that the affine morphisms U → Uγ and V → Vγ glue to an
affine morphism Y → Z over X.
¤
Our proof is a simple analog of Raynaud’s theory. Thomason used the
first two steps (induction argument in the proof of C.9 and lemma C.6). Our
simplification of his proof is due to the third step. The same arguments are
used in Raynaud’s theory, see [BL], the end of the proof of 4.1, statement
(d) from that proof and lemma 2.6 (a). In our paper, they also appear in
the proofs of 1.1.3, 3.2.1 (i) and 3.2.3.
Next, we recall the Nagata compactification theorem, see [Nag]. A scheme
theoretic proof may be found in [Con] or [Lüt]. Recall that a morphism
f : Y → X is called compactifiable if it may be factored as a composition
of an open immersion g : Y → Z and a proper morphism h : Z → X.
Nagata proved that a finite type morphism f : Y → X of qcqs schemes is
compactifiable iff it is separated. Actually, Nagata considered noetherian
schemes. The general case is due to [Con]. Let I ⊂ OZ be an ideal with
support Z \ Y , and Z 0 the blow up of Z along I. We can choose a finitely
generated I, because the morphism Y ,→ Z is quasi-compact. The open
immersion g 0 : Y → Z 0 is affine, because Z 0 \ Y is a locally principal divisor.
It follows that g is a composition of an affine morphism g 0 of finite type
and a proper morphism Z 0 → X. Conversely, assume that g : Y → Z
is affine of finite type and Z → X is proper. Then Y is quasi-projective
over Z, hence there is an open immersion of finite type Y ,→ Y with Zprojective and, therefore, X-proper Y . Thus, Nagata’s theorem may be
reformulated as follows: a finite type morphism is separated iff it may be
represented as a composition of an affine morphism of finite type and a
proper morphism. Now, one sees that a weak form of theorem 1.1.2 (f is
separated and Z → X is of finite type) and Nagata’s theorem are together
equivalent to the following decomposition theorem, which will be proven in
section 3.4.
Theorem 1.1.3. A morphism f : Y → X of quasi-compact quasi-separated
schemes is separated if and only if it may be factored as a composition of an
affine morphism Y → Z and a proper morphism Z → X.
2. Preliminary description of Riemann-Zariski spaces and
applications
Throughout this chapter, f : Y → X denotes a separated morphism of
qcqs schemes. We are going to recall some notions introduced in [Tem2].
RELATIVE RIEMANN-ZARISKI SPACES
5
Consider the family of all factorizations of f into a composition of a schematically dominant morphism fi : Y → Xi and a proper morphism gi : Xi → X,
it has an initial object corresponding to the schematic image of Y in X. We
call the pair (fi , gi ) a Y -modification of X, usually it will be denoted simply
as Xi . Given two Y -modifications of X, we say that Xj dominates Xi , if
there exists an X-morphism gji : Xj → Xi compatible with fi and fj . A
standard argument shows that if gji exists, then it is unique (one uses only,
that fj is schematically dominant and Xi is X-separated). The family of
Y -modifications of X is filtered, because any two Y -modifications Xi , Xj
are dominated by the scheme-theoretic image of Y in Xi ×X Xj .
A relative Riemann-Zariski space X = RZY (X) is defined as the projective
limit of the underlying topological spaces of Y -modifications of X. If X is
integral and Y is its generic point, then one recovers the classical RiemannZariski spaces. A slightly more general case, when Y is a dominant point,
was considered in [Tem1], Sect. 1. We have the obvious projections πi : X →
Xi and a map η : Y → X, and we provide X with the sheaf MX = η∗ (OY ) of
meromorphic functions. We will immediately see that the sheaf πi−1 (OXi )
embeds into MX , and we define a subsheaf OX ⊂ MX of regular functions
by OX = inj lim πi−1 (OXi ) = ∪πi−1 (OXi ).
Lemma 2.0.4. The natural homomorphism πi−1 (OXi ) → MX is monomorphism.
Proof.
¤
By schematical dominance of fi ,
2.1. Riemann-Zariski space of an affine morphism. Let ValY (X) be
the set of triples y = (y, R, φ), where y ∈ Y is a point, R is a valuation
ring of k(y) (in particular Q(R) = k(y)) and φ : S = Spec(R) → X is a
morphism compatible with y = Spec(k(y)) → Y and such that the induced
morphism y → S ×X Y is a closed immersion. Let Oy denote the preimage
of R in OY,y . For any Y -modification Xi → X, φ uniquely factors through
Xi by the valuative criterion of properness. Let xi ∈ Xi be the center of R
on Xi , i.e. the image of the closed point of S, then the family of points (xi )
defines a point x ∈ X. Thus, we obtain a map ψ : ValY (X) → X. For any i,
the image of OXi ,xi in k(y) lies in R. Therefore, the image of OXi ,xi in OY,y
lies in Oy , and we obtain a homomorphism OX,x = inj lim OXi ,xi → Oy .
Proposition 2.1.1. Suppose that f is decomposable. Then for any point
x ∈ X, there exists a unique point y = λ(x) ∈ ValY (X), such that ψ(y) = x
and Ox f
→Oy . In particular, λ is a section of ψ.
Proof. Factor f into a composition of an affine morphism Y → Z and a
proper morphism Z → X. After replacing X by the scheme-theoretic image of Y in Z, we may assume that f is affine. Let x and xi ’s be the
images of x in X and the Xi ’s. The schemes Ui = Spec(OXi ,xi ) ×Xi Y and
U = Spec(OX,x ) ×X Y are affine, because f and, therefore, fi ’s are affine.
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M. TEMKIN
Each Ui consists of points y ∈ Y , such that xi is a specialization of fi (y), the
morphisms Ui → Y are topological embeddings and OY |Ui f
→OUi . Clearly, U
and x satisfy similar properties. Notice that the schemes Ui = Spec(Bi ) form
a filtered family and U∞ := proj lim Ui = Spec(B∞ ), where B∞ = inj lim Bi .
By [EGA IV], ch. 8, U∞ = ∩Ui set-theoretically. Recall that schematical density is preserved under base changes with respect to schematically
dense morphisms, therefore the morphism Ui → Spec(OXi ,xi ) is schematically dominant. Thus, OXi ,xi ,→ Bi and we obtain an embedding of direct
limits OX,x = inj lim OXi ,xi ,→ B∞ .
Lemma 2.1.2. Suppose that elements g, h ∈ B∞ do not have common zeros
on U∞ , then either g ∈ hOx or h ∈ gOx .
Proof. Find i such that g and h are defined and have not common zeros
on Ui , and replace X by Xi . Notice that U∞ = ∩f −1 (Vj ), where Vj runs
over affine neighborhoods of x. Hence we can choose a neighborhood X 0 =
Spec(A) of x, such that g, h ∈ B and gB + hB = 1, where Y 0 = Spec(B) is
the preimage of X 0 in Y . Now, the pair (g, h) induces a morphism λ0 : Y 0 →
P1A = Proj(A[Tg , Th ]), whose scheme-theoretic image Xi0 is a Y 0 -modification
of X 0 .
It would suffice to extend the Y 0 -modification Xi0 → X 0 to a Y -modification
fi : Xi → X, where Xi0 is an open subscheme of Xi . Indeed, either
Tg ∈ Th Ox0 or Th ∈ Tg Ox0 , where x0 is the center of x on Xi0 . If fi exists, then we deduce that either g ∈ hO0 or h ∈ gO0 , where O0 denotes the
image of Ox0 in B∞ .
Probably such an extension fi does not exist in general, but fortunately
the following trick works. Given elements a, b ∈ B with (a, b) = 1, the morphism Y 0 → Proj(A[Tag , Tah , Tbg , Tbh ]) factors through Proj(A[Tg , Th ]) ×
Proj(A[Ta , Tb ]). Therefore, Xj0 is dominated by the scheme-theoretic image Z of Y 0 in Proj(A[Tag , Tah , Tbg , Tbh ]). If we choose a, b such that ag +
bh = 1, then Z is isomorphic to the scheme-theoretic image of Y 0 in P 0 =
Proj(A[T1 , Tag , Tah , Tbg , Tbh ]). Finally, the morphism Y 0 → P 0 may be extended as follows. By [EGA I], 6.9.7, the A-submodule E ⊂ B generated
by ag, ah, bg, bh may be extended to a finitely generated OX -submodule
E ⊂ F = f∗ (OY ). Replacing E by E + OX , we can achieve that the morphism E ⊗F → F is onto and still E(X 0 ) = E. Since f is affine, we obtain an
epimorphism f ∗ (E) → OY . Applying [EGA II], 4.2.3, we obtain a morphism
Y → PX (E) = Proj(S(E)), which obviously extends Y 0 → P 0 .
¤
Let us prove that B∞ is a local ring. Suppose conversely that the affine
U∞ contains two different closed points y and z, so there exist g, h ∈ B∞
such that g(y) = h(z) = 0 and g and h have no common zeros on U∞ . Now,
the lemma implies that either g|h or h|g, contradicting our assumptions.
We see that U∞ possesses a unique closed point y. In particular, it follows
immediately that U∞ = Spec(OY,y ) and OX,x ⊂ OY,y . Moreover, the lemma
implies that my ⊂ OX,x and the ring R = OX,x /my is a valuation ring of
RELATIVE RIEMANN-ZARISKI SPACES
7
k(y). To prove the first claim we pick up an element g ∈ my and apply the
lemma to g and h = 1. Since 1 ∈
/ my ⊃ gOX,x , we obtain that h ∈ OX,x . To
prove the second claim we pick up a non-zero element ge ∈ k(y) and apply the
lemma to any its lifting g ∈ B∞ and h = 1. It follows that either ge or ge−1 is
contained in R, thus proving the second claim. Define φ : S = Spec(R) → X
as the composition of the closed immersion S → Spec(OX,x ) and the natural
morphism Spec(OX,x ) → X.
For any i, Ui = Spec(Oxi )×Xi Y is a closed subscheme of Spec(Oxi )×X Y ,
hence the closure of the image of y in Spec(Oxi ) ×X Y coincides with the
closure of y in Ui . It follows that the closure of the image of y in Spec(Ox )×X
Y coincides with the closure of y in U∞ = ∩Ui . Since y is closed in U∞ ,
we obtain that the morphism y → Spec(Ox ) ×X Y is a closed immersion.
Hence the morphism y → S ×X Y is a closed immersion too, and we obtain
that y = (y, R, φ) is the required element of ValY (X).
¤
In the sequel, by a semi-valuation ring we mean a ring O with a valuation
| | such that for any pair g, h ∈ O with |g| ≤ |h| 6= 0, one has h|g. Notice that
the prime ideal m = Ker(| |) is (O \ m)-divisible, so O embeds into A = Om ,
mA = m, and O is composed from the local ring A and the valuation ring
R = O/m, i.e. O is the preimage of R ⊂ A/m in A. We say that A is a
quotient ring of O. Conversely, any ring composed from a local ring and a
valuation ring is easily seen to be a semi-valuation ring. In the situation of
the above proposition, Ox is a semi-valuation ring composed from Oy and R.
Semi-valuation rings play the same role in the theory of relative RZ spaces
as valuation rings play in the theory of usual RZ spaces.
2.2. Applications. A preliminary description of relative Riemann-Zariski
spaces, obtained in the previous section, suffices for some applications. In
this section we fix a qcqs scheme S with a schematically dense quasi-compact
subset U closed with respect to generalizations. An S-scheme X is called
U -admissible if the preimage of U in X is schematically dense. By a U etale covering we mean a separated morphism h : S 0 → S, such that h is
etale over U , S 0 is U -admissible and for any valuation ring R, any morphism
Spec(R) → S lifts to S 0 . Note that in [BLR] one considers a more restrictive
class of U -etale coverings S 0 → S, which split to a composition of a surjective
flat U -etale morphism and a U -modification. However, it follows from the
flattening theorem of Raynaud-Gruson, see [RG], 5.2.2, that the coverings
of the latter type form a cofinal subclass.
Let us consider the following situation: (A, m) is a local ring, U =
Spec(A), K = A/m, R ⊂ K is such that Q(R) = K, S = Spec(O) and
O is composed from A and R, i.e. O is the preimage of R in A. Notice that
S is covered by U and T = Spec(R), and T ∩ U = η = Spec(K).
Lemma 2.2.1. Suppose that we are given an η-admissible T -scheme XT , a
U -scheme XU and an isomorphism XT ×T ηf
→XU ×U η, then:
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M. TEMKIN
(i) there exists a unique way to glue XT and XU to a U -admissible Sscheme X;
(ii) X is of finite type (resp. finite presentation) iff XU and XT are so.
Proof. Let us prove first that an affine U -admissible S-scheme X = Spec(C)
is uniquely determined by XU = Spec(CU ) and XT = Spec(CT ). By admissibility, C ⊂ C ⊗O A = CU and C/mC = CT ⊂ CU ⊗O K, hence C is
the preimage of CT in CU . Moreover, if Y is another affine U -admissible
S-scheme, then the same argument shows that any morphism Y → X is
uniquely defined by its restrictions YU → XU and YT → XT , i.e.
MorS (X, Y )f
→MorU (XU , YU ) ×Morη (Xη ,Yη ) MorT (XT , YT ).
The remaining part is very standard, so we only outline it. Using the above
bijection, one proves that it holds for any pair of qcqs U -admissible Sschemes X and Y . Then (i) is easily deduced, because the required scheme
X is glued from affine pieces in an essentially unique way. The second part
of the lemma is exactly the Step 3 of proposition 2.4.4 from [Tem2].
¤
Now we prove a stable modification theorem, which generalizes its analog
from [Tem2]. We refer to loc.cit., 1.6 and 1.7, for terminology.
Theorem 2.2.2. Let C be an S-multipointed curve with semi-stable U fibers, then there exists a U -etale covering S 0 → S, such that the curve
C ×S S 0 admits a stable U -modification Cst .
Proof. Step 1. Reduction to the case of S = Spec(O) and U = Spec(A),
where O is a semi-valuation ring and A is its quotient ring. Since C is
semi-stable over an open subscheme of S, we can enlarge U to an open
schematically dense qcqs subscheme. Then by noetherian approximation,
we can assume that S is of finite Z-type. Set S = RZU (S), it is a qcqs
topological space by [Tem2], B.2.3. For any point x = (y, R, φ) ∈ S, set
Sx = Spec(OS,x ), Ux = Spec(OY,y ) and Cx = C×S Sx . If the case mentioned
in the formulation of the step is known, then there exists a Ux -etale covering
Sx0 → Sx , such that the Sx0 -multipointed curve Cx ×Sx Sx0 admits a stable
Ux -modification.
Let xi be the centers of x on U -modifications Si of S, then ∪OSi ,xi = OS,x .
By approximation, there exists i = i(x) and a U -etale morphism hx : Si0 →
Si , such that xi lies in its image and C ×S Si0 admits a stable U -modification.
Furthermore, by théorème de platification par éclatements, see [RG], 5.2.2,
we can assume that hx is flat: replace Si by some its U -modification and
replace Si0 by its strict transform with respect to that modification. Then
the preimage of hx (Si0 ) in S is an open subset. Since S is quasi-compact,
we need only finitely many points x1 , . . . , xn to cover the whole S. Then
0
tnj=1 Si(x
is a required U -etale covering of S.
j)
Step 2. The theorem holds over a semi-valuation ring O. We assume that
O is composed from a local ring (A, m) and a valuation ring R ⊂ K = A/m,
S = Spec(O) and U = Spec(A). Set also T = Spec(R) and η = Spec(K). By
RELATIVE RIEMANN-ZARISKI SPACES
9
[Tem2], 1.7.5, the theorem is known in the case of a valuation ring, i.e. the
case when m = 0. Find a finite separable extension K 0 /K with a valuation
ring R0 lying over R, such that C ×S T 0 admits a stable modification, where
T 0 = Spec(R0 ). Let A0 be an A-etale local ring, such that A0 /mf
→K 0 , and
0
0
0
let O be the semi-valuation ring composed from A and R . Replacing O
by O0 , we can assume that already C ×S T admits a stable modification CT0 .
By lemma 2.2.1, we can glue CU and CT0 to a U -modification C 0 → C. It
remains to show, that C 0 is an S-curve, then it is clear that C 0 is a stable U modification of C. By 2.2.1, C 0 is of finite presentation over S. Notice, that
S has no non-trivial U -modifications, because T is the only modification of
itself. Applying the flattening theorem, see [RG], 5.2.2, we obtain that C 0
is S-flat.
¤
A scheme version of the reduced fiber theorem of Bosch-LütkebohmertRaynaud, see [BLR], 2.1’, may be proven absolutely analogously.
Theorem 2.2.3. Let X → S be a schematically dominant finitely presented
morphism, whose U -fibers are geometrically reduced. Then there exists a
U -etale covering S 0 → S, and a finite U -modification X 0 → X ×S S 0 , such
that X 0 is flat, finitely presented and has reduced geometric fibers over S 0 .
Proof. If S is the spectrum of a valuation ring and U is its generic point,
then the theorem follows from [Tem2], 1.8.4 (actually it was the content of
Steps 1–4 of the loc.cit.). Acting as in Step 2 of the previous proof, we
deduce the case when S is the spectrum of a semi-valuation ring and U is
the corresponding local scheme. Then it remains to repeat the argument of
Step 1.
¤
3. Nagata compactification
3.1. Connection to adic spaces. Let A be a ring and B be an A-algebra.
Following [Hub1], we define a valuation on B as a commutative ordered
group Γ with a multiplicative map | | : B → Γ ∪ {0}, which satisfies the
strong triangle inequality and sends 1 to 1. Recall, that if B is a field,
then B ◦ = {x ∈ B| |x| ≤ 1} is a valuation ring, and the correspondence
B 7→ B ◦ gives a one-to-one correspondence between the equivalence classes
of valuations on B and valuation rings B ◦ with fraction field B. In general,
a valuation is defined up to an equivalence by its kernel p, which is a prime
ideal, and by the induced valuation on the residue field Q(B/p). By slight
abuse of language, we will often identify equivalent valuations. In loc.cit.,
the set Spv(B) of all equivalence classes of valuations of B is provided with
the weakest topology such that the sets of the form {| | ∈ Spv(B)| |a| ≤
|b| 6= 0} are open, where a, b ∈ B. Huber gives an elegant proof that the
resulting topological space is quasi-compact, see [Hub1], 2.2. Furthermore,
he considers the quasi-compact subspace Spa(B, A) ⊂ Spv(B) consisting of
valuations of B such that |A| ≤ 1, see the definition on p. 467 (here A and
B are considered as topological rings with discrete topology; note also that
10
M. TEMKIN
Huber actually considers the case when A is an integrally closed subring
of B, but it does not really restrict the generality, because replacing A by
the integral closure of its image in B does not change the space Spa(B, A)).
Actually, the topological space Spa(B, A) has a much finer structure of an
adic space, but we do not need it.
Let us generalize the above paragraph to schemes. If Y is a scheme,
then by a valuation on Y we mean a pair y = (y, R), where y ∈ Y is
a point and R is a valuation ring in k(y). One can define y by giving
a valuation | |y : Oy → Γy , whose kernel is my . By Oy we denote the
subring of Oy given by the condition | |y ≤ 1, it is the preimage of R
in Oy . Furthermore, if f : Y → X is a morphism of schemes, then by
an X-valuation on Y me mean a valuation y = (y, R) provided with a
morphism φ : S = Spec(R) → X which is compatible with the natural
morphism η = Spec(k(y)) → X. Recall that in the valuative criteria of
properness/separatedness, one considers commutative diagrams of the form
η
i
/Y
²
φ
²
S
(1)
f
/X
where S = Spec(R) is the spectrum of a valuation ring and η = Spec(K)
is its generic point, and studies liftings of S to Y . We will prove in 3.1.6
that it suffices to consider the only case when k(i(η))f
→K. The diagrams
of the latter particular type are exactly the diagrams which correspond to
X-valuations of Y . Note also, that a valuation y = (y, R, φ) gives rise to
the following finer diagram
η
/ Yy
/Y
²
²
/T
²
/X
S
(2)
where Yy = Spec(Oy ) and T = Spec(Oy ).
e = Spa(Y, X) denote the set of all equivalence classes of X-valuations
Let X
of Y . Given a morphism f 0 : Y 0 → X 0 and a morphism g : f 0 → f consisting
of a compatible pair of morphisms gY : Y 0 → Y and gX : X 0 → X, there is a
natural map Spa(f ) : Spa(Y 0 , X 0 ) → X which to a point (y 0 , R0 , φ0 ) associates
a point (y, R, φ), where y = gY (y 0 ),R = R0 ∩ k(y) and φ is defined as follows.
The morphism gX ◦ φ0 : Spec(R0 ) → X factors through Spec(OX,x ), where x
is the image of the closed point of the source, hence we obtain a homomorphism α : OX,x → R0 . Since the morphism Spec(k(y 0 )) → X factors uniquely
through Spec(k(y)), the image of α lies R = R0 ∩ k(y). So, gX ◦ φ0 factors
uniquely through a morphism φ : Spec(R) → X. We proved that Spa(Y, X)
depends functorially on f . Also, it is easy to check that Spa respects fibred
products, i.e. Spa(Y 0 ×Y Y 0 , X 0 ×X X 00 )f
→Spa(Y 0 , X 0 ) ×X Spa(Y 00 , X 00 ).
RELATIVE RIEMANN-ZARISKI SPACES
11
If Y 0 is a subscheme of Y and gX is separated, then φ is injective. Indeed,
e has a non-empty preimage in X
e 0 = Spa(Y 0 , X 0 ),
if a point y = (y, R, φ) ∈ X
then y ∈ Y 0 and any preimage of y is given by a lifting of φ : Spec(R) → X
to X 0 , which is unique by the valuative criterion of separatedness. If Y 0 is
e 0 is
open and affine, and X 0 is affine and of finite X-type, then we say that X
e If X1 , X2 ⊂ X are affine subsets, then X12 = X1 ∩ X2 is
an affine subset of X.
a finite union of affine subsets (use that X12 f
→X1 ×X X2 ), and X12 is affine if
e
X and Y are separated. We provide X with the weakest topology in which
all affine sets are open; obviously, affine sets form a basis of this topology.
The two following statements follows easily: if Y 0 ,→ Y is an open immersion
and X 0 → X is separated and of finite type, then Spa(Y 0 , X 0 ) ,→ Spa(Y, X)
is an open embedding, and the maps Spa(g) are continuous.
Lemma 3.1.1. If X = Spec(A) and Y = Spec(B) are affine, then the
canonical bijection X = Spa(Y, X) → Spa(B, A) is a homeomorphism.
Proof. It follows from the definitions of the topologies, that the map is
e 0 = Spa(Spec(C), Spec(A0 )) be an affine subset, where
continuous. Let X
Spec(C) is an open subscheme of Spec(B) and A0 is a finitely generated
e0 ⊆ X
e is a neighborhood of each point
A-algebra. It suffices to prove that X
0
z ∈ Spa(B, A) lying in it. Replacing A by its image in C, we can assume
that it is an A-subalgebra of C generated by h1 , . . . , hn ∈ C. Notice that if
{Ui } is an open covering of Spec(C), then the sets Spa(Ui , Spec(A0 )) cover
X. Therefore, shrinking Spec(C), we may assume that C = Bb for an
e 0 consists
element b ∈ B. Then hi = bi /bm , where bi ∈ B, m ∈ N, and X
e 0 is open in
of all valuations of B with |bi | ≤ |bm | =
6 0 for any i. Thus, X
Spa(B, A), and we are done.
¤
Corollary 3.1.2. If X and Y are qcqs schemes, then the space Spa(Y, X)
is qcqs.
Let B be a ring and | | : B → Γ ∪ {0} be a valuation. By y ∈ Spec(B) we
denote the point corresponding to the kernel of | |. We say that a convex
subgroup H ⊂ Γ bounds B, if for any element b ∈ B, there exists an element
h ∈ H with |b| < h. For any such subgroup, we can define a valuation
| |H : B → H by the rule |x|H = |x| if |x| ∈ H and |x|H = 0 otherwise.
Recall that | |H is called a primary specialization of | |, see [Hub1], 2.3. It
follows from H being convex, that the order on Γ induces an order on the
group Γ/H, and | | induces a valuation | |/H : B → Γ/H. Notice that
|B|/H ≤ 1 because H bounds B. Therefore the image of B in k(y) lies in
a valuation ring R0 corresponding | |/H, and the morphism y → Spec(B)
extends to a morphism Spec(R0 ) → Spec(B).
Let, more generally, (y, R) be a valuation on a scheme Y . Notice that
any ring R ⊆ R0 ⊆ k(y) is a valuation ring. Assume that the morphism
Spec(k(y)) → Y extends to a morphism S 0 = Spec(R0 ) → Y and let y be
the image of the closed point s ∈ S 0 . The image of R in k(s) is a valuation
12
M. TEMKIN
ring whose intersection with k(y) will be denoted R. The pair (y, R) is a
valuation on Y , which is called a primary specialization of (y, R). In general,
by a primary specialization of an X-valuation y = (y, R, φ), we mean an Xvaluation y = (y, R, φ), such that (y, R) is a primary specialization of (y, R)
and the image of φ in X is contained in the image of φ in X. Primary
specialization is a particular case of a specialization relation in Spa(Y, X),
i.e. any neighborhood of y contains y. A valuation (y, R, φ) is called minimal
if it has no non-trivial primary specializations. In the sequel we assume that
f : Y → X is a separated morphism of qcqs schemes, unless the contrary is
stated explicitly.
Lemma 3.1.3. Keep the above notations and let y = (y, R, φ) be an Xvaluation on Y , then:
i) the set of primary specializations of y is non-empty, totally ordered and
contains a minimal element;
(ii) y is minimal if and only if the morphism h : Spec(k(y)) → Y ×X
Spec(R) is a closed immersion.
Proof. Let X 0 be a separated neighborhood of the image of φ and Y 0 be
its preimage. Replacing X and Y by X 0 and Y 0 , we can achieve that Y
is separated. It follows from the valuative criterion of separatedness, that
for any valuation ring R ⊆ R0 ⊆ k(y), there is at most one morphism
Spec(R0 ) → Y compatible with y → Y . Since the set of valuation rings
R0 as above is totally ordered with respect to inclusion, we obtain a total
order on the set {(yi , Ri }i∈I of primary specializations of y. Provide I with
the corresponding order, then I possesses a maximal element corresponding
to the trivial specialization (y, R). Choose elements i ≤ j in I, and let
R ⊆ Ri0 ⊆ Rj0 ⊆ k(y) be the corresponding valuation rings. Then the
morphism Spec(Rj0 ) → Y is a composition of Spec(Rj0 ) → Spec(Ri0 ) and
Spec(Ri0 ) → Y , in particular yi specializes yj . By quasi-compactness of Y ,
there exists a point y ∈ Y which specializes all yi ’s. Each (yi , Ri ) induces a
valuation | |i on Oy , and | |i is a primary specialization of | |j for i ≤ j. Then
| |i = (| |0 )Hi , for some convex groups Hi bounding Oy . Hence H = ∩Hi
bounds Oy and (| |0 )H defines a primary specialization of y, which specializes
all valuations (yi , Ri ).
Let us prove (ii). Notice that y is not minimal iff there exist open affines
X 0 ⊂ X and Y 0 ⊂ Y , such that f (Y 0 ) ⊂ X 0 , y ∈ Y 0 and y is not minimal as
an X 0 -valuation on Y 0 . Analogously, h is not a closed immersion iff for some
choice of X 0 and Y 0 as above, the morphism Spec(k(y)) → Y 0 ×X 0 Spec(R) is
not a closed immersion. Thus, we may assume that our schemes are affine,
say X = Spec(A) and Y = Spec(B). The morphism from the lemma is
not a closed immersion iff the homomorphism B ⊗A R → k(y) is not onto.
The latter happens iff B is bounded by an element r ∈ |k(y)|, where we
provide k(y) with a valuation corresponding to R. Finally, such r exists iff
the valuation is not minimal.
¤
RELATIVE RIEMANN-ZARISKI SPACES
13
It follows from the second part of the lemma that the set X = ValY (X)
e consisting of minimal
defined in section 2.1 coincides with the subset of X
X-valuations. Note that in affine situation, such subsets were considered by
Huber, see [Hub1], 2.6 and 2.7. We provide X with the induced topology.
The following lemma follows easily from the valuative criterion of properness,
so we omit the proof.
Lemma 3.1.4. (i) If X 0 is a Y -modification of X, then there are natural
homeomorphisms Spa(Y, X 0 )f
→Spa(Y, X) and ValY (X 0 )f
→ValY (X).
0
(ii) If X is an open subscheme of X, then its preimage in ValY (X) (resp.
Spa(Y, X)) is canonically homeomorphic to ValY 0 (X 0 ) (resp. Spa(Y 0 , X 0 )),
where Y 0 = X 0 ×X Y .
Caution: if Y 0 is an open qcqs subscheme of Y and the morphism Y 0 → X
factors through an X-separated scheme X 0 of finite type, then Spa(Y 0 , X 0 )
e However, it may (and
maps homeomorphically onto an open subspace of X.
e is not contained in
usually does) happen that the image of ValY 0 (X 0 ) in X
X. This problem originates from the fact that a valuation, which has no
proper primary specializations on Y 0 , may have such a specialization on Y .
This observation enforces us to introduce a notion of open subdomain of X
in section 3.3.
Proposition 3.1.5. Assume that f : Y → X is a separated morphism of
qcqs schemes. Then the spaces ValY (X) and RZY (X) are quasi-compact.
e i ⊂ X,
e such
Proof. Let {Xi }i∈I be an open covering of X. Find open sets X
e i ∩ X. Since any point of X
e has a specialization in X, {X
e i }i∈I
that Xi = X
e By quasi-compactness of X,
e we can find a subcovering
is a covering of X.
e
{Xi }i∈J with finite J, and then {Xi }i∈J is a finite covering of X. Thus, X is
quasi-compact.
Let us prove that for any Y -modification f : X 0 → X, the map φ : X → X 0
is continuous. It suffices to show that φ−1 (U ) is open, where U ⊂ X 0 is
e is covered by open
an open affine subscheme. The preimage of U in X
0
e
sets Xi = Spa(Yi , X ), where {Yi } is an open affine covering of f −1 (X 0 ).
e i ∩X, as required.
Therefore, the preimage of U in X is covered by open sets X
It follows, that the map ψ : X → RZY (X) is continuous. Since X is quasicompact and ψ is surjective, the space RZY (X) is quasi-compact.
¤
In the sequel, we will need to strengthen the classical valuative criterions
of separatedness, see [EGA II], 7.2.3, and properness, see loc.cit., 7.3.8. Our
aim is to show, that it suffices to consider valuative diagrams of specific
types. We say that a morphism is compatible with a commutative diagram,
if the diagram remains commutative after adjoining this morphism. Until
the end of the section, f is not assumed to be separated.
Lemma 3.1.6. Keep the notations of diagram (1) and set K 0 = k(i(η))
and R0 = R ∩ K. Then diagram (1) completes uniquely to a commutative
14
M. TEMKIN
diagram
Spec(K)
/ Spec(K 0 )
/Y
²
²
/ Spec(R0 )
²
/X
Spec(R)
and any morphism h : Spec(R) → Y compatible with the above diagram is
induced by a morphism h0 : Spec(R0 ) → Y compatible with the diagram and
determines h0 uniquely.
Proof. The morphism Spec(K) → Y obviously splits through Spec(K 0 ). The
morphism Spec(R) → X factors through Spec(OX,x ), where x is the image
of the closed point of Spec(R). The image of Ox in R ⊂ K is contained in
K 0 , hence the morphism Spec(R) → X factors through Spec(R0 ). By the
same reasoning, a morphism h : Spec(R) → Y compatible with the diagram
factors through h0 : Spec(R0 ) → Y , and they both are determined uniquely
by the image of the closed point in Y .
¤
Lemma 3.1.7. Keep the notations of diagram (1) and assume that R ⊂
R0 ⊂ K is such that the morphism Spec(R0 ) → X admits a lifting g :
e be the
Spec(R0 ) → Y compatible with the morphism Spec(K) → Y . Let K
0
e
e
residue field of R and R the image of R in K.
/Y
w;
w
g www
ww
w
ww
Spec(K)
e
Spec(K)
²
e
Spec(R)
²
/ Spec(R0 )
²
/ Spec(R)
f
²
/X
e → Y compatible with the above diagram
Then any morphism e
h : Spec(R)
is induced from a morphism h : Spec(R) → Y compatible with the diagram
and determines h uniquely.
e → Y compatible with the diagram.
Proof. Consider a morphism e
h : Spec(R)
It suffices to show that it factors through Spec(R), the uniqueness is again
e and x = f (y), then
trivial. Let y be the image of the closed point of Spec(R)
e
h induces a homomorphism Oy → R. Since y specializes the image of the
closed point of Spec(R0 ), we have also a homomorphism Oy → R0 . Finally,
e = R0 /mR0 lies in R.
e
the compatibility implies that the image of Oy in K
e
Therefore, the image of Oy in R0 lies in R, and the homomorphism Oy → R
factors through R. It gives the desired morphism h : Spec(R) → Y .
¤
Proposition 3.1.8. Let h : Z → Y and f : Y → X be morphisms of qcqs
schemes, and ψ : Spa(Z, Y ) → Spa(Z, X) and ψ : ValZ (Y ) → ValZ (X) be
the natural maps. Assume that h is dominant and g = f ◦ h is separated.
RELATIVE RIEMANN-ZARISKI SPACES
15
(i) The morphism f is separated if and only if ψ is injective if and only
if ψ is injective.
(ii) Let us assume in addition that f is of finite type, then f is proper if
and only if ψ is bijective if and only if ψ is bijective.
Proof. The direct implications follows from the standard valuative criterions.
Suppose that f is separated and not proper (resp. not separated), then by
the standard valuative criterion and lemma 3.1.6, there exists an element y =
(y, Ry , φy ) ∈ Spa(Y, X), such that the number of liftings of the morphism
φy : Spec(Ry ) → X to Y is zero (resp. at least two). Let x denote the center
of y on X.
By [EGA I], 6.6.5, there exists a point z ∈ Z, such that y specializes
h(z). Let R0 be any valuation ring of k(z), which dominates the image of
Oy . It gives rise to an element (z, R0 , φ0 ) ∈ Spa(Z, Y ) centered on y. Choose
e of the residue field K
e of R0 , such that R
e dominates the
a valuation ring R
e and define a valuation ring R of k(z) as the
valuation ring Ry of k(y) ⊂ K,
0
e
composition of R and R. The compatible homomorphisms Ox → Oy → R0
e induce a homomorphism Ox → R, and we obtain the
and Ox → Ry → R
following commutative diagrams.
e
Spec(K)
²
e
Spec(R)
Spec(k(z))
/Z
²
0
/ Spec(R0 ) φ
²
/Y
²
/ Spec(R) φx
²
/X
e
Spec(K)
/ Spec(k(y))
/Y
²
²
/ Spec(Ry )
²
/X
e
Spec(R)
By lemma 3.1.6, there is a one-to-one correspondence between morphisms
e → Y compatible with the right diagram. By
Spec(Ry ) → Y and Spec(R)
lemma 3.1.7, the latter morphisms are in one-to-one correspondence with
the morphisms φ : Spec(R) → Y compatible with the left diagram. Notice
that z = (z, R, φx ) is an element in Spa(Z, X), and any morphism φ as above
gives a preimage of z in Spa(Z, Y ). We obtain that in the case (i), z has at
least two preimages, and ψ is not injective.
To prove (ii), we notice that the morphism φ0 is uniquely determined
by morphisms Spec(k(z)) → Y and Spec(R0 ) → X, because Y → X is
separated by our assumption. Therefore, any preimage of z in Spa(Z, Y )
comes from a morphism φ : Spec(R) → Y compatible with the left diagram.
We obtain that z has no preimages under ψ, in particular ψ is not bijective.
It remains to relate the fibers of ψ and ψ. Consider the minimal primary
specialization z0 ∈ ValZ (X) of z. Then lemma 3.1.7 implies that the sets
−1
ψ (z) and ψ −1 (z0 ) are naturally bijective.
¤
16
M. TEMKIN
3.2. Y -blow ups of X. In this section we assume that f is affine. Under
this assumption, there exists a large family of projective Y -modifications
of X having good functorial properties. A Y -modification gi : Xi → X is
called a Y -blow up of X if there exists a gi -ample OXi -module L provided
with a homomorphism ε : OXi → L, such that fi∗ (ε) : OY f
→fi∗ (L). It will
be more convenient to say X-ample instead of gi -ample in the sequel. We
call ε a Y -trivialization of L.
Lemma 3.2.1. The following statements hold.
(i) Suppose that Xj → Xi and Xi → X are Y -modifications such that Xj
is a Y -blow up of X, then Xj is a Y -blow up of Xi .
(ii) The family of Y -blow ups of X is filtered.
(iii) The composition of Y -blow ups gij : Xj → Xi and gi : Xi → X is a
Y -blow up.
Proof. The first statement is obvious, because any X-ample OXj -module L
is Xi -ample, and the notion of Y -trivialization of L depends only on the
morphism Y → Xj .
(ii) Let Xi , Xj be two Y -blow ups of X. Find X-ample sheaves Li , Lj with
Y -trivializations εi , εj . Then the X-proper scheme Xij = Xi ×X Xj possesses
an X-ample sheaf L = p∗i (L1 ) ⊗ p∗j (L2 ), where pi , pj are the projections.
The natural isomorphism OXij f
→OXij ⊗ OXij followed by fi∗ (εi ) ⊗ fj∗ (εj ) :
OXij ⊗ OXij → L provides a Y -trivialization of L. Consider the schemetheoretic image X 0 of Y in Xij , and let L0 and ε0 be the pull backs of L and
ε. Then (X 0 , L0 , ε0 ) is a Y -blow up of X which dominates Xi and Xj .
(iii) Choose an X-ample OXi -sheaf Li and an Xi -ample OXj -sheaf Lj with
∗ (L⊗n ) is
Y -trivializations εi nd εj . By [EGA II], 4.6.13 (ii), the sheaf Lj ⊗ gij
i
X-ample for sufficiently large n. It remains to notice that the composition
⊗n
∗ (ε⊗n ) is a Y -trivialization.
of OXj f
→OXj ⊗ OX
with εj ⊗ gij
¤
i
j
We will need an explicit description of Y -blow ups. Let F = f∗ (OY ), and
assume that E ⊂ F is a finitely generated OX -submodule containing the
image of OX , the canonical homomorphism of OY -modules µY : f ∗ (E) →
OY is surjective. By [EGA II], 4.2.3, we obtain a morphism h : Y →
PX (E) = P , and the unit section homomorphism e : OX → E induces a
homomorphism εP : OP → OP (1). Define XE to be the scheme-theoretic
image of h and define L and ε to be the pull backs of OP (1) and εP to XE .
Obviously, XE is a Y -modification of X and L is X-ample. It remains to
notice that ε is a Y -trivialization, because µY ◦ f ∗ (e) : OY → OY is the
identity.
Lemma 3.2.2. Any Y -blow up of X is isomorphic to some XE .
Proof. Let gi : Xi → X be a Y -blow up. Find an X-ample OXi -module L
and its Y -trivialization ε. Replacing L by L⊗n and changing ε accordingly,
we may assume that L is very ample, i.e. there exists a finitely generated
quasi-coherent OX -module E and a closed immersion h : Xi → P = PX (E)
RELATIVE RIEMANN-ZARISKI SPACES
17
∗
such that φ : Lf
→h (OP (1)). The morphism h with isomorphism φ correspond to an epimorphism H : gi∗ (E) → L by [EGA II], 4.2.3. Set E =
E ⊕ OX , then the epimorphism H = H + ε : gi∗ (E) → L defines a morphism h : Xi → P = PX (E), which is a closed immersion because already
E ⊂ E induces a closed immersion h. The second summand embedding
e : OX → E defines a homomorphism εP : OP → OP (1), whose pull back
to Xi is equal to ε; in particular, εP is a Y -trivialization of OP (1). The
morphism λ = h ◦ fi : Y → P corresponds to a morphism E ⊗ F → F. Let
E 0 be the image of E ⊗ OX in F, then λ factors through PX (E 0 ), and one
can easily seethat Xi is isomorphic to the scheme-theoretic image of Y in
PX (E 0 ), and ε is induced by the section e : OX → E 0 .
¤
Corollary 3.2.3. Assume that X 0 is an open subscheme of X and Y 0 =
f −1 (X 0 ). Then any Y 0 -blow up Xi0 → X 0 extends to a Y -blow up Xi → X.
Proof. Let f 0 : Y 0 → X 0 be the restriction of f , so F 0 = f∗0 (OY 0 ) is the restriction of F = OY on X 0 . By the lemma, a Y 0 -blow up of X 0 is determined
by a finitely generated OX 0 -submodule E 0 ⊂ F 0 containing the image of OX 0 .
By [EGA I], 6.9.7, one can extend E 0 to a finitely generated OX -submodule
E ⊂ F. Replacing E by E + OX , if necessary, we can achieve that E contains
the image of OX . Now, E defines a required extension of the blow up.
¤
The extension property of the corollary dictated our choice of the class of
Y -blow ups of X. The following lemma is an analog of [BL], 4.4.
Lemma 3.2.4. Given a quasi-compact open subset U ⊂ X = ValY (X), there
exists a Y -modification X 0 → X and an open subscheme U ⊂ X 0 , such that
U is the preimage of U in X.
Proof. If X1 , . . . , Xn form a finite open affine covering of X and Yi =
f −1 (Xi ), then Xi = ValYi (Xi ) form a covering of X. It suffices to solve
our problem for each Xi separately, because any Yi -blow up of Xi extends
to a Y -blow up of X, and Y -blow ups of X form a filtered family. Thus, we
can assume that X = Spec(A), and then Y = Spec(B). We can furthermore
assume that U = X ∩ Spa(Bb , A[a/b]), where a, b ∈ B, because as we saw in
the proof of lemma 3.1.1, U is covered by finitely many subsets of this form.
Now, the morphism Y → Proj(A[x0 , x1 , x2 ]), defined by (1, a, b), determines
a required Y -blow up X 0 → X with U given by the condition x2 6= 0.
¤
Corollary 3.2.5. The map ψ : ValY (X) → RZY (X) is a homeomorphism.
Proof. We already know that ψ is surjective and continuous. From other
side, the lemma implies that ψ is injective and open. Indeed, for any pair
of different points of X, there exists an open quasi-compact set U which
distinguishes the points, and the lemma implies that they are distinguished
already in some Y -modification X 0 of X. Hence ψ is injective, and its
openness follows obviously.
¤
18
M. TEMKIN
We use the corollary to identify X with RZY (X), in particular, X is provided with a sheaf OX of regular functions. As another corollary, we obtain
the following version of Chow lemma.
Corollary 3.2.6. Any Y -modification X → X is dominated by a Y -blow
up of X.
Proof. Let U 1 , . . . , U n be an affine covering of X. By Yi and Ui we denote
the preimages of U i in Y and X, respectively. By the lemma, we can find
a Y -blow up X 0 → X and a covering {Ui0 } of X 0 , whose preimage in X
coincides with {Ui }. Consider the scheme X 00 , which is the scheme-theoretic
image of Y in X ×X X 0 . The preimages of U i and Ui0 in X 00 coincide, because
their preimages in X coincide. Let Ui00 denote the corresponding subscheme
of X 00 . For any 1 ≤ i ≤ n, the morphism hi : Ui00 → Ui0 is affine, because the
morphism U i → X is affine (U i is affine and X-separated). From other side,
hi is proper because it is a base change of the Y -modification h : X 00 → X 0 .
Thus, hi is finite, and therefore h is finite.
We claim that h is a Y -blow up (this claim is an analog of [BL], 4.5).
Indeed, OX 00 is very ample relatively to h, because h is affine, and the
identity homomorphism gives its Y -trivialization. By lemma 3.2.1 (iii), X 00
is a Y -blow up of X, and obviously X 00 dominates X.
¤
3.3. Affinoid domains. Let f 0 : Y 0 → X 0 be a separated morphism of
e 0 = Spa(Y 0 , X 0 ). Suppose that we are
qcqs schemes, X0 = ValY 0 (X 0 ) and X
given a compatible pair of morphisms gY : Y 0 → Y , gX : X 0 → X, then
e 0 → X,
e which takes (y 0 , R0 , φ0 ) to
there is a natural continuous map ge : X
0
0
0
(y = gY (y ), R = R ∩ k(y ), φ).
Proposition 3.3.1. Suppose that gY is an open immersion and gX is separated. Then ge(X0 ) ⊂ X if and only if the natural morphism h : Y 0 → Y ×X X 0
is a closed immersion.
Proof. Suppose that h is a closed immersion. Let (y 0 , R0 , φ) ∈ X0 be a point
e Consider the schemes η 0 = Spec(k(y 0 )) and
and (y, R, φ) its image in X.
0
0
S = Spec(R ), the natural morphism η 0 → Y 0 ×X 0 S 0 is a closed immersion.
Therefore the composition morphism η 0 → Y 0 ×X 0 S 0 → (Y ×X X 0 ) ×X 0
S0f
→Y ×X S 0 induced by h is a closed immersion. It follows that the morphism Spec(k(y)) → Y ×X Spec(R) is a closed immersion too, as required.
Suppose now that h is not a closed immersion. Notice, that h is a locally
closed immersion, because gY is an open immersion. It follows that we can
find a Y ×X X 0 -valuation y0 = (y 0 , R0 , φ) of Y 0 , such that the morphism
φ : Spec(R0 ) → Y ×X X 0 cannot be lifted to a morphism Spec(R0 ) → Y 0 .
Replacing y0 by its minimal primary specialization, we achieve that y0 is
minimal and R0 ( k(y 0 ). Now, it is clear that y0 defines a minimal X 0 e is not minimal, because the morphism
valuation on Y 0 , but its image in X
0
Spec(R ) → X lifts to a morphism Spec(R0 ) → Y .
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RELATIVE RIEMANN-ZARISKI SPACES
19
Let us assume that gY is an open immersion and gX is separated and of
finite type. We saw that, if h is a closed immersion, then via ge X0 is naturally
identified with a quasi-compact open subset of X. In this case we say that
the map X0 → X is an open immersion and X0 is an open subdomain of X.
If, in addition, X 0 and Y 0 may be chosen to be affine, then we say that X0
is an affinoid subdomain of X. Note also, that the situation described in the
proposition appears in Deligne’s proof of Nagata compactification theorem
under the name of quasi-domination. (Recall that by a quasi-domination of
Y over X 0 one means an open subscheme Y 0 ⊂ Y and a morphism Y 0 → X 0 ,
such that the morphism Y 0 → Y ×X X 0 is a closed immersion, see [Con],
section 2.) The notion of quasi-domination plays a central role in Deligne’s
proof.
Lemma 3.3.2. Given two open subdomains Xi = ValYi (Xi ) of X, their
intersection X12 is an open subdomain of the form ValY1 ∩Y2 (X1 ×X X2 ). In
particular, if X is separated and Xi are affinoid, then X12 is affinoid.
Remark 3.3.3. Our definition of RZ spaces is a straightforward generalization of the classical one. It is also possible to define RZ spaces directly
as follows: an affinoid space is a topological space X = ValB (A)f
→RZB (A)
provided with two sheaves of rings OX ⊂ MX , general spaces are pasted
e 0 = Spa(B 0 , A0 ) of
from affinoid ones. One can check that an affine subset X
e = Spa(B, A) induces an open immersion ValB 0 (A0 ) → X iff X
e 0 ∩ X is an
X
affinoid subset of X.
We do not develop the outlined approach to Riemann-Zariski spaces in
this paper, because it seems worth a separate study. The remark explains
a motivation for defining affinoid subsets. The following example illustrates
a difference between adic and Riemann-Zariski spaces. Let k be a field,
e X
e 0 are as above. Then X
e0
A = B = A0 = k[T ], B 0 = k[T, T −1 ] and X, X0 , X,
e in particular X
e0 → X
e is an open embedding,
is a rational subdomain in X,
0
see [Hub2]. From other side, X is not an affinoid domain in X. One can
show that the map X0 → X is bijective, but the sheaves MX and MX0 are
not isomorphic.
Theorem 3.3.4. The affinoid subdomains of X form a basis of its topology.
Proof. If f is affine, then for any Y -modification X 0 → X, the morphism
f 0 : Y → X 0 is affine. Therefore, for any open affine subscheme X 00 ⊂ X 0
and its preimage Y 0 = f −1 (X 00 ), ValY 0 (X 00 ) is an affinoid subset of X. Now,
the case of affine f follows from corollary 3.2.5. We see also, that in the
general case, it suffices to prove that X admits an affinoid covering, because
being affinoid subset is a transitive property.
If X is an open affine subscheme of X and Y = f −1 (X), then any affinoid
subset in ValY (X) is affinoid in X. Thus, we can replace X and Y by X and
Y and achieve that X is affine. By Thomason’s noetherian approximation
theorem, there exists an affine morphism h : Y → T , whose target is of
20
M. TEMKIN
finite Z-type. Consider the induced map λ : X → X0 = ValT (Z). We
claim that, if ValT 0 (Z 0 ) is an affinoid subset of X0 , then its preimage in X
is an affinoid subset ValY 0 (X 0 ), where Y 0 = T 0 ×T Y and X 0 = Z 0 ×Z X.
Indeed, X 0 is obviously affine and of finite X-type, and the open subscheme
Y 0 ⊂ Y is affine because the morphism Y → T is affine. Furthermore, since
T 0 → T ×Z Z 0 is a closed immersion, the composition morphism
Y 0 = Y ×T T 0 → Y ×T T ×Z Z 0 f
→Z 0 ×Z Y f
→Z 0 ×Z X ×X Y f
→X 0 ×X Y
is a closed immersion. It follows that it suffices to find an affinoid covering
of X0 , and we can assume that X = Spec(Z) and Y is of finite Z-type.
Since X is qcqs, it suffices to prove that a point y = (y, R, φ) ∈ X possesses
an affinoid neighborhood X0 . Fix a non-zero element ω ∈ R with Rω = k(y),
it exists because R is of finite height. Let g ∈ Oy be any lifting of ω. We
will construct X0 in two steps. As a starting point, we consider the closed
immersion α0 : Spec(k(y)) → Spec(R)×X Y . By diagram (2), there is a finer
morphism α1 : Yy → T ×X Y , where T = Spec(Oy ) and Yy = Spec(Oy ).
Step 1. The morphism α1 is a closed immersion. Notice that (Oy )g = Oy ,
hence the morphism Yy → T is an open immersion. The morphism α1 is a
locally closed immersion, because it is a composition of a closed immersion
Yy → Yy ×X Y with an open immersion Yy ×X Y → T ×X Y . Let Z be the
scheme-theoretic image of α1 , then it suffices to prove that Yy f
→Z. Observe
that Z is a T -scheme and Yy is a schematically dense open subscheme. It
follows that Z is glued from schemes Spec(Ci ) with Oy ⊆ Ci ⊆ Oy . Now,
it is clear that either Yy f
→Z, or Z contains a non-trivial specialization of y.
The latter is impossible, because α1 (y) = α0 (y) is closed.
Step 2. We can approximate Oy and Oy by finitely generated rings.
Consider pairs (Yβ , Tβ ), where Yβ = Spec(Bβ ) is a neighborhood of y, such
that g ∈ Bβ∗ , Tβ = Spec(Aβ ) and Aβ is a finitely generated subring of Bβ ,
such that the image of Aβ in k(y) lies in R, g ∈ Aβ and Bβ = (Aβ )g .
It is easily seen, that ∪β Bβ = B and ∪β Aβ = A. We say that γ ≥ β
if Yγ ⊂ Yβ and the image of Aβ in Bγ is in Aγ . For any such pair, we
obtain the following commutative diagram, whose middle and left squares
are Cartesian
/ Yγ
/Y
/ Yβ
Yy
²
T
²
/ Tγ
²
/ Tβ
²
/X
For any β, consider the morphism hβ : Yβ → Tβ ×X Y of finitely presented
Tβ -schemes. It is a locally closed immersion, because Yβ → Tβ is an open immersion. Since the middle and the left squares of the diagram are Cartesian,
we have hβ ×Tβ Tγ = hγ and hβ ×Tβ T = α1 . Notice that proj lim Tβ = T ,
hence we can apply approximation results of [EGA IV], ch. 8. Since α1 is
a closed immersion, hβ is a closed immersion for sufficiently large β. Then
ValYβ (Tβ ) is a required affinoid neighborhood of y.
¤
RELATIVE RIEMANN-ZARISKI SPACES
21
Corollary 3.3.5. The space X is qcqs.
Proof. Any open subdomain is quasi-compact by 3.1.5, hence it suffices to
prove that the intersection of open subdomains is quasi-compact. The latter
follows from 3.3.2.
¤
e → X, which to any X-valuation
Let us define a retraction map π = πX : X
e
y ∈ X associates its minimal primary specialization. Then for any pair of
schemes Y 0 , X 0 as in the beginning of the section, we can associate a map
e0 → X
e with π.
g : X0 → X by combining the natural map X0 ,→ X
e → X is continuous. In particular, g
Corollary 3.3.6. The retraction π : X
is continuous.
Proof. It is easy to see, that for any open subdomain X0 = ValY 0 (X 0 ) of X,
e It
the set π −1 (X0 ) coincides with Spa(Y 0 , X 0 ), in particular it is open in X.
remains to use the theorem.
¤
3.4. Decomposable morphisms.
Proof of theorem 1.1.3. We will prove a slightly more general result. Namely,
we will prove that any quasi-compact open subset X of X = ValY (X) is an
open subdomain of the form ValY (X), where the morphism Y → X is affine
and schematically dominant. Applying this statement to X, one deduces
the theorem, because then Y = Y and X is proper over X by the valuative
criterion 3.1.8. By theorem 3.3.4, X admits a finite affinoid covering. In
particular, X is covered by open subdomains Xi = ValYi (Xi ), 1 ≤ i ≤ n,
where Yi are Xi -affine. Using induction on n we reduce the problem to the
case of n = 2. Without loss of generality, we can assume that the morphisms
Yi → Xi are schematically dominant.
Set X12 = X1 ∩ X2 and Y12 = Y1 ∩ Y2 . In the sequel, we will act as
in Step 3 of the proof of 1.1.2, the main difference is that we will use Y∗ blow ups instead of affine morphisms. For reader’s convenience, we give a
commutative diagram containing the objects which were and will be defined.

Y1 o
? _ Y12 ²
² 
? _ X12 ²
/ X2
²
? _ Z12 
CC
CC
{{
CC
{{
{
CC
{
!
}{{
²
/ Z2
X1 o
²
Z1 o
²
X1 o
? _X0
1
/ Y2
X20 
²
/ X2
By lemma 3.2.4, we can replace Xi ’s by their Yi -blow ups, such that each
Xi contains an open subscheme Xi0 , whose preimage in Xi coincides with
22
M. TEMKIN
X12 . It may be impossible to glue Xi ’s along Xi0 ’s, but we know by lemma
3.1.4 (ii), that at least ValY12 (Xi0 )f
→X12 for i = 1, 2. Let T be the schemetheoretic image of Y12 in X10 ×X X20 . Then ValY12 (T )f
→X12 by lemma 3.3.2,
0
and, therefore, T is a Y12 -modification of Xi ’s by the valuative criterion.
By 3.2.6, we can find a Y12 -blow up T 0 → X10 , which dominates T . It still
may happen that T 0 is not a Y12 -blow up of X20 , but it is dominated by a
Y12 -blow up Z12 → X20 . Then Z12 → T 0 is a Y12 -blow up by lemma 3.2.1 (i),
hence Z12 → X10 is a Y12 -blow up by lemma 3.2.1 (iii). By lemma 3.2.3, we
can extend the Y12 -blow ups Z12 → Xi0 to Yi -blow ups Zi → Xi . Then, we
can glue the X-schemes Zi along the subschemes X-isomorphic to Z12 to a
single scheme X, and the affine morphisms Yi → Zi glue to a single affine
morphism Y → X. Notice that the space ValY (X) is obtained by gluing
ValYi (Zi ), i = 1, 2, along ValY12 (Z12 ), hence ValY (X)f
→X.
¤
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