IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER
GABBER)

arXiv:1609.09246v1 [] 29 Sep 2016

KAZUHIKO KURANO AND KAZUMA SHIMOMOTO
Abstract. In this paper, we give a detailed proof to a result of Gabber (unpublished)
on the ideal-adic completion of quasi-excellent rings, extending the previous work on
Nishimura-Nishimura. As a corollary, we establish that an ideal-adic completion of an
excellent ring is excellent.

1. Introduction
Throughout this paper, we assume that all rings are commutative and possess an identity. The aim of this article is to give a detailed proof to the following theorem (see
Theorem 5.1).
Main Theorem 1 (Nishimura-Nishimura, Gabber). Let A be a Noetherian ring, and I
an ideal of A. Assume that A is I-adically complete. Then, if A/I is quasi-excellent, so is
A.
This result was proved in characteristic 0 by Nishimura-Nishimura in [12], using the
resolution of singularities. More recently, the general case was settled by Gabber, using
his theorem of local uniformization of quasi-excellent schemes. The idea of his proof is
sketched in a letter [16] from Gabber to Laszlo. The above theorem is a special and
difficult case of the Lifting Problem, which was formulated by Grothendieck [5, Remarque
(7.4.8)]. For the precise definition of lifting problem as well as its variations, we refer the
reader to Definition 2.4. As an important corollary, we obtain the following theorem (see
Corollary 5.5).
Main Theorem 2. Let A be an excellent ring with an ideal I ⊂ A. Then the I-adic

completion of A is an excellent ring. In particular, if A is excellent, then the formal power
series ring A[[x1 , . . . , xn ]] is excellent.
Here is an outline of the paper.
Key words and phrases. Ideal-adic completion, lifting problem, local uniformization, Nagata ring, quasiexcellent ring.
2010 Mathematics Subject Classification: 13B35, 13F25, 13F40.

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K. KURANO AND K. SHIMOMOTO

In § 2, we fix notation and definitions concerning excellent rings for which we refer the

reader to [9] and [10]. Then we introduce terminology related to the class of ring theoretic
properties preserved under ideal-adic completion and make a table on the known results.
In § 3, we begin with the notions of quasi-excellent schemes and alteration coverings.

Then we recall the recent result of Gabber on the existence of local unifromization of a
quasi-excellent Noetherian scheme as an alteration covering.
In § 4, using Gabber’s local uniformization theorem, we give a proof to the classical

theorem of Brodmann and Rotthaus in the full generality. This result is an important
step to prove the main result of this paper.
In § 5, we finish the proof of the main result after proving a number of intermediate

lemmas based on the ideas explained in [12] and [16]

2. Notation and conventions
We use the following notation. Let I be an ideal of a ring A. Then denote by V (I) the
set of all prime ideals of A containing I. Let D(I) := Spec A \ V (I). Let A be a ring with
bI the I-adic completion lim A/I n . If (A, m) is a local ring,
an ideal I. We denote by A
←−n
b or A∧ the m-adic completion lim A/mn . Let A be an integral
we simply denote by A
←−n
domain. Denote by Q(A) the field of fractions of A.
Now let P be a ring theoretic property of local Noetherian rings (e.g., regular, normal,

reduced,. . .). Let A be a Noetherian algebra over a field K. We say that A has geometrically P over K, if every local ring of A ⊗K L is P for every finite field extension L/K.

Let k(p) denote the residue field of A at p ∈ Spec A. For the following definition, we refer

the reader to [5, (7.3.1)].

Definition 2.1. Let P be a ring theoretic property of local Noetherian rings.
(1) A homomorphism of Noetherian rings ψ : A → B is said to be a P-homomorphism,

if ψ is flat and the fiber B ⊗A k(p) is geometrically P over k(p) for any p ∈ Spec A.
A homomorphism of Noetherian rings is said to be a regular homomorphism, if

it is a P-homomorphism with P being regular.
(2) A Noetherian ring A is said to be a P-ring, if for any p ∈ Spec A, the natural
cp is a P-homomorphism.
homomorphism Ap → A

For the following definition, we refer the reader to [9] and [10].
Definition 2.2. Let A be a Noetherian ring.

(1) We say that A is a G-ring (resp. Z-ring), if A is a P-ring with P being regular
(resp. normal).
(2) We say that A is a J2 -ring, if the regular locus of every finitely generated A-algebra
is Zariski open.

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(3) We say that A is catenary, if for any pair of prime ideals p ⊂ q of R, let p = p0 ⊂
p1 ⊂ · · · ⊂ pm = q and p = p′0 ⊂ p′1 ⊂ · · · ⊂ p′n = q be saturated strictly increasing
chains of prime ideals between p and q, then we have m = n. We say that A is
universally catenary, if every finitely generated A-algebra is catenary.
A Noetherian ring is called quasi-excellent, if it is a G-ring and a J2 -ring. A Noetherian
ring is called excellent, if it is a G-ring, a J2 -ring and universally catenary.
Any field, the ring of integers and complete local rings are excellent rings. Polynomial
rings over a G-ring are also G-rings in [9, Theorem 77]. One can prove the same for Z-rings
in the same way. Therefore, the category of G-rings (that of Z-rings, that of J2 -rings or
that of universally catenary rings) is closed under polynomial extensions, homomorphic
images and localizations.
It is known that any semi-local G-ring is a J2 -ring (cf. [9, Theorem 76]). Hence semilocal G-rings are quasi-excellent.
Definition 2.3. Let A be a Noetherian ring.
(1) We say that A is an N2 -ring (or Japanese ring), if A is an integral domain and
the integral closure of A in any finite field extension of Q(A) is module-finite over
A, where Q(A) is the field of fractions of A.
(2) We say that A is a Nagata ring (or universally Japanese ring), if for any P ∈ Spec A
the integral domain A/P is an N2 -ring.

It is known that quasi-excellent rings are Nagata rings (cf. [9, Theorem 78]). The
category of Nagata rings is closed under polynomial extensions, homomorphic images and
localizations (cf. [9, Theorem 72]).
Definition 2.4. Let P′ be a ring theoretic property of Noetherian rings.
(1) We say that the Lifting Property (LP for short) holds for P′ , if the following
condition holds. For any Noetherian ring A with an ideal I such that A is Iadically complete, if A/I is P′ , then A is P′ .
(2) We say that the Local Lifting Property (LLP for short) holds for P′ , if the lifting
property holds for any semi-local ring A.
(3) We say that the Power Series Extension Property (P SEP for short) holds for P′ ,
if the following condition holds. For any Noetherian ring A, if A is P′ , then the
formal power series ring A[[x]] is P′ .
(4) We say that the Ideal-adic Completion Property (ICP for short) holds for P′ , if
the following condition holds. For any Noetherian ring A with an ideal I, if A is
P′ , then the I-adic completion of A is P′ .

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K. KURANO AND K. SHIMOMOTO

For any property P′ , it is easy to see the following implications (see Corollary 5.5 and
its proof):
LLP ⇐= LP =⇒ P SEP.
If the category of Notherian rings having the property P′ is closed under homomorphic
images, then the implication
P SEP =⇒ ICP
holds.
If the category of Notherian rings having the property P′ is closed under polynomial
extension, then the implication
P SEP ⇐= ICP
holds.
In this article, we consider P′ as one of the following properties:
G-ring, quasi-excellent, excellent, universally catenary, Nagata, Nagata Z-ring.
Let us remark that the category of Noetherian rings with P′ as above is closed under
polynomial extensions and homomorphic images.
In 1970, Seydi [15] proved that PSEP holds for P′ = universally catenary. In 1975,
Marot [8] proved that LP holds for P′ = Nagata. In 1979, Rotthaus [13] proved that LLP
holds for P′ = G-ring. Hence, LLP holds for P′ = quasi-excellent. In 1981, Nishimura [11]
found an example, and proved that ICP does not hold for P′ = G-ring. In 1982, Greco [4]
found an example, and proved that LLP does not hold for both P′ = universally catenary
and P′ = excellent. In 1987, Nishimura-Nishimura ([12, Theorem A]) proved that LP
holds for P′ = Nagata Z-ring. They also proved that LP holds for P′ = quasi-excellent,
if the ring contains a field of characteristic 0 ([12, Theorem B]). Recently, Gabber [16]
proved that LP holds for P′ = quasi-excellent in general. The aim of this paper is to
complete the following table by giving the details of Gabber’s theorem.
LLP
G-ring
quasi-excellent
excellent
universally catenary
Nagata

×
×

LP

PSEP

ICP

×

×

×

×

×

Nagata Z-ring
Here, remark that PSEP and ICP hold for P′ = excellent, since they hold for both P′ =
quasi-excellent and P′ = universally catenary.

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3. Gabber’s local uniformization theorem for quasi-excellent schemes
In this section, let us recall a recent result on the existence of local uniformizations for
quasi-excellent schemes, due to Gabber.
Definition 3.1. A Noetherian scheme X is quasi-excellent (resp. excellent), if X admits
an open affine covering, each of which is the spectrum of a quasi-excellent ring (resp. an
excellent ring). Once this condition holds, then any other open affine covering has the
same property.
We introduce the notion of alteration covering of schemes.
Definition 3.2. In this definition, we assume that all schemes are Notherian, and all
morphisms are dominant, generically finite morphisms of finite type between Notherian
schemes.
Let Y be a Noetherian integral scheme. We say that a finite family of scheme maps
{φi : Xi → Y }i=1,...,m is an alteration covering of Y , if there exist a proper morphism
S
f : V → Y , a Zariski open covering V = m
i=1 Vi , together with a family of scheme maps
{ψi : Vi → Xi }i=1,...,m such that the following diagram commutes for each i = 1, . . . , m
Vi
(3.1)

↓ ψi

Xi

−→
φi

−→

V
↓f

Y

where Vi → V is the natural open immersion.

If Xi is a regular integral scheme for each i = 1, . . . , m, we say that {φi : Xi → Y }i=1,...,m

is a regular alteration covering of Y .

We refer the reader to [7] for alteration coverings or the alteration topology in the
general situation. The definiton adopted in [7] looks slightly different from the above
one. However, we may resort to [7, Théorèm 3.2.1; EXPOSÉ II]. For the convenience
of readers, notice that the alteration topology is the same as Voevodsky’s h-topology
in the Noetherian case (see [2] for the proof of this fact). Let us state Gabber’s local
uniformization theorem for which we refer the reader to [7].
Theorem 3.3 (Gabber). Assume that Y is a quasi-excellent Noetherian integral scheme.
Then there exists a regular alteration covering of Y .
Using the valuative criterion for proper maps and Gabber’s theorem, we obtain the following corollary. That is the reason why the above theorem is called “local uniformization
theorem” for quasi-excellent schemes.

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K. KURANO AND K. SHIMOMOTO

Corollary 3.4. Assume that A is a quasi-excellent domain. Then there exists a finite
field extension K/Q(A) such that if R is a valuation domain satisfying Q(R) = K and
A ⊂ R, then there exists a regular domain B such that A ⊂ B ⊂ R and B is a finitely
generated A-algebra.

4. A generalization of a theorem of Brodmann and Rotthaus
The purpose of this section is to prove the following theorem. It was proved by Brodmann and Rotthaus [1] for rings containing a field of characteristic zero, using the resolution of singularities by Hironaka. Let rad(A) be the Jacobson radical of a ring A.
Theorem 4.1. Let A be a Noetherian ring with an ideal I ⊂ rad(A). Assume that A/I

is quasi-excellent and A is a G-ring. Then A is a J2 -ring. In other words, A is quasiexcellent.
We need to prove a number of lemmas before proving this theorem. Let X be a Noetherian scheme. We denote by Reg(X) the regular locus of X, and put Sing(X) = X \Reg(X).

Let us recall that a subset of a Noetherian scheme is open if and only if it is constructible
and stable under generalization of points [6, II. Ex. 3.17, 3.18].

Lemma 4.2. Let A be a Noetherian ring with an ideal I ⊂ A and let π : X → Spec A be

a scheme map of finite type. Assume that A/I is a J2 -ring. Then Reg(X) ∩ π −1 V (I) is

open in π −1 V (I) .
Proof. For the proof, we may assume that X is an affine scheme. Let B be an A-algebra

of finite type and put X = Spec B. First, let us prove the following lemma:
Claim 4.3. Assume that Z ⊂ V (IB) is a closed subset. Then Reg(X) ∩ Z is constructible

in Z.

Proof of the claim. We prove it by Noetherian induction. So let us suppose that any
proper closed subset of Z satisfies the conclusion of the claim. We may assume that
Z = V (q) for some prime ideal q ⊃ IB. Since A/I is a J2 -ring, we have

(4.1)
Reg Spec(B/q) is a non-empty open set in Spec(B/q).

Assume that q ∈
/ Reg(X). Then we have Reg(X) ∩ Z = ∅ and this is evidently con-

structible. Next, assume that q ∈ Reg(X). Then since Bq is a regular local ring, the

maximal ideal qBq is generated by a regular sequence. This together with (4.1) implies
the following:

- There exists an element f ∈ B \ q such that qB[f −1 ] is generated by a B[f −1 ]regular sequence and B[f −1 ]/qB[f −1 ] is a regular ring.

IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

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Hence if p ∈ Spec B is taken such that p ∈ V (q) and p ∈
/ V (q + f B), then Bp is regular.

We have the decomposition:

 

Reg(X) ∩ V (q) = Reg(X) ∩ V (q + f B) ∪ Reg(X) ∩ V (q) \ V (q + f B)

By Noetherian induction hypothesis, we see that Reg(X) ∩ V (q + f B) is constructible.

On the other hand, we have Reg(X) ∩ V (q) \ V (q + f B) = V (q) \ V (q + f B) and this

is clearly open in V (q). We conclude that Reg(X) ∩ V (q) is constructible, which finishes
the proof of the claim.



We can finish the proof of the lemma in the following way. It is easy to show that


Reg(X) ∩ π −1 V (I) is closed under taking generalizations of points inside π −1 V (I) .



Combining Claim 4.3, we conclude that Reg(X) ∩ π −1 V (I) is open in π −1 V (I) .
Using this lemma, we can prove the following crucial fact.

Lemma 4.4. Let B be a Noetherian domain and let us choose q ∈ Spec B and an ideal

J ⊂ B. Assume that Bq is a G-ring and B/J is a J2 -ring. Then there exists b ∈ B \ q
together with an alteration covering {φb,i : Xb,i → Spec B[b−1 ]}i such that


B[b−1 ]
(4.2)
φ−1
Spec
⊂ Reg(Xb,i )
b,i
JB[b−1 ]

for all i.
Proof. Since Bq is a quasi-excellent local domain by assumption, there exists a regular
alteration covering {φi : Xi → Spec Bq }i=1,...,m , together with a proper map f : V →
S
Spec Bq with V := m
i=1 Vi and {ψi : Vi → X}i by Theorem 3.3. By Chow’s lemma, we

may assume that f : V → Spec Bq is projective. Then we can find an element b̃ ∈ B \q and

an alteration covering {φb̃,i : Xb̃,i → Spec B[b̃−1 ]}i=1,...,m such that φi = φb̃,i ⊗B[b̃−1 ] Bq .

Remark that, if φb̃,i (s) is a generalization of qB[b̃−1 ], then we have s ∈ Xi . In particular,
OXb̃,i ,s is regular.

(4.3)
Let us put



B[b̃−1 ]
Zb̃ := Spec
, which is a closed subset of Spec B[b̃−1 ].
JB[b̃−1 ]
Then we find that
Wb̃,i := φ−1 (Zb̃ ) \ Reg(Xb̃,i ) is closed in φ−1 (Zb̃ )
b̃,i

b̃,i

by Lemma 4.2. Hence Wb̃,i is a closed subset of Xb̃,i and thus, φb̃,i (Wb̃,i ) is a constructible
subset of Spec B[b̃−1 ] by Chevalley’s theorem (cf. [9, Theorem 6]). From this, it follows
that the Zariski closure φb̃,i (Wb̃,i ) of φb̃,i (Wb̃,i ) is the set of all points of Spec B[b̃−1 ] that
are obtained as a specialization of a point of φb̃,i (Wb̃,i ).

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K. KURANO AND K. SHIMOMOTO

Assume that we have q ∈ φb̃,i (Wb̃,i ). Then there is a point s ∈ Wb̃,i such that q ∈

{φb̃,i (s)}. By (4.3), the local ring OXb̃,i ,s is regular. Hence s ∈ Reg(Xb̃,i ), which is a
contradiction to s ∈ Wb̃,i . Thus, we must get q ∈
/ φb̃,i (Wb̃,i ).

Let us choose an element 0 6= b ∈ B such that q ∈ Spec B[b−1 ] ⊂ Spec B[b̃−1 ] and

Spec B[b−1 ] ∩ φb̃,i (Wb̃,i ) = ∅ for all i. Consider the fiber square:
φb,i

−→ Spec B[b−1 ]

Xb,i
↓

↓

φb̃,i

−→ Spec B[b̃−1 ]

Xb̃,i

Then we have Wb̃,i ∩ Xb,i = ∅. Let us put



B[b−1 ]
and Wb,i := φ−1
Zb := Spec
b,i (Zb ) \ Reg(Xb,i ).
JB[b−1 ]

Since Zb = Zb̃ ∩ Spec B[b−1 ], we get



−1
−1
(Zb̃ ) ∩ Xb,i \ Reg(Xb̃,i ) = Wb̃,i ∩ Xb,i = ∅.
Wb,i = φb̃,i
(Zb̃ ) ∩ Xb,i \ Reg(Xb,i ) = φb̃,i

This proves the assertion (4.2).



Lemma 4.5. Let {φi : Xi → Y }i=1,...,m be an alteration covering and let y1 , . . . , yl be a

sequence of points in Y such that yj+1 ∈ {yj } for j = 1, . . . , l − 1, where {yj } denotes the
Zariski closure of {yj } in Y . Then there exist i and a sequence of points x1 , . . . , xl in Xi
such that φi (xj ) = yj for j = 1, . . . , l and xj+1 ∈ {xj } for j = 1, . . . , l − 1.

Proof. By assumption, for i = 1, . . . , m, there is a commutative diagram:
Vi
↓ ψi

Xi
where V =

Sm

i=1 Vi

−→
φi

−→

V
↓f

Y

f

is a Zariski open covering and V −
→ Y is a proper surjective map. Let

us find a sequence v1 , . . . , vl in V such that vj maps to yj for j = 1, . . . , l and vj+1 ∈ {vj }
for j = 1, . . . , l − 1. First, lift y1 to a point v1 ∈ V via f . Suppose that a sequence
v1 , . . . , vt in V has been found such that f (vj ) = yj for j = 1, . . . , t and vj+1 ∈ {vj } for

j = 1, . . . , t − 1. So let us find vt+1 ∈ V with the required condition. Since f is a proper

map, f ({vt }) is equal to {yt }. Hence yt+1 ∈ f ({vt }) and there is a lift vt+1 of yt+1 such
that vt+1 ∈ {vt }. Here, we have vl ∈ Vi for some i. Since Vi is closed under generalizations,

v1 , . . . , vl are contained in Vi . Then we see that the sequence x1 := ψi (v1 ), . . . , xl := ψi (vl )

in Xi satisfies the required conditions.



IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

9

Proof of Theorem 4.1. In order to show that A is a J2 -ring, it is enough to prove that the
regular locus of any finite A-algebra is open (cf. [9, Theorem73]). Let B be a Noetherian
domain that is finitely generated as an A-module. It suffices to prove that Reg(B) contains
a non-empty open subset of Spec B by Nagata’s topological criterion (cf. [10, Theorem
24.4]). Since A → B is module-finite, we have IB ⊂ rad(B) and B/IB is quasi-excellent.

Since B is a G-ring, Bq is quasi-excellent for any q ∈ Spec B by [9, Theorem 76]. By Lemma
4.4, there exists bq ∈ B \ q and an alteration covering {φbq ,i : Xbq ,i → Spec B[b−1
q ]}i such
that

(4.4)

φ−1
bq ,i



Spec

B[b−1
q ]
IB[b−1
q ]


⊂ Reg(Xbq ,i ) for all i.

There exists a family of elements b1 , . . . , bs ∈ {bq | q ∈ Spec B} such that
(4.5)

−1
Spec B = Spec B[b−1
1 ] ∪ · · · ∪ Spec B[bs ].

There exists a finite family of alteration coverings {φbj ,i : Xbj ,i → Spec B[b−1
j ]}i with the
same property as (4.4) by letting bj = bq . By the lemma of generic flatness (cf. [9, (22.A)]),
we can find an element 0 6= c ∈ B such that the induced map

−1 −1
−1 −1
φ−1
bj ,i Spec B[bj c ] → Spec B[bj c ] is flat for all i and j.
If we can show that B[c−1 ] is regular, then the proof is finished.

Let us pick p ∈ Spec B such that c ∈
/ p. Then we want to prove that Bp is regular. Let

us choose a maximal ideal m ∈ Spec B such that p ⊂ m. Since IB ⊂ rad(B), it follows

that IB ⊂ m. By (4.5), there exists j such that p, m ∈ Spec B[b−1
j ]. By Lemma 4.5,

there exist x1 , x2 ∈ Xbj ,i for some i such that x2 ∈ {x1 }, φbj ,i (x1 ) = p and φbj ,i (x2 ) = m.

By (4.4) together with the fact IB ⊂ m, the local ring OXbj ,i ,x2 is regular. Since x1 is a

generalization of x2 , OXbj ,i ,x1 is also a regular local ring. Here, Bp → OXbj ,i ,x1 is flat, as
c∈
/ p. Therefore, Bp is regular, as desired.



5. Lifting problem for quasi-excellent rings
In this section, we shall prove the main theorem:
Theorem 5.1 (Nishimura-Nishimura, Gabber). Let A be a Noetherian ring, and I an
ideal of A. Assume that A is I-adically complete. Then, if A/I is quasi-excellent, so is A.
Proof. Assume the contrary. Let A be a Noetherian ring, and I an ideal of A. Suppose
that A is I-adically complete, A/I is quasi-excellent, but A is not quasi-excellent.
Step 1. We shall reduce this problem to a simpler case as long as possible.
(1-1) The following are well-known facts.

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K. KURANO AND K. SHIMOMOTO

• Let R be a Noetherian ring and I1 , I2 be ideals of R such that I1 ⊃ I2 . If R is
I1 -adically complete, then R is I2 -adically complete.

• Let R be a Noetherian ring and J1 , J2 be ideals of R. If R is J1 -adically complete,
then R/J2 is ((J1 + J2 )/J2 )-adically complete.

We shall use these facts without proving them.
Suppose I = (a1 , . . . , at ). Put Ii = (a1 , . . . , ai ) for i = 1, . . . , t and I0 = (0). Remark
that A/Ii is (Ii+1 /Ii )-adically complete, and
(A/Ii )/(Ii+1 /Ii ) = A/Ii+1
for i = 0, 1, . . . , t − 1. Here, Ii+1 /Ii is a principal ideal of A/Ii . Remember that A/It is

quasi-excellent, but A/I0 is not so. Therefore, there exists i such that A/Ii+1 is quasiexcellent, but A/Ii is not so.
Replacing A/Ii and Ii+1 /Ii with A and I respectively, we may assume that
(A1) I is the principal ideal generated by some x 6= 0, that is, I = (x).
(1-2) We put
F = {J | A/J is not quasi-excellent}.
Since F contains (0), the set F is not empty and there exists a maximal element J0 in F.
Replacing A/J0 by A, we may assume that

(A2) if J 6= (0), then A/J is quasi-excellent.
(1-3) By Theorem 4.1, A is not a G-ring. There exist prime ideals P and Q of A such
that P ⊃ Q, and the generic fiber of
AP /QAP −→ (AP /QAP )∧
is not geometrically regular. On the other hand, if Q 6= (0), the above map is a regular
homomorphism by (A2). Therefore we know Q = (0), that is, A is an integral domain.

Since quasi-excellent rings are Nagata [9, Theorem 78], we know that A/xA is a Nagata
ring. Since the lifting property holds for Nagata rings by Marot [8], A is a Nagata domain.
Let A be the integral closure of A in Q(A). Then A is module-finite over A.
By Greco’s theorem [3, Theorem 3.1], A is not quasi-excellent, since A is not so. Here
A is xA-adically complete and A satisfies (A2). Replacing A with A, we further assume
that
(A3) A is a Nagata normal domain.
(1-4) Since A/xA is quasi-excellent, A/xA is a Nagata Z-ring. Since the lifting property
holds for Nagata Z-rings by Nishimura-Nishimura [12, Theorem A], A is also a Nagata
cP is a local normal domain for any prime ideal P of
Z-ring. Since A is a normal Z-ring, A
A. Then by [10, Theorem 31.6], A is universally catenary. Thus we know

IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

11

(A4) A is a Z-ring and universally catenary.
Step 2. We assume (A1), (A2), (A3) and (A4).
By Theorem 4.1, A is not a G-ring. By [9, Theorem 75], there exists a maximal ideal
cm is not regular. Consider the fibers of
m of A such that the homomorphism Am → A
cm . By (A2), the fibers except for the generic fiber are geometrically regular. So
Am → A
we concentrate on the generic fiber.

Let L be a finite algebraic extension of Q(A), where Q(A) is the field of fractions of A.

Let BL be the integral closure of A in L. Remark that BL is a finite A-module by (A3).
Consider the following fiber squares:
L

=

L

↑

↑

↑

↑

−→

cm
L ⊗A A
↑

=

cm =
BL −→ BL ⊗A Am −→ BL ⊗A A
A

−→

Am

−→

↑
cm
A

Q

nL

\
⊗BL (B
L )n

Q \
n (BL )n

Here n runs over all the maximal ideals of BL lying over m.
\
We want to discuss whether L ⊗BL (B
L )n is regular or not for L and n.

Recall that (BL )n /x(BL )n is a G-ring since A/xA is quasi-excellent. Let (BL )∗n be the

x(BL )n -adic completion of (BL )n . By the local lifting property for G-rings (Rotthaus [13]),
(BL )∗n is a G-ring.

(5.1)


∗
\
∗
\
Hence, the homomorphism (BL )∗n → (B
L )n = (BL )n is regular and Sing (BL )n is a closed
\
subset of Spec(BL )∗ . Therefore, L ⊗B (BL )∗ is regular if and only if L ⊗B (B
L )n is
n

L

n

L

regular. For a finite algebraic extension L of Q(A), we put
)
(

∗ is a minimal prime ideal of Sing (B )∗ such that q∗ ∩ B = (0),
q
L
L
n
n
n
SL = q∗n
where n is a maximal ideal of BL .

Here, L ⊗BL (BL )∗n is not regular for some L and n, since A is not a G-ring. Therefore,
SL is not empty for some L. We put
(5.2)

h0 = min{ht Q | Q ∈ SL for some L}.

Remember that A is a Z-ring by (A4). Since BL is a finite A-module, BL is also a
∗
\
Z-ring. It is easy to see that (B
L )n is the completion of both (BL )n and (BL )n . Since
(BL )n and (BL )∗n are Z-rings (cf. (5.1)),
(5.3)

\
(BL )n , (BL )∗n and (B
L )n are local normal domains.

Therefore we know
(5.4)

h0 ≥ 2.

12

K. KURANO AND K. SHIMOMOTO

We shall prove the following claim in the rest of Step 2.
Claim 5.2. Let p be a prime ideal of A. If ht p ≤ h0 , then Ap is excellent.
Let p be a prime ideal of A such that 0 < ht p ≤ h0 . By (A4), A is universally catenary.

It is enough to prove that Ap is a G-ring. By (A2) and [9, Theorem 75], it is enough to
cp is geometrically regular. Let L be a finite algebraic
show that the generic fiber of Ap → A

extension of Q(A), and BL be the integral closure of A in L. Consider the following fiber
squares.

L

=

L

↑

↑

↑

↑

−→

cp
L ⊗A A

=

↑

cp =
BL −→ BL ⊗A Ap −→ BL ⊗A A
A

−→

Ap

−→

Q

qL

↑
c
Ap

\
⊗BL (B
L )q

Q \
q (BL )q

Here q runs over all the prime ideals of BL lying over p. Since A is normal, we have
0 < ht q = ht p ≤ h0 .

(5.5)

\
It is enough to show that L ⊗BL (B
L )q is regular. Let n be a maximal ideal of BL such

that n ⊃ q. Let q∗ be a minimal prime ideal of q(BL )∗n . Since (BL )n → (BL )∗n is flat, we

have q∗ ∩ BL = q. Consider the commutative diagram:
(BL )∗n −→ ((BL )∗n )q∗
↑

BL −→ (BL )n −→

↑α

(BL )q

β

−→ (((BL )∗n )q∗ )∧
−→

↑ α̂
\
(BL )q

The map α as above is a flat local homomorphism. Since the closed fiber of α is of
dimension 0, we have dim(BL )q = dim((BL )∗n )q∗ . In particular
ht q = ht q∗ .

(5.6)


By (5.1), β is a regular homomorphism and Sing (BL )∗n is a closed subset of Spec(BL )∗n .

√ ∗
Let c∗n be the defining ideal of Sing (BL )∗n satisfying c∗n =
cn . We put T =

(((BL )∗n )q∗ )∧ . Since β is a regular homomorphism, c∗n T is a defining ideal of Sing(T ).
Suppose
c∗n = q∗n,1 ∩ · · · ∩ q∗n,s ,
where q∗n,i ’s are prime ideals of (BL )∗n such that q∗n,i 6⊃ q∗n,j if i 6= j.
One of the following three cases occurs:

Case 1. If none of q∗n,i ’s is contained in q∗ , then c∗n T = T .
Case 2. If q∗n,i = q∗ for some i, then c∗n T = q∗ T .
Case 3. Suppose that q∗n,1 , . . . , q∗n,t are properly contained in q∗ , and q∗n,t+1 , . . . , q∗n,s are
not contained in q∗ for some t satisfying 1 ≤ t ≤ s. Then, c∗n T = q∗n,1 T ∩ · · · ∩ q∗n,t T .

IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

13

In any case as above, we can verify c∗n T ∩ BL 6= (0) as follows. In Case 1, we have

c∗n T ∩ BL = BL 6= (0). In Case 2, we have q∗ T ∩ BL = q∗ T ∩ (BL )∗n ∩ BL = q∗ ∩ BL =
q 6= (0) by (5.5). In Case 3, suppose that q∗n,i is properly contained in q∗ . Then by (5.5)

and (5.6), we have ht q∗n,i < h0 . Since q∗n,i is in Sing (BL )∗n , we have q∗n,i ∩ BL 6= (0) by

the minimality of h0 .

Take 0 6= b ∈ c∗n T ∩ BL . Since c∗n T is the defining ideal of Sing(T ), T ⊗BL BL [b−1 ] is

regular. Since

−1 α̂⊗1
−1
\
(B
L )q ⊗BL BL [b ] −→ T ⊗BL BL [b ]

−1
\
\
is faithfully flat, (B
L )q ⊗BL BL [b ] is regular. Hence, (BL )q ⊗BL L is regular. We have

completed the proof of Claim 5.2.

Step 3. Here, we shall complete the proof of Theorem 5.1.
First of all, remember the following Rotthaus’ Hilfssatz (cf. [12, Theorem 1.9 and
Proposition 1.18] or originally [14]).
Theorem 5.3 (Rotthaus’ Hilfssatz). Let B be a Noetherian ring and n ∈ Max B. Assume
that B is xB-adically complete Nagata ring.
We put
Γ(n) = {γ | n ∈ γ ⊂ Max B,

#

γ < ∞}.

For γ ∈ Γ(n), we put Sγ = B \ ∪a∈γ a and Bγ = Sγ−1 B. Consider the homomorphism
Bγ∗ → Bn∗ induced by Bγ → Bn , where ( )∗ denotes the (x)-adic completion.

Let q∗n be a minimal prime ideal of Sing(Bn∗ ). For each γ ∈ Γ(n), we put q∗γ := q∗n ∩ Bγ∗ .

We define

n
[
o
∆γ (x) := Q ∩ B | Q ∈ Min(B ∗ /q∗ ) (Bγ∗ /q∗γ )/(x) , ∆(x) :=
∆γ (x)
γ

γ

γ∈Γ(n)

where (Bγ∗ /q∗γ ) is the normalization of Bγ∗ /q∗γ in the field of fractions.
Assume the following two conditions:
(i) for each γ ∈ Γ(n), ht q∗γ > 0,

(ii)

# △(x)

< ∞.

Then q∗n ∩ B 6= (0) is satisfied.
We refer the reader to [12] for the proof of this theorem. We deeply use it in our proof.
Now, we start to prove Theorem 5.1.
Let L be a finite algebraic extension of Q(A). Let B be the integral closure of A in L.
Let n be a maximal ideal of B. Suppose that q∗n is a minimal prime ideal of Sing(Bn∗ ) such
that
(5.7)

q∗n ∩ B = (0)

14

K. KURANO AND K. SHIMOMOTO

and ht q∗n = h0 . We remark that such L, B, n, q∗n certainly exist by the definition of h0
(see (5.2)).
We define Bn∗ , Bγ , Bγ∗ , q∗γ , ∆γ (x), ∆(x) as in Theorem 5.3. Recall that Bγ is a semi-local
ring satisfying Max(Bγ ) = {aBγ | a ∈ γ}. Since B is xB-adically complete, any maximal

ideal of B contains x. Since Bγ /xBγ is isomorphic to Bγ∗ /xBγ∗ , there exists one-to-one

correspondence between γ and the set of maximal ideals of Bγ∗ . Let n∗ be the maximal
ideal of Bγ∗ corresponding to n, that is, n∗ = nBγ∗ .
In the rest of this proof, we shall prove the conditions (i) and (ii) in Theorem 5.3. Then,
it contradicts (5.7) and this completes the proof of Theorem 5.1.
Put Cγ = Bγ∗ /q∗γ . Since A is a Nagata ring, Cγ is a Nagata ring, too. Therefore, the
normalization Cγ is a finite Cγ -module. Note that x 6∈ q∗γ since q∗n ∩ B = (0). Take

Q ∈ MinCγ (Cγ /xCγ ). Since Cγ is a universally catenary Nagata Z-ring *, we find that
Q∩Cγ is a minimal prime ideal of xCγ . Therefore, △γ (x) defined in Theorem 5.3 coincides
with


{Q̃ ∩ B | Q̃ ∈ MinBγ∗ Bγ∗ /q∗γ + xBγ∗ }.

Here, we shall prove the following claim.
Claim 5.4.

(1) For each γ ∈ Γ(n), ht q∗γ = h0 .

(2) For any Q ∈ △(x), ht Q = h0 + 1.

First, we shall prove (1). Consider the following homomorphisms.
(5.8)

f

g

cn ,
(Bγ∗ )n∗ −→ Bn∗ −→ B

where n∗ = nBγ∗ . By the local lifting property for G-rings, Bγ∗ is a G-ring. Since
\
∗
cn = (B
B
γ )n∗ , gf is a regular homomorphism. Since g is faithfully flat, f is a regular
homomorphism by [9, (33.B)] or [10, Theorem 32.1]. Since q∗γ (Bγ∗ )n∗ = q∗n ∩ (Bγ∗ )n∗ ,

(5.9)

q∗γ (Bγ∗ )n∗ is a minimal prime ideal of Sing((Bγ∗ )n∗ ).

Furthermore, q∗n is a minimal prime ideal of q∗γ Bn∗ . Then, we have
ht q∗γ = ht q∗γ (Bγ∗ )n∗ = ht q∗n = h0 .
The assertion (1) has thus been proved. Since h0 ≥ 2 as in (5.4), the condition (i) in

Theorem 5.3 follows from the above assertion (1).

Next, we prove (2). Take Q ∈ △γ (x) for some γ ∈ Γ(n).

We shall prove that Bγ∗ is normal. For a ∈ γ, let a∗ denote the maximal ideal aBγ∗ of
ca is a normal local domain. By the
Bγ∗ . Here B is a normal Z-ring, since A is. Therefore, B

* Assume that C is a universally catenary Nagata Z-domain. Let C be the normalization of C. Let Q

(resp. P ) be a prime ideal of C (resp. C). It is easy to see, if Q ∩ C = P , then ht Q = ht P .

IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

15

\
∗
c
local lifting property for G-rings, (Bγ∗ )a∗ is a G-ring. Therefore, (Bγ∗ )a∗ → (B
γ )a∗ = Ba is

a regular homomorphism. Thus, (Bγ∗ )a∗ is normal. Hence, we know that Bγ∗ is normal.

By definition, there exists Q̃ ∈ MinBγ∗ Bγ∗ /q∗γ + xBγ∗ such that Q = Q̃ ∩ B. Since
Bγ → Bγ∗ is flat and QBγ∗ = Q̃, we have

(5.10)

ht Q̃ = dim(Bγ∗ )Q̃ = dim(Bγ )Q = ht Q.

Since Bγ∗ is a Noetherian normal ring, Bγ∗ is the direct product of finitely many integrally
closed domains. Remember that Bγ∗ is universally catenary. Then we have
ht Q̃ = ht q∗γ + 1.

(5.11)

By (5.10), (5.11) and the assertion (1) together, we obtain ht Q = h0 + 1.
We have completed the proof of Claim 5.4.
Let q be a prime ideal of B such that ht q ≤ h0 . Since A is normal, ht(q∩A) = ht q ≤ h0 .

Hence, A(q∩A) is excellent by Claim 5.2. Therefore, Bq is excellent. Let us remember that
B/xB is quasi-excellent. Then by Lemma 4.4, there exists bq ∈ B \q such that there exists
an alteration covering
(5.12)

φbq ,i

{Xbq ,i −→ Spec(B[b−1
q ])}i

such that
(5.13)

φ−1
bq ,i



for each i. We put
Ω=

Spec

[
q

B[b−1
q ]
xB[b−1
q ]



⊂ Reg(Xbq ,i )

Spec(B[b−1
q ]) ⊂ Spec(B).

By definition, Ω is an open set that contains all the prime ideals of B of height less than or
equal to h0 . Hence, the complement Ωc contains only finitely many prime ideals of height
h0 + 1. If △(x) is contained in Ωc , then △(x) must be a finite set by Claim 5.4 (2).
Thus, it suffices to prove

△(x) ⊂ Ωc .
Assume the contrary. Take Q ∈ △(x) ∩ Ω. Since Q ∈ △(x), there exists Q̃ ∈

MinBγ∗ Bγ∗ /q∗γ + xBγ∗ such that Q̃ ∩ B = Q for some γ ∈ Γ(n). Since Q ∈ Ω, we
find that Q ∈ Spec(B[b−1
q ]) for some q with ht q ≤ h0 .

Since (5.12) is an alteration covering, we have a proper surjective generically finite
S
map π : V → Spec(B[b−1
q ]), together with an open covering V =
i Vi and a morphism
ψi : Vi → Xbq ,i for each i with commutative diagrams as in (3.1). Consider the following

16

K. KURANO AND K. SHIMOMOTO

diagram:
Vi′
↓ ψi′
Xb′ q ,i
↓g
Xbq ,i

V′

⊂

↓ f′

φ′bq ,i

∗
−→ Spec(Bγ∗ [b−1
q ]) −→ Spec(Bγ )

φbq ,i

−→

↓h

↓

Spec(B[b−1
q ])

−→

Spec(B)

We put ( )′ = ( ) ×Spec B Spec Bγ∗ . Then, both Q̃ and q∗γ are contained in Spec(Bγ∗ [b−1
q ]).

′
Since f ′ : V ′ → Spec(Bγ∗ [b−1
q ]) is proper surjective, there exist ξ1 , ξ2 ∈ V such that
S
f ′ (ξ1 ) = Q̃, f ′ (ξ2 ) = q∗γ , and ξ1 is a specialization of ξ2 . Since V ′ = i Vi′ is a Zariski open

covering, we have ξ1 ∈ Vi′ for some i. Since Vi′ is closed under generalization, both ξ1 and

ξ2 are contained in Vi′ . Here, we put η1 = ψi′ (ξ1 ), η2 = ψi′ (ξ2 ) and ζ1 = g(ψi′ (ξ1 )).

Since x ∈ Q and φbq ,i (ζ1 ) = Q, we know that OXbq ,i ,ζ1 is a regular local ring by (5.13).

Since B → Bγ∗ is flat, OXbq ,i ,ζ1 → OXb′

q ,i

,η1

is a flat local homomorphism. Its closed fiber

is the identity since the maximal ideal of OXbq ,i ,ζ1 contains x. Thus, OXb′

q ,i

local ring. Since η2 is a generalization of η1 , OXb′

q ,i

(Bγ∗ )q∗γ

→ OXb′

q ,i

,η2

is flat, since

h(q∗γ )

,η2

,η1

is a regular

is also a regular local ring. Here,

= (0) by (5.7). Therefore, (Bγ∗ )q∗γ is a regular local

ring. It contradicts (5.9). The condition (ii) in Theorem 5.3 has been proved.
We have completed the proof of Theorem 5.1.



Now we obtain the following corollary.
Corollary 5.5. Let A be an excellent ring with an ideal I ⊂ A. Then the I-adic completion

of A is an excellent ring. In particular, if A is excellent, then the formal power series ring

A[[x1 , . . . , xn ]] is excellent.
bI denote the I-adic completion of A. We want to prove that A
bI is excellent.
Proof. Let A
bI /I A
bI is also excellent. In particular, it is
As A is excellent by assumption, A/I ∼
= A
bI is quasi-excellent. So it suffices to prove that A
bI is
quasi-excellent. By Theorem 5.1, A

universally catenary. For this, let I = (t1 , . . . , tm ) with ti ∈ A. By [15, Théorèm 1.12], the

formal power series ring A[[T1 , . . . , Tm ]] is universally catenary. As there is an isomorphism

[10, Theorem 8.12]
bI ∼
A
= A[[T1 , . . . , Tm ]]/(T1 − t1 , . . . , Tm − tm ),

bI is universally catenary and hence excellent.
we see that A

Finally, assume that A is excellent. Then the polynomial algebra A[x1 , . . . , xn ] is excel-

lent, and the (x1 , . . . , xn )-adic completion of A[x1 , . . . , xn ] is A[[x1 , . . . , xn ]]. Hence it is

excellent. This proves the corollary.



IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER)

17

Acknowledgement . The authors are grateful to Professor O. Gabber for permitting us to
write this paper. The authors are also grateful to Professor J. Nishimura for listening to
the proof of the main theorem and providing us with useful suggestions kindly.
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[1] M. Brodmann and C. Rotthaus, Ü ber den regulären Ort in Ausgezeichneten Ringen, Math. Z. 175
(1980) 81–85.
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[3] S. Greco, Two theorems on excellent rings, Nagoya Math. J. 60 (1976), 139–149.
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[5] A. Grothendieck, Élements de Géométrie Algébrique IV, Publications Math. I.H.E.S. 24 (1965).
[6] R. Hartshorne, Algebraic Geometry, Springer-Verlag (1977).
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[9] H. Matsumura, Commutative algebra. Second edition, . Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.
[10] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge
University Press, Cambridge (1986).
[11] J. Nishimura, Ideal-adic completion of Noetherian rings, J. Math. Kyoto Univ. 21 (1981), 153–169.
[12] J. Nishimura and T. Nishimura, Ideal-adic completion of Noetherian rings II, Algebraic Geometry
and Commutative Algebra in Honor of Masayoshi Nagata (1987) 453–467.
[13] C. Rotthaus, Komplettierung semilokaler quasiausgezeichneter Ringe, Nagoya Math. J. 76 (1979),
173–180.
[14] C. Rotthaus, Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253 (1980), 213–226.
[15] H. Seydi, Anneaux henséliens et conditions de chaı̂nes, Bull. Soc. Math. France 98 (1970) 9–31.
[16] A letter from O. Gabber to Y. Laszlo (2007).
Department of Mathematics, School of Science and Technology, Meiji University, Higashimata 1-1-1, Tama-ku, Kawasaki 214-8571, Japan
E-mail address: kurano@isc.meiji.ac.jp
Department of Mathematics, College of Humanities and Sciences, Nihon University,
Setagaya-ku, Tokyo 156-8550, Japan
E-mail address: shimomotokazuma@gmail.com

