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Local Etaie Extensions
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1!” COMM“

 

Local Etale Extensions And N ormalizations

By

Mark .S'. M c Cormz'ck

A Dissertation

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

_ DOCTOR OF PHILOSOPHY

Department of Mathematics

1998

ABSTRACT

Local Etale Extensions And Normalizations

By

Mark S . M c Cormz'ck

An easy generalization of a result of C. Rotthaus yields that for A an excellent
normal local domain with strict Henselization Ah“, any local normal intermediate
ring A Q D g A’” dominated by A’“ is a direct limit of local etale extensions of A.
In a more general setting, we use this fact to characterize etaleness of intermediate
rings in terms of properties of their normalizations. Specifically, let A be an excellent
reduced local ring with strict Henselization A’”, and let D be an intermediate ring
A g D g; A” dominated by A’” and having total quotient ring finitely generated over
that of A. Then D is etale over A if and only if the normalization of D is semilocal
and has an appropriate residual field structure.

Using analyses of normalizations, we study intersection rings under the Henseliza-
tion and under the strict Henselization. We find that for A an excellent reduced local
ring and Q(A) <_I L Q Q(A") an intermediate field obtained by adjoining idempotent
elements to Q(A), the intersection L n A” is a local etale extension of A. We present
an example to show that this does not hold for A’” in place of A". Then turning

to the strict Henselization, we find that for A an excellent Henselian reduced local

ring, I E Min(A’”) and Q(A/(I n A)) <_Z L C_: Q(A"‘/I) an intermediate field which

is finitely generated over Q(A/(I (i A)), the intersection L n (Aha/I) is finite and
unramified over A / (I HA). We use this intersection theorem to characterize etaleness
and Noetherianness for rings between a Henselian ring A and its strict Henselization

in terms of the residual fields of their normalizations.

ACKNOWLEDGMENTS

I would like to thank my advisor, Christel Rotthaus, for her patience with my all
too fathomable ignorance, Michigan State University’s commutative algebra group
for always letting me tag along, my father for his perseverence in encouraging me to
persevere, and my wife for love and for laughter.

iv

TABLE OF CONTENTS

INTRODUCTION 1
1 Preliminaries 5
2 Etale Extensions and Their Normalizations 17
2.1 The Splitting of Minimal Prime Ideals Across A" ¢—+ A’” ....... 17
2.2 Characterizing Etale Intermediate Rings ................ 22
3 Etale and Unramified Intersections 41
3.1 Etale Intersections Using Idempotents ................. 41
3.2 Intersections over Henselian Rings .................... 54
BIBLIOGRAPHY 80

Introduction

In this work we study the etaleness of intermediate rings A g D g A’” through an
analysis of properties of their normalizations, where A is an excellent reduced local
ring with strict Henselization A’” and where D is a local ring dominated by A’”. C.
Rotthausbegan the study of such rings, as part of a general project with W. Heinzer
and S. Wiegand, in [R03] where she looked at intersections LflA" under the Henseliza-
tion A" of A, where L is an intermediate ring Q(A) g L g Q(A") which is finitely
generated over Q(A). Her work utilizes some powerful connections between interme-
diate rings and their normalizations. In this work we generalize such techniques in
the strict Henselization setting and prove that some nice properties of intermediate
rings are determined by the residual field structure of their normalizations.

We prove in section 2.1 that the residual fields of the normalization A of A de-
termine the splitting of minimal prime ideals across the extension A” —> A’” and
observe that there is no splitting of minimal prime ideals if and only if the residual
field of A is purely inseparable in each of the residual fields of A. While it is known
that normalization and Henselization commute for A, Ah E” A’I. This is not so for
strict Henselization. The most that can be said in general for an excellent reduced
local ring A is that A“ is a localization of [A]; at certain maximal ideals (18.8.10 of
EGAN). However, we observe in section 2.1 that A’” E” A75 exactly when there is no
splitting of minimal prime ideals across the extension A" —> A’”

In section 3.1, we generalize a result of C. Rotthaus to state that for A an excel-

 

lent reduced normal local domain, any normal local intermediate ring A Q D Q A’”
dominated by A’” is a direct limit of local etale extensions of A (that is, of etale, local,
and essentially of finite type extensions). This is an astonishingly powerful statement
when one considers the strong hypotheses necessary in broader situations for an ex-
tension of normal domains to be etale, typically requiring at least unramifiedness in
codimension one as in the body of work on purity of branch locus. We use this state-
ment to prove that for A an excellent reduced local ring and A Q D Q A’” a local
intermediate ring dominated by A’” and having total quotient ring finitely generated
over that of A, D is a local etale extension of A if and only if the normalization of D
has an appropriate residual field structure.

In chapter 3 we return to the issue of the intersection rings of C. Rotthaus. In
section 3.1 we prove that for Q(A) Q L Q Q(A") an intermediate ring obtained by
adjoining finitely many idempotent elements to the total quotient ring Q(A) of A,
the intersection LflA" is a local etale extension of A. This says that there is a unique
minimal local etale extension A Q E Q A" having any possible splitting of minimal
prime ideals. We present an example to show that this statement does not hold with
A“ in place of A". Thus the minimal prime ideal structure of an intermediate ring
A Q D Q A’” can be an obstruction for such a ring to be etale. This suggests that
the theory for A’” is nicer upon reduction to the case of domains.

In section 3.2 we prove in a difficult theorem that for an intermediate ring A“ Q
D Q A’” finitely generated over A" and p E Min(D), D/ p is birationally dominated
by an unramified extension of A which is exhibited as an intersection. Specificaly,
we prove the following. Let A be an excellent Henselian reduced local ring, P 6
Min( A’” ) and p = P n A’”. Then for k(p) Q L Q k(Q) an intermediate field which
is finitely generated over k(p), the intersection k(p) fl (AM/ Q) is a local domain which
is finite and unramified over A.

We use this intersection theorem to characterize etaleness and Noetherianness of

intermediate rings Ah Q D Q A’” and give an easy application to homomorphic
images of power series rings over Q.

Intersections of the form studied in chapter 3 are closely related to a broad class
of rings studied by W. Heinzer, C. Rotthaus, J. Sally and S. Wiegand. They observe
that many of the well-known examples of rings with bad properties are intersections
of the form Q(S) (‘1 (S /a), where S is a localization of a polynomial ring over a
field and where a Q S is an ideal. Indeed, Nagata’s first examples of non—excellent
rings and Ogama’s example of a noncatenary pseudogeometric normal domain are
obtained in this way ([N], [01]), as well as many other other examples ([Rol], [R02],
[BRl], [BR2], [W]). In light of this, the four are developing an extensive theory of
such intersections in their recent papers [HR], [HRS], [HRWl], [HRW2] and [R4].

Conventions and Notations:

All rings are commutative with unity. For any ring R, Min(R) represents the set
of minimal prime ideals of R and Max(R) represents the set of maximal ideals of R.
For any prime ideal P of R, k(P) is the quotient field of R/ P. If S is an overring of
R then IntCloss(R) represents the integral closure of R in S. If R is a reduced ring
with finitely many minimal prime ideals, K R is used to represent the total quotient
ring of R, and R represents the integral closure of R in K 3. Note that K R is a finite
product of fields. If R is a local ring, then m}; is the maximal ideal and k3 := R/ m R
(where the symbol :2 always means is defined to be or is by definition).

A morphism R —¢-) S of rings is said to be essentially of finite type if S is isomor-
phic as an R-algebra to a localization of a finitely generated R-algebra. If R and S
are semilocal rings, then 45 is said to be a semilocal morphism if for every maximal
ideal N of S, ¢‘1(N) is a maximal ideal of R, and if every maximal ideal of R can
be obtained in this way.

For a semilocal ring ( R, (m1, . .. ,mn), (k1, . . . , kn) ) we define the residual field

 

 

product to be
k3 := Hk, g R/Rad(R)

i=1

A semilocal morphism R ——) S of semilocal rings induces a canonical morphism

kR g) kg of residual field products given by

R/Rad(R) «a S/Rad(S).

CHAPTER 1

Preliminaries

Definition 1.1

i) A morphism A ——> B is said to be etale if it is O-unramified (unramified) and
O-smooth (smooth) as in chapter 9, section 25 of [M]. This definition imposes

no requirement on the manner in which B is generated over A.

ii) A local morphism A —-> B of local rings is said to be a local etale extension
if it is etale and essentially of finite type. If in addition A ——> B is residually

trivial, we say that B is an etale neighborhood of A.

iii) A semilocal morphism A ———> B of semilocal rings is said to be a semilocal etale

extension if it is etale and essentially of finite type.

iv) A local morphism A ——) B of Noetherian local rings is said to be regular if
it is flat and if for each prime ideal p Q A and each finite field extension L
of Q(A/p), the ring B ®A L is regular. In'particular, the fibers of a regular

morphism A -——) B are regular.

Remark 1.2

i) There is little consistency in the literature about the definitions of unramified

and etale, some authors requiring additionally that B be of finite type over A.

5

iii)

The above definition 2') of Matsumura is the most general and makes no such

assumptions.

Unramifiedness is defined in [M] by the uniqueness of lifts of certain morphisms
and this definition is equivalent to the vanishing of the module of differentials.
That is a morphism A ———> C of rings is unramified if and only if SIC/A = 0. If
an unramified morphisms A ——> C factors through an A-algebra B, then the

morphisms A —> B -——> C induce an exact sequence

QB/A ®B C —-> QC/A —> QC/B —> 0

of modules of differentials (Thm. 25.1 of [M]). Then since SIC/A = 0, we also
have (20/3 = 0. Therefore C is unramified over B. Thus we may always lift

unramifiedness to unramifiedness over intermediate rings.

An unramified, essentially of finite type morphism of Noetherian rings is etale

if and only if it is flat (Prop. 111.23 of [1]).

For a morphism A ——) B of Noetherian local rings smoothness is a stronger
condition than regularity, but if in addition B is essentially of finite type over

A then the notions of smoothness and regularity are equivalent. (Thm. 28.7 of

[M] and Prop. 111.33 of [1]).

For a semilocal etale extension A ——) B, A is normal (reduced) if and only if B
is normal (reduced). To see this observe that for each maximal ideal N of B,

the morphism ANnA —> BN is regular and then apply Theorem 32.2 of [M].

Let A be a ring and let

(R).

where f E A[X] is a monic polynomial and f’ denotes its formal derivative. Then
B is an etale A-algebra (ch. 2, Prop. 8 of [Ra]). The following important theorem
states that all local etale extensions are localizations of etale A-algebras of this form.
In particular, local etale extensions are all essentially finite extensions. The proof

relies on Zariski’s Main Theorem and can be found in [Ra].

Theorem 1.3 (Local Structure Theorem) Let ( A, m ) be a local ring and let A —>
B be an essentially of finite type morphism of local rings. Then B is a local etale

extension of A if and only if there is an isomorphism of A-algebras

B (LXI) ,
(f ) Q
where f E A[X] is a monic polynomial and where Q is a maximal ideal of A[X]

containing m but not containing f’. More generally, B is unramified over A if and

only if B is a homomorphic image of such a structure.

Definition 1.4

i) A local ring ( A, m, k ) is said to be Henselian if it satisfies Hensel’s lemma,
that is if whenever a monic polynomial f E k[X] factors f = g7: as a product
of monic relatively prime polynomials g and h, then there exist monic lifts f,

g, and h E A[X] of f, g, and h respectively such that f = gh.

ii) A local ring is said to be strictly Henselian if it is Henselian and if its residual

field is separably closed.

iii) A semilocal ring is said to be Henselian (resp. strictly Henselian) if it is a finite

product of local Henselian rings (resp. local strictly Henselian rings).

Remark 1.5 A definition equivalent to i) above says that a local ring A is Henselian

if every finite A-algebra decomposes as a product of local rings (ch. I of [Ra]).

If A is a Henselian local ring, B a finite A-algebra and B,- one of the local factors
of B, then any finite Bi-algebra is also a finite A-algebra and so decomposes as a
product of local rings since A is Henselian. It follows that each local factor B,- of B
is Henselian and thus that B is a Henselian semilocal ring.

If A is an excellent Henselian local domain, then its normalization A is a finite
extension of A and thus a product of Henselian local rings by the above discussion.

Since A is also a domain, it must be local.

We give a brief summary of the construction and basic proporties of Henselizations
and strict Henselizations. For a more detailed treatment, see [Ra].

A local ring is Henselian if and only if it has no nontrivial etale neighborhoods,
and strictly Henselian if and only if it has no nontrivial local etale extensions. Thus
to construct a canonical Henselian (resp. strictly Henselian) ring from a local ring
( A, m, k ), it is reasonable to try to close it with respect to etale neighborhoods (resp.
local etale extensions). This is possible because the Local Structure Theorem gives a
system of representatives of the etale neighborhoods (resp. local etale extensions) of
A.

It can be shown that the system of representatives of etale neighborhoods forms a
direct system whose limit is a Henselian local ring. This ring is called the Henseliza-
tion of A and is denoted A". The extension A —) Ah is local, flat, regular (if A is
Noetherian), residually trivial, and m generates the maximal ideal of A“. It follows
that A and Ah have the same completion A.

To speak of a direct limit of local etale extensions of A, we must consider the
representatives of the local etale extensions together with morphisms of residual fields.
Let Q be a separable closure of k in some algebraic closure. Then the system of couples

(EA, {Ml/\em where E',\ is a local etale extension of A of the form given in the Local

Structure Theorem and having residual field k5,, and where

C-YAIkEAHQ

is a k-morphism, is a direct system whose limit is a local, strictly Henselian ring with
residual field {2, called the strict Henselization of A and denoted A’”. Note that for
each /\ the canonical morphism EA —-) A’” induces the k-morphism c’z,\ on residual
fields. For an arbitrary local etale extension E of A there is no canonical morphism

E ——> A’“. Indeed, by Proposition 1.10 we have an isomorphism

HOIIIIOCA(E, AM) ":1! Homk(kE, 9).

Thus to specifiy a A—morphism E ——) A’” one must choose a morphism of residual
fields.

Though the construction of A’“ seems to depend on the choice of (2, any two
such constructions are isomorphic. The morphism A —) A’” is similarly local, flat,
regular (if A is Noetherian), and m generates the maximal ideal of A”. We also have
that A and Ah have the same strict Henselization A’”, so that A ——> A’“ factors
through A".

We now turn to some properties of local etale extensions which will be very useful

in proving the results of the following sections.

Proposition 1.6 Let A be a reduced semilocal ring and E be a semilocal etale emten-

sion of A. Then

i) The total quotient ring K E of E is finite over K A and there is an isomorphism

KE§E®AKA.

ii) The normalization E of E is given by E 91 E (8A A.

10

Proof:

i) By Remark 1.2, E is reduced. By flatness of E over A, every regular ele-
ment of A is regular in E. Thus E Q E (8,; KA. By base change E ®A KA is
etale over K A and hence reduced. Let Q be a prime ideal of E ®A K A and set
q := Q 0 KA. Then ring (E (8,4 KA)Q is a local etale extension of the field (KA)q.
Since local etale extensions are essentially finite extensions, (E ®A K A)Q has di-

mension zero. It follows that E ®A K A is zero—dimensional and thus isomorphic to K E.

ii) Now E —~> E ®A A is injective by faithful flatness of E over A. Since A is an
integral extensions of A, E (8,; A is an integral extension of E. But since E ®A A is
etale and essentially of finite type over the normal ring A (ch. II, Prop. 2 of [Ra]),
E ®A A is normal (ch. VII, sect. 2, Prop. 2 of [Ra]). Thus since by i) we have

injections

EHE®AAL+E®AKA§KE

we must have E (8),; A E’ E. E]

Remark 1.7

i) It follows from i) of the above proposition that for A a reduced semilocal ring and
B a direct limit of semilocal etale extensions of A having finitely many minimal
prime ideals, the extension of total quotient rings K A —+ K B E B (8),; K A
is integral. Fhrthermore, for any intermediate ring A Q D Q B, the induced
morphism K A -—> D ®A K A —> B (8),; K A ’5 K B is an integral extension of
reduced zero dimensional semilocal rings. Hence the total quotient ring K D of

D is given by K D E’ D (8,; K A and is also an integral extension of K A.

11

ii) By ii) of the above proposition, it follows that for A a reduced semilocal ring,
the normalizations of Ah and A’“ are obtained by application of ®AA. If
in addition A is Noetherian, then A is semilocal by the Mori—Nagata integral

closure theorem (sect. 33 of [M]). Then by 18.6.8 of [EGAN],

~
~

Athh®AA

II?

A”.
It follows that there is a bijection

Min(Ah) <——> Max(A).

Proposition 1.8 Let A be a Henselian local ring. Then every local etale extension

E of A is finitely generated as an A-module.

Proof:

By Theorem 1.3, E is a localization at a maximal ideal of a finite extension F of
A, E = Fq. Since A is Henselian, F is a finite product of local rings, F = 1'12;le
Thus E = FQ = F, for some i. Then since E is a summand of a finite extension of

A, E is finite over A. E]

Remark 1.9 Let ( A, m, k ) be a Noetherian local ring and let E be a local etale

extension of A. Since
(Ah®AE) ®Ak§E®Ak§kE

is a field, m generates a maximal ideal Q of Ah (8),; E. Then (Ah ®A E) Q is a local
etale extension of Ah and hence by the previous proposition, a finite extension of

A”. Thus (Ah (8,; E) Q is a Henselian ring. Since it is also a direct limit of etale

12

neighborhoods of E, we must have
h Q h

In particular, Eh is etale and finite over A“.

The following useful proposition (ch. VIII, sect. 1, Prop. 1 of [Ra]) is used
again and again in this work to obtain morphisms from local etale extensions of a
Henselian local ring to other finite extensions, or to prove that two such morphisms

are isomorphic.

Proposition 1.10 Let A be a Noetherian local ring, and let E and C be local A-
algebras dominating A and with E a direct limit of local etale extensions of A. Let k ,4,

kg and kc be the respective residual fields. Then the canonical map
(I) I HomzocA(E,C) —) HomkA (kg, 160)

is injective. Furthermore, if C is Henselian, then (I) is an isomorphism.

Proposition 1.11 Let ( A, m, k ) be a Noetherian local ring. Then for any finite
separable field extension I of k, there is a finite local etale extension E of A with
residual field kg 9—“ l as k-algebras. Furthermore, if A is Henselian then E is unique

up to isomorphism.

Proof:

Write l E“ k[X]/(f) where f E k[X] is a monic, irreducible and separable polyno-
mial. Let f 6 A[X] be a monic preimage of f. Set E := A[X] / (f) Since E is finite
over A, every maximal ideal of E lies over m. But by construction, E (8),; k ’3—_’ l. Thus

E is local with residual field l.

13

Let Q be the preimage in A[X] of the maximal ideal of E so that

Since f is separable, f and f’ are relatively prime in k[X]. Thus for some a and o in
k[X] we have 11f + of' = 1. Then taking arbitrary preimages u and v in A[X] of a

and a respectively, there exists w E mA[X] such that
u f + v f’ = 1 + w

Since 10 and f are contained in Q, 1 — v f’ E Q. Thus f’ is not in Q. By the Local
Structure Theorem, E is a local etale extension of A.
Now suppose A is Henselian and that F is another local etale extension of A with

residual field kp E’ l as k-algebras. Then there is a k-isomorphism

By Proposition 1.10, since E and F are Henselian and local etale over A, we have

isomorphisms

HomzocA(E,F) E Homk(kE,kF) and

HomlocA(F, E) ”E Homk(kp,kE).

Thus {/3 lifts to a local A -morphisms ¢ : E —> F, and 4—5“ lifts to a local A-
morphism 2b : F ——> E. Then by construction, the composite 1de : E ——) E induces

the identity on kg, and (151/) : F —> F induces the identity on kp. But since again by

14

Proposition 1.10

HOIIIIOCA(E,E) 9—" Homk(kg,kE) and

HOIIIlocA(F, F) '5 Homk(kp, kp)

it follows that wcb is the identity on E, and ow is the identity on F. Therefore d) is

an isomorphism. [:1

Remark 1.12 More generally, one can argue in a similar fashion that two direct
limits of local etale extensions of a Henselian Noetherian local ring ( A, m, k ) having

k-isomorphic residual fields are isomorphic.

The following proposition uses the module of differentials and André homology to

characterize certain etale extensions.

Proposition 1.13 Let R 32—) S be an essentially of finite type morphism of Noethe-
rian rings. Then gt is etale if and only if 95/3 = 0 and H1(R, S, M) = 0 for all
S -modules M.

Proof:

Suppose S is etale over R. Then since S is unramified over R, 95/12 2 0 so that

for all S—modules M we have
0 = HOIDR(QS/R, M) g DerR(S, M) ’5 H0(R, S, M)

(ch. VI, Prop. 3 of [A1]). By smoothness of S over R we also have that H1 (R, S, M) =
0 for all S modules M (ch. XVI, Prop. 17 of [A1]). Since S is essentially of finite

type over R, so is S (83 S and hence is Noetherian. Then since

0 = H°(R, s, M) = H1(R, s, M)

15

for all S—modules M, we may apply (ch. XV, Prop. 21, 2 => 4 of [A1]) to obtain that
H1(R, S, M) = 0 for all S-modules M.

Conversely, assume 95/1: = O and H1(R, S, M) = 0 for all S-modules M. The
former condition is equivalent to the unramfiedness of S over R. The latter condition
implies that S is flat over R (ch. XV, Cor. 20 of [A1]). Thus S is etale over R by
Remark 1.2. C]

Proposition 1.14 Let A be a Noetherian ring, and let E and F be two etale, essen-

tially of finite type A-algebras. Then any morphism E ——> F of A-algebras is etale.

Proof:
We consider the Zariski-Jacobi sequence induced by the morphism of A-algebras

E ———> F and by an arbitrary F—module M:

H1(A,F, M) ——> H1(E,F, M) ——> {IE/A @913 M ——->

QF/A ®FM Hap/E ®FM —) 0

(ch. V, Thm. 1 of [A1]). By unramifiedness of F over A, we have QF/A = 0. Thus
Sip/E 8);» M = O for all F-modules M, implying that OF”; 2 0. Using the previous
proposition, by etaleness of F over A and of E over A we have H1(A, F, M) = O and
STE/A = 0. Thus

0 ——) H1(E,F,M) —> 0

is exact and so H1(E, F, M) = 0 for all F-modules M. Then since

0 = 52m = H1(E, F, M),

16

E —> F is etale by the previous proposition. Cl

Remark 1.15 Let ( A, m, k ) be a complete Noetherian local ring. A coefficient
ring R Q A is a field or a homomorphic image of a complete p-ring (a DVR whose
maximal ideal is generated by a prime number p), which satisfies A = R+m and such
that the local morphism R e—> A induces an isomorphism on residual fields. Every
complete Noetherian local ring A has a coefficient ring. For a detailed discussion of

this material, see section 29 of [M].

Proposition 1.16 Let ( A, m, k ) be a complete Noetherian local ring and let R be
a coefficient ring for A. Then for any local etale extension ( E, mE, kg ) of A there

is a local etale extension Sg of R with residual field kg and an isomorphism

Under this isomorphism, Sg is identified with a coefl‘icient ring of E.

Proof:

Fix a local etale extension E of A. By Proposition 1.11, there is a local etale
extension Sg of R with residual field kg. Since R is also Henselian, Sg is a finite
R—module by Proposition 1.8. Thus A 83 S g is finite and etale over A. In particular,
every maximal ideal of A 8);; S g lies over m. Since R has residual field k and since

the maximal ideal of R generates that of SE, we have

k®A(A®RSE)gk®RSEng.

Thus A (8);; S g is a local etale extension of A with residual field kg. Then by Propo-
sition1.11, E 34 A ®R SE. B

CHAPTER 2

Etale Extensions and Their

N ormalizat ions

2.1 The Splitting of Minimal Prime Ideals Across
Ah <—> Ahs

Lemma 2.1.1 Let k be a field, and let K and L be two algebraic field extensions of
k with K purely inseparable over k and L finite separable over k. Then K (8),, L is a

field of degree over k given by
[K®kL:k]=[L:k][K:k]

Proof:

First we argue that K 8),, L is a field. We may assume neither K nor L is k.
Since L is a finite separable field extension of k we may write L = k[X ] / (g), where
g E k[X] is a monic, irreducible and separable polynomial of positive degree. Let F
be a splitting field for g over k in some algebraic closure of K. Since K is purely
inseparable over k we must have K n F = k.

Now if g factors over K, g = hq where h, q E K [X] are monic polynomials of

17

18

positive degree, then the coefficients of h and q are in K n F = k, implying that 9

must factor over k, a contradiction. Therefore g is irreducible over K, and so

as.

K® LE
k (g)

is a field.

Now to see the statement about degree, observe that
[K®kL:k] =[K®kL:K][K:k].

But by the above discussion [K (8);, L : K] = deg(g) = [L : k]. E]

Lemma 2.1.2 Let ( A, m, k ) be an excellent Henselian domain with normalization

( A, rh, k ) The following are equivalent:
i) k is purely inseparable over k.
a) 2i": 2’ 2171? .
iii) A’” is a domain.
Proof:
(i 4:) iii) : This is ch. IX, sect. 1, Cor. 1 of [Ra].
(ii => iii) : Since A is a Henselian local domain, A is a normal local domain (ch.

IX, sect. 1, Cor. 1 of [Ra]). Then A’” is also a normal local domain (ch. VII, sect.

2, Prop. 2 of [Ra]). So by ii), AT"; is a domain and hence A“ is a domain.

19

(iii => ii) : If A’” is a domain, then A773 is a local domain. By 18.8.10 of [EGAN],
A’” is a localization of A}; at some maximal ideals. Since AI”: is already local these

rings are isomorphic. Cl

Theorem 2.1.3 Let { A, m, k ) be an excellent reduced local ring with normalization

(A, (m1, ,mn), (k1, ,kn) ). The following are equivalent;

i) For each i, k,- is purely inseparable over k.
a) A“ 9: I178.
iii) The minimal prime ideals of A’” are extended from A”.

Proof:

(ii => iii) : Note that A71; is a finite products of normal local domains and thus has
the same number of maximal ideals as minimal prime ideals, which also correspond
bijectively with the minimal prime ideals of A’”. By definition of the strict Henseliza-
tion of a semilocal ring, A’” and A have the same number of maximal ideals. Thus

by ii) we have the following bijections:
Min( A’” ) <—+ Max( 21753 ) <—> Max( 21'") ) 4—) Max( A ).

But then using the bijection Max( A ) <—> Min(Ah ) (ch. IX, sect. 1, Cor. 1 of [Ra]),
we obtain that A’” and Ah have the same number of minimal prime ideals. Thus for
P a minimal prime ideal of A”, PA’” is contained in exactly one minimal prime ideal

of A’”. But

IIZ

Aha A ’13
PA“ (F)

20

is reduced (ch. VII, sect. 2, Prop.1 of [Ra]). Therefore PA’” is a minimal prime
ideal of A’”.

(iii => ii) : Arguing as above using iii) for the second bijection, we have

Max( A’gh—a Min(A’” ) <—> Min(A")<—>

Max( A ) <—> Max( A’” ).

By 18.8.10 of [EGAN], A’” is a localization of A7; at some maximal ideals. Since

both rings have the same number of maximal ideals, they are isomorphic.

(i 4:} iii) : By Remark 1.7, A; ’5 A”. Thus the maximal ideals of A; correspond
bijectively with those of A and furthermore, A; has residual fields k1, . . . ,kn. Thus
we may assume A is Henselian.

Since A is Henselian, A is a product of normal local domains. Thus there is a

~

bijective correspondence Min( A ) <———) Max( A ). Let Min( A ) = {P1, . .. ,Pn}.
Then Ai/P, ’E Am, and has residual field lei. By Lemma 2.1.2, k,- is purely inseparable

over k if and only if
(A/Pi)hs g Airs/RAM;
is a domain, which holds exactly when RAM is a minimal prime ideal of A’”. The

result follows. C]

Corollary 2.1.4 Let ( A, m, k ) be an excellent Henselian local domain with normal-
ization ( A, m, k ). Let k’ be the separable closure of k in k. Write k‘ :2 k[X]/(f),

where f E k[X] is a manic, irreducible and separable polynomial. Then

21

i) Ah“ has d = deg(f) minimal prime ideals.

ii) Let E be a local etale extension of A and fix a morphism E <—> A’”. Then the
minimal prime ideals of A’” are extended from E if and only if f splits into
a product of linear factors over the residual field kg of E. In this case, kg is

separably closed in each residual field of E.

Proof:

It is enough to show ii) and that such an E has d minimal prime ideals. Let E
be a local etale extension of A such that f splits as a product of linear factors over
kg. Now E E’ E ®A A is a semilocal etale extension of A. Thus E has residual field

product given by

E/fiiE E (A®AE)®AE
g E®AE

'5 E®kkE

IIZ

ken. k8 (59,, kg
But since f splits over kg, we have
k3 69,. kg 2 (kgy’.
Thus the residual field product of E is given by
- - d - a
k ®ka ks ®k ICE g k ®ks (’63) g (k ®ka kg) .

By Lemma 2.1.1, since it is purely inseparable over k3 while kg is separable over k‘,

k (8),. kg is a field. Therefore E has exactly d maximal ideals and the corresponding

22

residual fields are all isomorphic to k®ks kg. Since E is Henselian, E is a product of
local domains. Thus E and hence E has d minimal prime ideals
Since k ®ka kg is purely inseparable over kg, Theorem 2.1.3 applies for E so that
the minimal prime ideals of EM ’:_‘—’ A’“ are extended from E. This shows i).
Conversely, suppose the minimal prime ideals of A“ are extended from E. Since

E is Henselian, E is a product of local domains. So there are bijections
Max(E) <—-> Min(E) <——> Min(A’”).

Thus E has exactly d maximal ideals.

Arguing as before, E has residual field product k ®ka k3 (8);, kg. Since k is purely
inseparable over It”, for any separable field extension l of k’, k (8),. l is a field by
Lemma 2.1.1. Thus since k’ (8,, kg is a product of finite separable field extensions of
k‘, the number of fields in the product k®ka (k3 (8);, kg) is the same as the number of
fields in the product ks (8v. kg. Since E has d maximal ideals, k3 <81: kg is a product

of d fields. But this can only happen if f splits as a product of linear factors over kg.

E]

2.2 Characterizing Etale Intermediate Rings
The following lemma relates some of the hypotheses which appear in this section.

Lemma 2.2.1 Let A be a reduced local ring, and let A Q D Q A'” be local intermedi-
ate ring dominated by A’”. Then the total quotient ring K g of D is finitely generated
over K A if and only if the containment D Q A” factors through some local etale
extension of A. Furthermore, the above equivalent conditions hold if D is essentially

of finite type over A.

23

Proof:

Suppose that Kg is finitely generated over K A. Then since K Ah. 9—: A’” 69,; K A
is a direct limit of the total quotient rings of local etale extensions, Kg is contained
in the total quotient ring of some local etale extension E of A. But then by faithful

flatness of E —> A’”, we have

DQKgflAhszE.

On the other hand, suppose D is contained in some local etale extension E of A.
Then we have containments K A Q Kg Q Kg of total quotient rings where Kg is
finite over K A by Proposition 1.6. Thus Kg is also finite over K ,4.

Suppose that D is essentially of finite type over A. Let d1, . .. ,d, E D be such
that D is a localization of A[d1,... ,dr], and let E be a local etale extension of
A such that A[d1,. .. ,dr] Q E. Then since D is dominated by A’”, if we localize
A[d1,. .. ,d,] at the preimage of the maximal ideal of E, we obtain D. Therefore,
D Q E. El

Remark 2.2.2 Let A be a reduced local ring and A Q D Q A’” an intermediate
ring with total quotient ring Kg finitely generated over K A. Since by Remark 1.7,

Kg ’.-‘=’ K A 8),; D is integral over K A, we may write

Kg = KA[d1,... ,d.)

where the d.- E D are integral over A. In particular it follows that the integral closure

of A in D contains the elements at,- and hence has total quotient ring Kg.

The following proposition is a generalization of (1.3) of [R03], and although the

statement below seems much more general, the proof is essentially the same.

24

Proposition 2.2.3 Let ( A, m, k ) be an excellent normal local domain, and let
A Q D Q A’” be a normal local intermediate ring dominated by A’”. Then D is a
direct limit of local etale extensions of A. Furthermore, if in addition the quotient

field Kg of D is finitely generated over K A, then D is a local etale extension of A.

Proof:

First we assume that Kg is finitely generated over K A.

Let Co be the integral closure of A in Kg. Then by Remark 2.2.2, Co is a normal
domain with quotient field Kg. Since D is normal, we also have Co Q D and so we
may let C Q D be the localization of Co at the maximal ideal which lies under the
maximal ideal of D. Then C also has quotient field Kg. Since A is excellent, C0 is a
finite A-module and hence C is a local normal domain which is essentially finite over
A.

By Lemma 2.2.1, D is contained in some local etale extension E of A. So we now

have containments
A Q C Q D Q E.

Set r :2 dim(A). Since C is essentially finite over A, dim(C) S r. On the other
hand, since E is essentially finite over A, E is also essentially finite over C. Thus
dim(C) Z dim(E) = r and so dim(C) = r.

Considering the canonical surjection C ®A E —> E, we localize at the preimage

Q of the maximal ideal of E to obtain a surjection
(C ® A E)Q it E.

Note that since E is etale over A, (C ®A E)Q is a local etale extension of C. Then

since C is a normal domain of dimension r, (C ®A E)Q is also a normal domain of

25

dimension r. Thus ()5 is a surjection between local domains of the same dimension r
and so must be an isomorphism. So E E“ (C ®A E)Q is a local etale extension of C.
Finally observe that by faithful flatness of C —-+ E and since C has quotient field

Kg,wehave
C=KgflEQD.

Therefore C = D, concluding the proof in the case where Kg is finitely generated
over K A.
For the general case, making no assumption about how Kg is generated over K A,

we write D as a direct limit of local, essentially of finite type A-subalgebras dominated

by D,

D = 1199 DA = U 0..
AEA AEA
Since A is excellent, so are the DA. Hence the normalization D; is a finite extension

of DA, and so is also essentially of finite type over A. Since D is normal we have

D = lim D).
)3
Then localizing each D; at the maximal ideal m) which lies under the maximal ideal

of D, we have

0:11.. (51)

AEA m

Now for each A, (D3,) is essentially of finite type over A and thus by Lemma 2.2.1

m)

has quotient field finitely generated over K A. Thus by the previous case, since the

26

(D?) are also normal and dominated by A’”, these rings are local etale extensions
mi

ofA. [3

We now generalize to the semilocal case to obtain a result which can be applied
to normal intermediate rings A Q D Q A75, where A is an excellent reduced local
ring with normalization A. Note that since A’” 1‘:’ A’” ®A A with A’” a direct limit

of local etale extensions of A, A’” is a direct limit of semilocal etale extensions of A.

Proposition 2.2.4 Let A be an excellent normal semilocal ring, let B be a semilocal
ring which is a direct limit of semilocal etale extensions of A, and let A Q D Q B be
a normal semilocal intermediate ring such that the inclusion D ‘—> B is a semilocal
morphism. Then D is a direct limit of semilocal etale extensions of A. Furthermore,
if in addition the total quotient ring Kg of D is finitely generated over K A, then D is

an essentially finite, semilocal etale extension of A.

Proof:

As with Proposition 2.2.3, it suffices to prove the case where we assume that Kg
is finitely generated over K A. The general statement then follows using an analogous
direct limiting argument.

Let Co be the integral closure of A in Kg. By Remark 2.2.2, Co is a normal ring
having total quotient ring Kg. Since D is normal, Co Q D.

Now for M any maximal ideal of B, set n 2: MflCo and m := M n A. Then BM

is a direct limit of local etale extensions of Am and so has strict Henselization
h ~ It
(BM) 3 = (Am) 8°
Thus (00),, is a a normal local intermediate ring

A... c (00),. g (A...)"‘

27

dominated by (A...)’”. By Proposition 2.2.3, (C0)” is a local etale extension of Am.
Let C be the localization of Co at the maximal ideals lying under the maximal

ideals of B (hence also at the maximal ideals lying under D), so that
A ——> C —> D ——> B

are semilocal morphisms. Then for any maximal ideal n of C, C" is a local etale
extension of AnnA. Thus C is a semilocal etale extension of A. Since for any maximal
ideal M of B, Cram: and BM have the same strict Henselization, CMnc —-) BM is
faithfully flat. Hence C —-> B is faithfully flat. Since C has total quotient ring Kg,

wehave
C=KDnB2D.

Therefore D = C is a semilocal etale extension of A. El

Lemma 2.2.5 Let A be a reduced local ring and E be a local etale extension of A.

Let L be an intermediate ring K A Q L Q Kg. Then

i ) The intersection D := L F) E is a local ring with maximal ideal lying under that

of E and having total quotient ring Kg = L.

ii) The intersection C := L (i E is a normal semilocal ring with maximal ideals the

preimages of those of E and having total quotient ring K (5 = L.

Proof:
Note that since K A Q Kg is finite by Proposition 1.6, so is K A Q L. Thus L is

zero dimensional.

28

To prove i), let l E L. Since Kg 2’ E ®A KA, we may write l = e/s where e E E
and s is a nonzerodivisor of A. Then e E L n E = D. Thus I = 6/3 6 Kg and
therefore Kg = L. If (1 E D is a unit of E, then at“1 E L n E = D. Therefore every
nonunit of D is a nonunit of E, so that every nonunit of D is in the prime ideal
mg 0 D. Hence this ideal is the unique maximal ideal of D.

To prove ii), observe that since we also have Kg ’5 E (8,; K A the argument that
K5. = L follows just as above. If c E C is a unit of E, then c‘1 E L F) E = C. Thus
every nonunit of C is a nonunit of E and so any maximal ideal N of C is contained
in the union (m1 0 C) U ...U (m, m C) where Max(E) 2 {m1, . .. ,ms}. Then by

prime avoidance we must have N = m,- D C for some i. The statement follows. D

The following theorem, which generalizes some ideas of C. Rotthaus, exhibits the
strong connection between intermediate rings A Q D Q A’“ and their normalizations.
Under certain circumstances the intersection rings of the previous lemma are etale
extensions. The local etale extension of iii) in the following theorem is obtained by
such an intersection. The issue of etale intersections is explored in greater detail

Chapter 3.

Theorem 2.2.6 Let ( A, m, k ) be an excellent reduced local ring, E be a local etale
extension of A, and let A Q D Q E be a local intermediate ring dominated by E.

Suppose D is a semilocal ring and D L) E is a semilocal morphism. Then

i) D is an essentially finite, semilocal etale extension of A.
ii) D is essentially of finite type over A.

iii) If D L-) E is residually trivial, then D is birationally dominated by a local etale

extension of A which has the same normalization D.

29

Proof:

i) This follows immediately by Lemma 2.2.1 and Proposition 2.2.4.

ii) Let C be the integral closure of A in D localized at the preimage of the maximal
ideal mg of D. By Remark 2.2.2, C has total quotient ring Kg. Then since nor-
malization and localization commute, C is a localization of IntClos Kn (A) Since D is
essentially finite over A, D is also a localization of IntClosKD(A). It follows that D

is a localization of C at certain maximal ideals. Let
M3.X( é ) = {m1,... ,mt}

where for some 1 S s S t and

~

S:=C—(mlu...Um,)

we have D = S‘IC. Then for each s + 1 S i g t there is an element c.- E m,- with
cfl E D. Thus the cfl are integral over D and we may let {d,,-} be the coefficients of

the corresponding integral equations. Set
012: C[ {dij }]M Q D

where M is the preimage of mg. Then cfl is in C1 for 3+1 3 i g t, so that considering
the containment C Q C1, no maximal ideals of C1 lie over m3+1,. .. ,mt. Therefore

S 'lCl = C1 and thus we have

 

 

30

Since C1 is excellent, 01 ‘—+ C1 = D is finite. Since this map factors through D,
D is finite over C1 which is itself essentially of finite type over A. Thus D is also

essentially of finite type over A.

iii) Set F 2 Kg 0 E. Then F is a local ring with maximal ideal lying under that
of E, and F has total quotient ring Kg by Lemma 2.2.5. Since D and E are both
semilocal etale extensions of A and since D ——> E is semilocal, E is a semilocal etale
extension of D, by Pr0position 1.14. Hence by faithful flatness we have D = Kg 0 E.
Thus D Q F Q D. Since D is excellent by ii), D is finite over D. Hence F is finite
over D and has normalization F = D. Since D is essentially of finite type over A
and F is finite over D, F is also essentially of finite type over A. In particular, F is
excellent.

Since by hypothesis D —-+ F ——) E is residually trivial and since the maximal
ideal mg of F generates the maximal ideal of E, we have a surjection on completions
F —-) E. To see that this surjection is also injective, consider the commutative

diagram

Dr)

‘11:)
P3) —-) Dill)

“lib—'9 H

——>

Since the morphisms D —> E is a semilocal etale extension and hence faithfully flat,
the induced morphism of completions (with respect to Jacobson radicals) is injective.
Hence the top morphism in the diagram is injective. Since F is excellent and reduced,

normalization and completion commute for F and thus the left morphism

>l

'11:)
H2

F—>

31

is injective. Then by commutativity, F —-> E is also injective and thus an isomor-
phism.

Since E E’ F is faithfully flat over both E and F, we also have that E is faithfully
flat over F. But E is also essentially of finite type and unramified over F by ii) of
Remark 1.2. Thus E is a local etale extension of F. Since E is etale over both F

and A and since F is essentially of finite type over A, F is a local etale extension of

A by (1.4) of [R03]. Cl

Theorem 2.2.7 Let ( A, m, k ) be an excellent reduced local ring, and let A Q D Q

A'” be a local intermediate ring dominated by A’”. Then the following are equivalent:

i) The Henselization D" of D is a direct limit of local etale extensions of A (so
that A and D have the same strict Henselization A’”, and hence A ———-) D is a

regular morphism.)

ii) The normalization D of D is a semilocal ring and the canonical morphisms

k1,; L—> kb and kg ‘—-) k 1') of residual field products induce an isomorphism

k, ®kkg ikb.

Proof:

(i => ii) : Suppose that D" is a direct limit of local etale extensions of A. Then D"
is excellent and by faithful flatness of D —-) D“, D is Noetherian. Also, since D" is

faithfully flat over both D and A, D is faithfully flat over A. Hence we have injections

DHA®AD%KA®AD§KD.

32

Since D" is a direct limit of local etale extensions of A, Dh has normalization given
by D7‘ ’.‘—_’ A (8,; D". But A ®A D" is a direct limit of semilocal etale extensions of
A 8),; D. Thus A 8),; D —) A ®A D" is regular and so A 68,; D is normal. Then we

must have

and so ii) follows.

(ii => i) : Applying ®gkg to D ®A A we obtain

(D®A/'i)®DkD g l‘IDQEDAI‘i

knskksAA

II?

II?

Similarly, since mg generates the maximal ideal of A’”, applying ®gkg to A’” (8),; A

we obtain

~

(Aha ®A/l) (EDD/CD E kAha ®AA
g kAha ®kk®A1a

g kAhs ®k (xi/mg).
Thus applying ®gkg to the morphism

D®A.zi——>A’”®AAEZ778,

33
we obtain a commutative diagram

D®Afl —a+Ahs®Ax‘if——\=’Z7‘;

.1 it

k0 @k (A/mA) —"—+ km. a, (A/mA)

Since D®AA is finite over D, every maximal ideal of D®A A contains the maximal
ideal mg of D. Since the kernel of 45 is generated by mg, no maximal ideals are lost
under the surjection (1). So qt is a semilocal morphism between rings with the same
number of maximal ideals. An analogous statement holds for «[2. Since fl is a faithfully
flat morphism of semilocal rings, hence also semilocal, a is a semilocal morphism by
commutativity.

Let r = |Max(D®AA)|. Since a factors through D to yield morphisms D®AA ——>
D —-> A73, it follows that the maximal ideals of Al; lie over at least r distinct maximal

ideals of D. But by hypothesis,
|Max(D)| = |Max(k,, @k kg)| = IMax(A 59,, D)| = r.

Therefore D —-> A713 is semilocal. By Proposition 2.2.4, D is a direct limit of semilocal
etale extensions of A. In particular, D is excellent.

Now the residual field kg of D is a direct limit of finite separable field extensions
of k. Let {(FA, fix) : A 6 A} be a system of representatives of all couples (F, p), where
F is a local etale extension of A and p : kp —-> kg is a k-morphism. Then exactly
as in the construction of the strict Henselization [Ra], {FA}; 6 A is a direct sytem.
Let F" be the limit of this system. Note that Fh is Henselian since it may also be
obtained by taking an analogous direct limit of local etale extensions of A”. Since F h
is a direct limit of local etale extensions of A it has strict Henselization Ah’, so that

the canonical morphism F" —> A’” is faithfully flat.

34

Now there exists canonical a k-morphism kph ——> kg induced by the morphisms
fix. Since kg is a separable algebraic over k, for any d 6 kg, k[d] is a finite separable
field extension of k. By Proposition 1.11, there is a local etale extension F of A with
residual field isomorphic to k[d]. Thus letting p be the inclusion k[d] Q kg, (F, p) is
then represented in the above system. Thus (I is in the image of k p}. in kg. It follows
that the canonical k-morphism kph ——> kg is an isomorphism. By Proposition 1.10,

there is a morphism F h —£—> D" of Ah-algebras which is injective since the composite
F" 13+ D" —+ A’”

is faithfully flat. To complete the proof, we argue that (b is an isomorphism.

We consider the following commutative diagram

Ah———>Fh———>Dh——>Ah8

l l l T

Ah ,Fh 4’ a Dh aAha:

By 18.8.6 of EGAN, since D is semilocal and integral over D, D has Henselization

given by

~

Dag D" 9—: Di.

ll?

Dh

In particular D7‘ is a product of finitely many local domains. Since D is a direct limit
of semilocal etale extensions of A, its Henselization D" is a direct limit of semilocal
etale extensions of A” ”-3 A7‘ (see for example ch. VIII, sect. 3, Prop. 5 of [Ra]).
Thus D7‘ is a direct limit of semilocal etale extensions of Ah C‘-=’ A’: and by the above
isomorphisms has the same residual field product as D”, hence also the same as D.

But by hypothesis D has residual field product k A (8,, kg.

35

Making similar considerations for F" we have
F h E Ah ®Ah Fh,

and thus F3 is a direct limit of semilocal etale extensions of :47. Since A7‘ E Ah
which has residual field product given by k A: it follows from the above that F71 also

has residual field product k A (8;, kg. Therefore the morphism
a _. a

induces an isomorphism of residual field products. In particular both rings are prod-
ucts of the same number of local normal domains.

Let M be a maximal ideal of DE, Q its preimage in F3, and P its preimage
in AZ. Then (13%),” and (F700 are both direct limits of local etale extensions of
(1:17)”. But by Proposition 1.10, two Henselian direct limits of local etale extensions

are isomorphic if their residual fields are isomorphic. Thus

(Elke (571))”.

Then since F71 and D7‘ are products of the same number of local domains, they are
isomorphic. In particular, D; ——> A7; is injective and so every morphism in the
previous commutative diagram is injective by commutativity.

Now the isomorphism F71 2' D71 yields an isomorphism of total quotient rings

upon application of K A®A . Thus applying K mm to the top row of the previ-

ous commutative diagram, we obtain the following commutative diagram of injective

36

morphisms

KAh —) KFh 4) KDh -—> K23;

[ l l l

A"———>F"——£—>D"——>A’;.

Thus considered as subrings of K A)”, F h and D" have the same total quotient ring.

Hence by faithful flatness of F h -—> A’” we have
F" = KF. n A’” = Km 0 A’” :_> D".

Therefore (15 must be an isomorphism. [3

Remark 2.2.8 From the proof of the previous theorem it follows that under the

equivalent conditions i) and ii), the induced morphism of normalizations D —> A’”

is semilocal.

One would like to say that for A an excellent reduced local ring and A H D <—> A’18
local regular morphisms, D is a direct limit of local etale extensions of A. Thus
the previous theorem would give a characterization of when local intermediate rings
A Q D Q A’“ are direct limits of local etale extensions of A. However the author
does not know how to prove this.

General Neron Desingularization ([Pl], [P2], [O2], [Sp], [Sw]) gives that in the
above situation D is a direct limit of smooth A-algebras of finite type. If in fact D is

a direct limit of smooth A-subalgebras of finite type,

Daya=U$,

AEA AEA

37

then we can conclude that D is a direct limit of local etale extensions of A as desired.
To see this, we only need to observe that in this situation each localization (.S',\)mA at
the preimage of the maximal ideal of D is a local etale extension of A. Indeed, since
(S A)”,A Q A’“ is essentially of finite type over A, it is contained in some local etale
extension E of A. Then by regularity of the morphisms (.S'g)"1A ——> D —-> A’” and
E ———> A“, the inclusion (.S'g)"1A ‘-) E is local and regular, thus faithfully flat. Since
E is also unramified over (63),“ by Remark 1.2, E is etale over (Sg)m’\. Thus (ng
is etale over A by (1.4) of [R03].

It is currently unknown whether a regular morphism A —) D is a direct limit
of smooth subalgebras, even in the algebraic situation of this work, where A ——> D
induces and integral extension of total quotient rings. Spivakovsky has a proof in
his preprint that under certain circumstances a regular morphism is a direct limit of
smooth subalgebras of finite type, but this preprint is rumored to contain errors. So
this author is uncertain of the veracity of that statement.

The only statement we currently make is the following characterization of etaleness
for intermediate rings A Q D Q A’” which are contained in a local etale extension of

A or which equivalently have total quotient ring finitely generated over A.

Corollary 2.2.9 Let ( A, m, k j be an excellent reduced local ring, and let A Q D Q
A’” be a local intermediate ring dominated by A’” and whose total quotient ring Kg
is finitely generated over K A. Then D is a local etale extension of A if and only if D
is a semilocal ring and the canonical morphisms kg <—) kb and kg % kb of residual

field products induce an isomorphism
kg a, kg 3:) k ,5.

Proof:

38

(=>): This is trivial since if D is a local etale extension of A then the normalization

DofDisgivenbyDT-1D®AA.

(<2): By Lemma 2.2.1, the containment D Q A’” factors through some local etale
extension E of A. By the Theorem 2.2.7 and the subsequent Remark, the morphism

D -——+ A’“ is semilocal. Thus since the composite
D —-+ E —-> A773

is semilocal and since E ——-+ A"; is semilocal, so is D ——> E. By Theorem 2.2.6, D
is essentially of finite type over A.

Now by Theorem 2.2.7, D has strict Henselization A’”. Since E also has strict
Henselization A’”, A"8 is faithfully flat over both D and E. Thus E is faithfully flat
over D. Since E is essentially of finite type over A, E is also essentially of finite type
over D. Then since E is unramified over D by Remark 1.2, E is etale over D. Thus

E is etale over both D and A, and so D is also etale over A by (1.4) of [R03]. C]

We conclude this section with another easy characterization of etaleness for inter-

mediate rings, which does not rely on an analysis of normalizations.

Proposition 2.2.10 Let ( A, m, k ) be a Noetherian local ring, and let A Q D Q A’”
be a local intermediate ring dominated by Ah‘and whose total quotient ring Kg is
finitely generated over Kg. Suppose m generates the maximal ideal mg of D and that

either of the following conditions holds.
i) D is essentially of finite type over A.
ii) A is excellent reduced and D is Noetherian.

Then D is a local etale extension of A.

39

Proof:

By Lemma 2.2.1, the containment D Q A” factors through some local etale
extension E of A. The morphisms A —+ D —-> E induce morphisms A % D ——> E
of completions. Let R be a coefficient ring for A. By Proposition 1.11, since R is
complete and hence Henselian, there is a unique local etale extension S of R with
residual field kg. Then A (8;; S is a local etale extension of A with residual field kg.

By Proposition 1.10, since D is Henselian, we have an isomorphism
Homlocg(A ®R 5,15) ’5’ Homk(kg, ’60).

Hence the identity morphism on residual fields lifts to a morphism A (8);; S —> D.
Since E is a localization of E (8),; A, E is a local etale extension of A. Thus since the

composite
A ® R S —-) D —-> E

is etale, A (8);; S ¢—> D is injective.

Now by hypothesis the maximal ideal of A @R S generates that of D. Thus
since A (8);; S ‘—> D is a residually trivial morphism of complete rings, it follows that
A (83 S 3) D is an isomorphism. Hence D —> E is etale. Note that with either
of the hypotheses i) or ii) we have that D is Noetherian, and so D ——> D ——> E is
faithfully flat. Since E is faithfully flat over both E and D, E is faithfully flat over
D. Since E is also unramified and essentially of finite type over D, D ——> E is a
local etale extension.

Assume that i) holds. Since E is etale over both A and D, D is a local etale

extension of A by (1.4) of [R03].

40

Now suppose ii) holds. Since E is etale over D, we have E E” E ® g D. So E is a
semilocal etale extension of D. By Theorem 2.2.6, D is essentially of finite type over

A. Thus again by (1.4) of [R03], D is a local etale extension of A. D

CHAPTER 3

Etale and Unramified Intersections

3.1 Etale Intersections Using Idempotents

Let A be an excellent reduced local ring. In this section we study intersections
D := L n A’” where K A Q L Q K Aha is an intermediate ring obtained by adjoining
finitely many idempotent elements of K ,4». to K A. We find that if L Q K Ah then the
intersection D is a local etale extension of A, but without this hypothesis, it need not
be the case that D is etale. The difference arises in whether A ———> D is residually
trivial.

We begin with some elementary results which are mostly due to

C. Rotthaus.

Proposition 3.1.1 Let ( A, m, k ) be a Noetherian reduced local ring which is not
Henselian and let F be an etale neighborhood of A. Then there exists a regular element

a 6 A such that for all n E N we have

A F
a"A anF'

IIZ

 

 

In particular, A and F have the same a-adic completion.

Proof:

41

42

By Theorem 1.3, there exists a E A” such that F is a localization of A[a]. By

(1.8) of [R03], there is an etale neighborhood

<——->

of A containing a and satisfying:

oxeo,

b) a := f(0) 6 m is a regular element,

deD¢m

Then modulo aA[X], f E X g for some g ¢ Q. It follows that E /aE E’ A/aA. Also,

localizing the inclusion
A[a] <—) E

at the preimage of the maximal ideal of E, we obtain F ¢—> E. Then since the

morphisms A ¢—> F H E are etale and hence faithfully flat, we obtain morphisms
A/aA g> F/aF H E/aE 2’ A/aA.

Therefore A/aA 21 F / aF . It follows that F is given by
F=A+aF=A+aA+a2F, etc.

Thus

“2

 

 

a"A anF'

43

Corollary 3.1.2 Let ( A, m, k ) be an excellent reduced local ring, and let K A Q L Q
K A» be an intermediate ring which is finitely generated over K A. Set D := L n K Ah.

If D is Noetherian, then D is an etale neighborhood of A.

Proof:

If A is Henselian, the result is trivial. So assume A is not Henselian. By
Lemma 2.2.5, D is a local ring with maximal ideal lying under that of A”, and D
has total quotient ring L. By Lemma 2.2.1, the containment D Q Ah factors through

some etale neighborhood F of A. Then by faithful flatness of F —> A", we have

D=LnAh=LnKpnAthflF

As in Proposition 3.1.1, let a E A be a regular element such that for all n 6 N we
have A/anA = F/anF. Since D = LnF, it follows that for all n E N, anFflD = anD.

Thus we have inclusions

A/a"A <—) D/a"D L) F/a"F = A/anA,

so that A/a"A = D/a"D = F/anF. Therefore F and D have the same a—adic
completion (A, a)". Since (A, a)" is faithfully flat over both F and D, F is also
faithfully flat over D (and hence, similarly, D is faithfully flat over A). Since F is

also unramified over D by Remark 1.2, D —-> F is etale. Then

~

D®Afl—)F®Afl§F

is a semilocal etale extension and hence is faithfully flat. Thus D ®A A is a normal

ring. Since we have inclusions

Dgoagfigpagma’m},

44

we must have D § D ®A A, and so D 9 F is etale and thus semilocal. By
Theorem 2.2.6, D is essentially of finite type over A. Then by (1.4) of [R03], since F

is etale over both D and A, D is etale over A. [3

Proposition 3.1.3 Let ( A, m, k ) be an excellent reduced local ring and let
61, . .. ,6, 6 Kg). be idempotent elements. Set L :2 KA[€1,... ,er] and D := L n A".

Then D is etale over A.

Proof:

It suffices to show this for r = 1 and we may assume that e = 61 is different from

Write
Min(A) = {P1, . .. ,Pn},
and for each i let k(P,-) be the quotient field of A/P,. Also write
Min(A") = {P11,... ,PM,... ,Pn1,... ,Pndn},

where Pij n A = P,, and similarly for each i and j let k(P,-J-) be the quotient field of
Ah/Pz'j. Then

n d.-
KA. r—z Huang,

i=1 j=1

and we may identify 6 E KAI: with an element (e11, . . . , e1“, . . . ,en1,. . . , end"), where
for each i and j, eij is either 1 or 0. Under this identification, eij = 0 if and only if

k(P,-J-)e = 0. We may adjust the indexing if necessary, so that for each 1 S i S n

45

there exists an s,- with 0 S s,- S d,- such that
en 2 ---=e,r,,i = 1 and e,-,,,+1 = -~=e,-d, =0.

Then KAI: E“ the x th(1 — e) where we now have

Kg... '5 Hflkmj)

i=1 j=l

and
n d,
KAh(1—€)~H I—I k(P (Pij)'
i=1 j: —s.+l
Now

L :2 KA[€]
E” KAexKA(1—e)
H[k(H) )(Pexk )(1—6)].

II?

It is possible in the above product that k(P,-)e = 0, that k(P,)(1 — e) = 0 or that

neither are zero. The case where k(P,-)e = 0 occurs if we have s, = 0 so that en =

- = eid, = 0, and analogously the case where k(P,-)(1 — e) = 0 occurs if s,- = d,- so

that en = = 8w,- = 1. Note that the inclusion L <—> K Ah is given by the product

of all the diagonal maps

46

and

di
k(HXl—e) —> II k(P.»
x(1—e) r—> (x,‘x,...,x).

Since the minimal prime ideals of A" correpond bijectively with the maximal ideals

of the normalization A of A by Remark 1.7, we may write
M83414) 2 {777.11, . . . ,m1d1,. . . ,mn1,. . . ,mndn},
where m,,- corresponds to the minimal prime ideal 13,-,- of Ah so that
"' h ~ h "
(Am) = (A /P,-,) .

Note that 71/73, has maximal ideals generated by the images of m,1,. . .mid“ so that
the maximal ideal mij of A contains only that minimal prime ideal of A which lies

over P,- Q A. Now set

S, =2 A — U mu
j=l
and
d."
7: = A — U m”
j=3i+1

If 3,- ¢ 0 then SflA is a normal domain with quotient field k(H) and Henselization

(57%),: g II (Amvlh g filmi-
j=1 j=1

47

Also, if s.- 75 0 then the image of the quotient field k(P,-) of SflA in the quotient field

of its Henselization is given by the diagonal map

k(Pz') "" HHPU')
j=l
x r—> (x,x,...,x).

Similarly if s,- 75 d,- then 7E‘1A is a normal domain with quotient field k(B) and

Henselization

_ .. h ‘1‘ ~ h d‘ N
(731A) '3 II (AW) 3 II Ah/Pij-
j=8g+l j=85+l

Also, if s,- 79 d,- then the image of the quotient field k(P,) of 7,7—1A in the quotient

field of its Henselization is given by the diagonal map

Define

Then C is an A-algebra via the diagonal map, C has Henselization AV" E” Ah and
furthermore, by construction the image of the total quotient ring of C in K Ah is
the same as the image of L in K Ah. Thus C has total quotient ring L. Since C is
a product of localizations of A which loses no maximal ideals, C is faithfullay flat,
unramified and essentially finite over A. Thus C is etale over A.

Suppose that D = C. Then by Proposition 2.2.6, D is essentially of finite type

48

over A, and then by Corollary 3.1.2, D is etale over A. Therefore, it suffices to show
mmbzé.
Since K Ah is a direct limit of the total quotient rings of etale neighborhoods of A,

we may choose an etale neighborhood E of A whose total quotient ring K 1.; contains

L. Then by faithful flatness of E ——> Ah,
D :=LnA" =LnKEnA" —_- LnE.

By Proposition 3.1.1 there exists a E A be a regular element such that A/aA E“ E / aE .

Since D = L n E, it follows that aE n D = aD. Thus we have injections
A/aA H D/aD H E/aE ’E A/aA.

So A/aA ’z-‘i D/aD E E/aE.
Consider D[A] Q D Q C. Since D[A] is a homomorphic image of D (8,1 A, we

have morphisms
Aaoagéflnm —)A-’;,

with the composite being faithfully flat (since AV" % Ah). Since D/aD E“ A/aA,

applying ®AA/aA we obtain morphisms

 

with the composite being faithfully flat. Thus the surjection

aA aD[A]

 

49

is also injective, so that we have an isomorphism

 

Since D[A] is also finite over D so that all of its maximal ideals contain the element a,
it follows that the maximal ideals of D[A] correspond bijectively with those of A (and
hence also with those of C). Since D[A] also has total quotient ring L, the minimal
prime ideals of D[A] and C also correspond bijectively.

For p, p’ 6 Min( D[A] ), if (p + p’ )D[A] is contained in some maximal ideal N
of D[A], then (p + p’ )C is contained in the maximal ideal of C which lies over N.
However, pC is primary to some minimal prime ideal q of C and p’ C is primary to
some minimal prime ideal q’ of C. Thus any prime ideal of C which contains (p+p’)C
also contains the minimal prime ideals q and q’. But since C is a product of domains,
q + q’ generates the ring, a contradiction. Therefore (p + p’ )D[A] is contained in no
maximal ideal of D[A] and so by the Chinese Remainder Theorem, D[A] is a product
of domains.

Write
Max(D[A])={N11,...,N1d1,...,Nn1,...Nndn},

where Nij = mijé fl D[A]. Then by the above discussion and the correspondences

between the maximal and minimal prime ideals of D[A] and C, we must have

D[A] = flu,"D[A] x V,’1D[A],

i=1

50

where as with S, and 7: we have

s,‘ d.‘
u,:: D[A]-UN,~,- and v,:: D[A]— U N,,-.
i=1 i=8i+l

Therefore we have injections

3,-1.4 9 aflopi] <—> u,-1(Z*=s,.-Ui, and

7E‘1A H V,“D[A] H V,‘1C = 7:414.
Thus D[A] = C and so D[A] = D = C. Cl

Example 3.1.4 For ( A, m, k) a complete Noetherian local ring and K A Q L Q K gh.
an intermediate ring obtained by adjoining idempotent elements to K A, it need not be

true that the local ring D := L n A’” is etale over A.

Proof:

Let A be the power series ring in two variables, X and Y, over Q modulo the ideal

generated by X 3 — 3X Y2 + Y3,

A: Q[[X.Y]]
' (X3-3XY2+Y3)°

 

If X 3 —3XY2+Y3 factors in the polynomial ring Q[X, Y], then substituting Y = 1
we obtain a factorization of X 3 — 3X + 1 over Q. However X 3 — 3X + 1 is easily
seen to be irreducible over Q. Thus X 3 — 3X Y2 + Y3 is an irreducible element of
Q[X, Y] and so (X 3 — 3XY2 + Y3)Q[X, Y] is a homogeneous prime ideal. It follows
that X 3 — 3X Y2 + Y3 generates a prime ideal of Q[[X, Y]]. Therefore A is a domain.

51

We show that the normalization A of A is given by A = A[X / Y]. Set

 

B-—( QlXJ’l )
'_ (X3-3XY2+Y3)' (XX)

Then A 9;“ B and A 2’ A (83 B. Thus it suffices to show that E is given by B =
B[X / Y].

Now in K B we have
(X/Y)3 — 3(X/Y) +1 = 0.

Thus X / Y is integral over B and so B[X / Y] Q B. Thus we need only show that

B[X / Y] is normal. For this we consider the surjection

 

 

 

._ B[Zl ¢ .
C'_(Z3—3Z+1,YZ—X) ——> B[X/Y], g1ven by
Z ——> X/Y.
Now
C E QiX’Yl(X,Y)[Zl ,Q, QiY](Y)[Z]

(Z3—3Z+1,YZ-X) — (Z3—3Z+1)'

Since C is finite over Q[Y](y), all of its maximal ideals contain Y. However

 

£2. 3 WI

YC — (Z3—3Z+1)
is a field. Therefore C is a DVR with maximal ideal generated by Y. Since (I) is
a surjection and since both C and B[X / Y] are one-dimensional domains, d is an

isomorphism, and thus B[X / Y] is normal.

52

Therefore A is given by A = A[X / Y]. Note that since A is an excellent Henselian

local domain, A is a local ring. Hence A has residual field

QM]
(£3 — 36 + 1)’

IIZ

kg

 

Since, using a discriminant argument, the Galois group of £3 — 35 + 1 is cyclic of
order three, kg is a normal field extension of k = Q. In particular, the polynomial

£3 — 36 + 1 splits as a product of linear factors over kg, so that

mom/sag)?

Define E := A ®Q kg. Then E is a local etale extension of A with residual field
kg = kg. Since every maximal ideal of E contains m, the residual field product of

EEE®AAisgivenby

(E®AA)®AQ g [$5369ng 1‘ kg®ng E (kg)3.

Since E is an excellent Henselian reduced local ring, E is a product of local domains.
Therefore E has three minimal prime ideals and hence so does E.

Write
Min(E) = {P1, P2, P3}.
Denoting the quotient field of E/H by k(P,), we have
Kg E k(Pl) x k(Pg) x k(P3).

Let n := (1, O, 0) 6 Kg and put L := KA[77]. Note that L ”-3 (KA)2 and maps into

53
KB via

L a (K,.,)2 —+ k(Pl) x k(Pz) x k(P3) 2% KB

(a. fl) *—> (a. 5, fl)-

In particular L has two prime ideals.
Define D := L n E. Then, fixing an inclusion E H A’”, we also have by faithful

flatness

D:=LnE=LnKEnA’”=LnA’”,

Since D has total quotient ring L, D has two minimal prime ideals.
If D is etale over A, then D is excellent Henselian and so D is a product of local
domains. Since D has two minimal prime ideals, D 95 D (8,; A is a product of two

local domains and hence has residual field product

(D®AA)®AngD®Qk/i

which is then a product of two fields, kg ®A kg 2’ k1 x k2 . It follows that kg is

neither Q nor kg and hence that the containments

ng0§k5=kj

are proper. This is a contradiction since by construction there are no intermediate

fields between Q and kg. Therefore D cannot be etale over A. E]

Remark 3.1.5 In the previous example, while the intersection ring D is not etale
over A, it is contained in a finite extension E of A. Thus D is finite over A and so is

excellent. Nevertheless, it seems difficult to exhibit D explicitly as a finite A-algebra

54

without a fair amount of computation. One wonders if there is a general technique

for exhibiting such intersection rings.

3.2 Intersections over Henselian Rings

Let A be an excellent reduced local ring, K A Q L Q K Ah. be an intermediate ring
which is finitely generated over K A, and let D be the intersection ring given by
D := LflAh". In section 3.1, we saw that the minimal prime structure of D can be an
obstruction to D being etale over A. However, if A is Henselian, then the extension
A H A’” is integral. Thus A H D H All; are integral extensions and hence are
semilocal. Then we know by Theorem 2.2.4 that D is a direct limit of semilocal etale
extensions of A. So D already has a great deal of structure through its normalization.

In Theorem 3.2.1 we avoid the minimal prime obstruction through modding out by
minimal prime ideals and consider corresponding intersections under a homomorphic
image of the strict Henselization. Making use of the integrality of the morphisms of
normalizations we conclude in this theorem that the resulting intersection rings are as
close to etale over A as we can hope for: they are unramified extensions. Specifically,

we give the following theorem.

Theorem 3.2.1 Let ( A, m, k ) be an excellent Henselian reduced local ring.

i) Let A Q D Q A’” be an intermediate ring which is module finite over A. Then
for every minimal prime ideal p of D, there exists a local, finite and unramified
extension D’ of A with D/p Q D’ Q k(p) = Q(D/p). More generally, the

following is true.

ii) Let Q E Min(A’”), q := Q 0 A and let k(q) Q L Q k(Q) be an intermediate

55

field which is finitely generated over k(q). Then the ring
D’ := L n (AM/Q)

is finite and unramified over A (equivalently over A / q ) and has strict Henseliza-

tion A” / Q.

In the situation where the base ring A is Henselian, we use this theorem to give
criteria for certain intermediate rings A Q D Q A’” to be etale over A. As another
interesting application, we characterize Noetherianness of arbitrary local intermediate
rings A Q D Q A’” dominated by A’” completely in terms of the residual extensions
arising from the morphism D —-> D.

Before proving Theorem 3.2.1 we need a few technical lemmas.

Lemma 3.2.2 Let ( A, m, k ) be an excellent Henselian reduced local ring with
strict Henselization A’”. Let Q E Min(A’”), q := Q n A, and let k(q) and k(Q)
be the quotient fields of A/q and Ans/Q respectively. Let k(q) Q L Q k(Q) be an

intermediate field which is finitely generated over k(q). Define
D’ := L n (Ahs/Q).

Then there is a local etale extension E Q A’” of A satisfying

i) The minimal prime ideals of A""/qA’m ’5 (A/q)hs are extended from E /qE .

ii) For P := Q 0 E we have

D’ = L n (E/P).

Proof:

Set B :22 A/q. Then B = ( B, mB, k ) is an excellent Henselian local domain,

56

and so its normalization B is a local domain. Let k [3 be the residual field of B and
k’ be the separable closure of k in kg. Write k’ = k[X]/(f), where f E k[X] is a
monic, irreducible and separable polynomial. Let G be a local etale extension of A
such that f splits as a product of linear factors over kg. Then G/qG’ is a local etale
extension of B and by Corollary 2.1.4, the minimal prime ideals of B’” = Ahs/qA’”
are extended from G/qG.

Since L is finitely generated over k(q) = Q(A/ q) we may write

L = Hafiz-1,... .15le k(Q)- (3-1)

Since k(Q) E (AM/Q) (8),; k(q) we may assume the x,- are in Ahs/Q. Let x1, . . . , xn be
preimages of the x, in A’”, and let E Q A’” be a local etale extension of G containing
the x,. Then E is also a local etale extension of A and has strict Henselization A’”.

Since

B Q G/qG g E/qE g Ahs/qA’” (3.2)

with the minimal prime ideals of Ahs/qA’” extended from G/qG, they are also ex-
tended from E / qE.

Let P := Q 0 E and k(P) be the quotient field of E/P. Then since Q contains
qA’”, Q = PA’” is extended from E. Thus E / P has strict Henselization A’” / Q, so
that the morphism E / P —-> A’” / Q is faithfully flat. Therefore E / P can be obtained

by intersecting its quotient field with A’" / Q,

E/P = k(P) n (AM/Q). (3.3)

57

Since by construction the x,- are in k(P) = Q(E/P), L Q k(P). Thus we have

D’ := L n (Aha/Q) = L n k(P) n (AM/Q) = L n (E/P) (3.4)

A A hs
Remark 3.2.3 By A’” in the following lemma, we mean (A) . For a Henselian

local ring ( A, m, k ),
.. hs ..
(A) a A (8,, A’”.
This is so because writing

Aha = lim AA
__§
AEA

as a direct limit of local etale extensions of A, we then have

fl ®A A,” g lLIIl/‘l ®A Ag.
AeA
Since A is Henselian, each Ag is a finite extension of A. Thus A (8)4 A), is a finite

extension of A, so that all of its maximal ideals contain mA. On the other hand
A ®A AA ®A (A/m) '5 k ®k kA, ’5 kA,

is a field. Thus each A ®A A), is local and so A (8),, Ah3 is a direct limit of local etale
extensions of A. Since it is Henselian and has a Separably closed residual field, it
must be the strict Henselization of A.

The ring A’“ need not be complete. If A is Noetherian or excellent then so is A’”.

58

Lemma 3.2.4 Let ( A, m, k ) be an excellent Henselian reduced local ring with strict
Henselization A’“. Let Q E Min(Ah’), q := Q 0 A and let k(q) g L Q k(Q) be an

intermediate field which is finitely generated over k(q). Then
13' == L n (AM/Q)
is an excellent Henselian local domain with quotient field L and completion given by
D, 2 Kb, 0 (Aha/Q)

where Q is a minimal prime ideal of AM ’-_‘-’ A®A A’” which is extended from Q Q A’”

along the canonical morphism.

Proof:
Replacing A by A/q we may assume that A is a domain, q = (0) and k(q) = K ,4.

Using Lemma 3.2.2 we may write
D’ = L 0 (E/P), (3.5)

where E is a local etale extension of A, P := Q n E, and such that the minimal prime
ideals of A’” are extended from B. By Theorem 2.1.3, the latter is equivalent to the
residual field of E being separably closed in each residual field of its normalization.
Since E is a local etale extension of the Henselian ring A, it is a finite extension
by Proposition 1.11. Thus A Q D’ g E/P are finite extensions, so that D’ must be

an excellent Henselian local domain. To see that D’ has quotient field L, write
L = KA[:T:1, . .. ,xn] g k(P). (3.6)

Since k(P) E“ (E/P) (8,; KA we may assume the x, are in E/P. Then the x,- are in

59

L F) (E / P) = D’. Thus D’ must have total quotient ring L.

Since the residual field of E is separably closed in each residual field of its nor-
malization, the same is true of E in its normalization since completions are obtained
merely by tensoring with ®AA. Thus again by Theorem 2.1.3, the minimal prime
ideals of E’” are extended from E. Since E ’5 E 8);; A is a local etale extension of A,

it has the same strict Henselization,

Eha z A’“. (3.7)

Furthermore, since E is excellent and Henselian, the minimal prime ideals of E are
extended from E. It follows that P g E extends to a minimal prime ideal in each

ring of the following commutative diagram

3 Ehs Ahs

(3.8)

II? ———> H?

Gil—H01)

3 Ehs Ahs.

Thus Q = PA’”. Set 15 :2 PE and Q :2 PA’“ = PA’” = QA’”. Applying ®AA to

the injections

A —> D’ ——> E/P —> Ahs/Q, (3.9)
we thus obtain the following injections of domains,

A -—> D’ —> E/f’ ——> Ahs/Q. (3.10)

We now argue that

A

D’ E’ Klj,fl(E/p) g KDHHE/T). (3.11)

60

To see this, first observe that

D’ D’ (8),; A

IIZ

[LO (E/P)] ®A A

H?

:L (8,, A] n [(E/P) ®A A] by flatness

HZ

:L ®D' D, (8.4 A] n [(E/P) ®A 14]

IIZ

 

PL e0, 13'] fl 5/79 (3.12)

where the last isomorphism follows by finiteness of D’ and E / P over A. Since for D’

the normalization of D’ we have

A

D, Q D, g D’ @0! D’ Q L ®D’ D’, (3.13)
we may replace L @0. D’ in 3.12 with D’ to write
13' 2’ D' n 5/7). (3.14)

To establish 3.11 we need to make some observations about normalizations.
Since E is finite over A, E ’-‘_-’ E (8),; A is a local etale extension of A. Thus its

normalization is given by

»l
I

II?

E 3,, A (3.15)

and is finite and etale over A. Since E is Henselian, semilocal and normal, it is a

product of local normal domains with each factor finite and etale over the local normal

NN

domain A. Thus E / P E E / 13 is one of these factors and so is a local etale extension

of A.

61

Applying ®AA to the morphisms
A '——> D’ L» E/P, (3-16)

we obtain morphisms of completions
A c—+ D' H 5/79. (3.17)

Then considering normalizations we obtain

N

A <——> D’ c—> E/T’, (3.18)

~

with the composite being finite, local and etale. By Proposition 2.2.4, D’ is etale over

A. So the second morphism above is etale by Proposition 1.14 and also faithfully flat.

N

In particular, D’ can be obtained by intersecting its total quotient ring with E / P,

N
~

D = K15, n 137/73. (3.19)
Substituting in 3.14 we obtain
D 2 K15, nE/TanE/TJ = K15, n 5/7), (3.20)

the desired isomorphism 3.11.
To finish, recall that E has strict Henselization E’” E’ A"3 and that P g E extends

to the prime ideals P g E and Q Q A’”. Therefore

 

 

E/P§ A —-) A, ”:3 A (3.21)

is a faithfully flat morphism of domains. Thus E/T’ ’5 E /P can be obtained by

62
intersecting its quotient field k(P) with the faithfullay flat extension A’” / Q,

13/73 = k(P) Q A; (3.22)

 

From 3.11 we now have

 

IIZ
>4
q.

 

(3.23)

where the last isomorphism follows since K15, Q k(P) by 3.10. [:1

Remark 3.2.5 By Proposition 1.8, every local etale extension of a Henselian local
ring A is a finite extension. Since the strict Henselization A’” is a direct limit of local
etale extensions, A’“ is integral over A. So for any intermediate ring A Q D Q A’”,

A’” is also integral over D. Since A’" is local, the intermediate ring D is also local.

Proof of Theorem 3.2.1:

The statement i) follows from ii) by setting L :2 k(p). To show i), fix Q E
Min(A’”). Replacing A by A/q we may assume that A is a domain, q = (O) and
k(q) = K A. Since A is a Henselian local domain, its normalization A is a local
domain. Let ‘24 be the residual field of A.

By Lemma 3.2.2, we may write
D’ = L n (E/P), (3.24)

where E is a local etale extension of A, P := Q0 E, and such that the minimal prime

ideals of A’” are extended from E. Since E is a local etale extension of the Henselian

63

ring A, it is a finite extension. Thus since the composite A ‘—> D’ L+ E / P is finite,
D’ is also finite over A.

Note that by Theorem 2.1.3, since the minimal prime ideals of A’” = E’” are
extended from E, the residual field of E is separably closed in each residual field of
its normalization. Let k3 be the separable closure of k in kg. Write k” = k[X]/(f)
where f is a monic, irreducible and separable polynomial of degree d. Then by

Corollary 2.1.4, f splits as a product of linear factors over kg.

A ) We may assume that A is complete:
Knowing the result for complete rings we can conclude, since D’ is an appro-
priate intersection by Lemma 3.2.4, that D’ is unramified over A. But since D’ is

finite over A, D’ ’é D’ (X) A A. Then D’ is unramified over A by (ch. II, Prop. 4 of [Ra]).

B) Expressing E713 as a tensor product:

Henceforth we assume that A is complete and that D’ is given as in the intersection
3.24.

Let R be a coefficient ring for A, R3 be the local etale extension of R with residual
field k‘, and let S E be the local etale extension of R with residual field ICE. Note that
since R is complete and hence Henselian, R’ and SE are finite extensions of R.

Now by choice of E, we have an k-morphism k3 L) kg. By Proposition 1.10, since
R3 is etale over R and SE is Henselian, k‘ ;+ k3 lifts to an R- morphism R’ —+ SE.
By Proposition 1.14, since both R’ and S E are etale over R, R3 —> SE is etale.

By Proposition 1.16

E f‘2’ A ®R SE, (3.25)

64

and so

E E” A®AE
g A®A(A®RSE)

E’ A®R SE. (3'26)

Since A is Henselian and R3 is etale over R, there is an isomorphism

Homzoc R(123", A) a! Homk(k’, k A). (3.27)

Thus there is a canonical morphism R3 —+ A induced by the inclusion k“J <—> k A- So

we may write

~

E '5 2169,; SE "a A®R. 12369383. (3.28)

Since R3 (29;; S E is a finite semilocal etale extension of the Henselian ring R“, it is a
product of local etale extensions of R’. Recall that k‘ = k[X]/(f) where f is a monic,
irreducible and separable polynomial of degree d and that by choice of E, f splits as
a product of linear factors over kg. Therefore the residual field product of R3 ®R S E

is given by

(R3 ®R SE) @128 k3 '5 ks ®k ks 3" (kEld- (3-29)

Thus R3 (83 SE is a product of d local etale extensions of R3, each with residual
field kg. Since (51,)" is also a product of d local etale extensions of R3, each having

residual field kg, and since local etale extensions of the Henselian local ring R8 are

65

uniquely determined by their residual fields, it follows that
R“ 3,. SE 2' (SE)", (3.30)

an isomorphism of Rs-algebras, and arguing similarly, also an isomorphism of SE-

algebras. Substituting in 3.28 yields
.. - d 2 d
E 9.1 Am. (SE) a (A 33.. SE) . (3.31)
Since A <83. S E is finite over A and since
(A 6%. SE) (3,, k A :3 1a,, (2%. kg (3.32)

is a field by Lemma 2.1.1, A ®R. SE is a local ring. Since E is excellent Henselian and
reduced, E is a product of local domains. Therefore 3.31 exhibits E as a product of

local domains. Thus E7;3 is given by
E713 2 A 3,... SE. (3.33)

Furthermore there is a commutative diagram of S E-algebras

... .. d
E§A®RSE —> EE’ (A®Rs SE)
C...) (C... (3.34)
E/P —) E773§A®Ra 83.

To see that the composite SE —-> A (83 SE ”:1“ E —> E /P is injective note
that since A is reduced, its coefficient ring R is a field or a complete p—ring. If R
is a field then so is 5;; since it is a local etale extension of R. Thus in this case

injectivity is trivial. Suppose R is a complete p—ring with uniformizing element

66

t E Z. Then so is SE. Since R Q A Q E/P, no power oft is in P. Therefore

since t also generates the maximal ideal of SE, the kernel of S E ——> E / P must be zero.

C) Expressing D’ as a tensor product:

Since A <—> D’ ‘—) E/P are finite extensions, so are the morphisms of normal-
izations A ‘——> D7 H E773. In particular, the second morphism is local. By Theo-
rem 2.2.6, D7 is a local etale extension of A.

Now kg ——-) k3, is a separable field extension. Let t be the separable closure of

k3 in kg (equivalently the separable closure of k in k5,, since k3 is by definition the

separable closure of k in kg). Then we have the following diagram of field extensions
kg —>s°" 1:5,

13:23) ($323 (335)
k3 & 1.

It follows that [k3, : kg] = [l : k3] (see for example ch. V, sect. 6, Cor. 13 of [Hu]).

Therefore by Lemma 2.1.1, since kg is purely inseparable over k3 and l is separable

over k3, kg (8),. l is a field of degree over k‘ given by

[kg®k.l:ks] —_- [kg:k3][l:k3]

[k5 = kAllkA = 193]

3, : k3]. (3.36)

[1:

Thus the canonical morphism k g (8),. l H k5 is a ks-morphism between fields of the

same degree over k3, hence an isomorphism

kg (3,... l 2 k5,. (3.37)

Let T be the unique local etale extension of R3 with residual field Z. Note that T

67

is also etale over R. Then A (293. T is a local etale extension of A with residual field

kg ®ks l ’5 k5. Since D7 is local etale over A, Proposition 1.11 gives

5' g A e... T (3.38)

D) We show that the inclusion
A <33. T a D’ L» E/P a A 3,... Sg (3.39)

sends T into Sg:

We have the following diagram of finite injective morphisms of complete local

domains
Aflfigg®RaTfllfigfi®RaSEj
A —+ D’ ——> E/P

The inclusion k5, ¢—> km, of residual fields induces a morphism between separable
closures over k’. The separable closure of k3 in k5 is by definition I. By choice of E
the minimal prime ideals of A’” are extended from E. Thus by Corollary 2.1.4, kg is
separably closed in each of the residual fields of k if:- Hence kg is separably closed in

kw. Therefore the morphism of separable closures gives

l 43> kg. (3.40)

68

Applying k g® R. we obtain the following commutative diagram of fields

19,393.: m kg®Rs kg

~l El (3.41)

inc
k5; -—-) km).

By Proposition 1.10, since T is local etale over R3 and SE is Henselian, there is

an isomorphism

Homlocga (T, Sg) ”:3 Homk.(l, kg) (3.42)
Therefore 43 lifts to an etale morphism
T i» Sg (3.43)
of Rs-algebras. Then applying A®gs we obtain a morphism
A®R.T m 31693. SE
)2 (a:
5 E779 (3.44)
which induces by 3.41 the inclusion map on residual fields
k5 L» kg, (3.45)
Then since there is an isomorphism
(3.46)

Homloc A17” E/P) 9—: Homkg (kg, leg/7.).

 

69

the composite
17' 31> A @R. T “5°13" A @913. SE 3+ E713 (3.47)

is the inclusion map. That is, there is a commutative diagram

A®R.T M. A®R.Sg

a) a!) (3.48)

~

D, inc E7?)

This establishes the claim.

E) Concluding the proof:
By part D) and we have T Q Sg Q E/‘7P. But by 3.34, considering E/P as a
subset of E/P, Sg Q E/P. Thus also T Q E/P, so that

T g 17' n (E/P) = D’. (3.49)

Then defining F := A ®g T, there is a canonical morphism F ——> D’. Let p0 be the
kernel of this morphism so that F/po L—) D’ is injective.

Note that F is a local etale extension of A with residual field

H?

kg E F®Ak = A®RT®Ak E’ k®kl (3.50)

Also F g F (3,4 A is a semilocal etale extension of A. Since F is Henselian, F is a
product of local domains with each factor being a local etale extension of the local
domain A. Thus F/po is one of these factors and so a local etale extension of A.

Now since F has residual field l and since A c-—> F7170, the residual field of F750

70

contains the field

kg ®ks l 9-" kg. (3.51)
Thus
1771;, <—> 5' (3.52)

is a residually trivial morphism of local etale extensions of A and also etale by Propo-
sition 1.14. But since F7110 is Henselian, it is closed with respect to residually trivial
local etale extensions. Therefore this morphism is an isomorphism. In particular,
F / p0 and D’ have isomorphic quotient fields, K p/po ’-‘_-’ L.

By Proposition 1.10, since E is Henselian, there is an isomorphism
HomlocA(F, E) ’5 Homk(l, kg). (3.53)

Therefore the the composite l ‘—> kg: g—) kg induced by F / p0 —> D’ —> E / P lifts
to an etale morphism F L—> E of local etale extensions of A. Then to see that the

resulting diagram

F —+ E
l l (3.54)
F/po —+ D’ —+ E/P (3.55)

is commutative, observe that by construction the two morphisms F ——> E / P obtained

in the diagram induce the same map on residual fields. Then use the fact that by

71

Proposition 1.10, since E / P is Henselian, we have
HomlocA(F,E/P) '3 Homk(l,kg), (3.56)

so that the two morphisms must be the same.

Since F/po has residual field 1 and Rip/0 has residual field kg (8),. l,
F/Po % F /P0 (3.57)

is residually purely inseparable. By Theorem 2.1.3, since F / p0 is Henselian, we have

that
(F/Polhs = Aha/p04“ (3.58)

is a domain. Thus poA”3 = Q and P := Q n E = poE. Therefore F/Po -—> E/P is

etale. Then by faithful flatness we have

F/PO = KF/pon(E/P)
= Lfl(E/P)

= D’, (3.59)

where the rings here are considered as subrings of k(P) = Q(E / P). Since F is etale

over A, this establishes that D’ is unramified over A. Furthermore we have
(Dl)hs = (F/p0)hs = Aha/Q (360)

as desired. Cl

72

Corollary 3.2.6 Let ( A, m, k ) be an excellent Henselian local domain such that
A’” is a domain (equivalently, by Theorem 2.1.3, such that the residual field of A is
purely inseparable over k). Then for any intermediate field K A Q L Q K Ah. which is

finitely generated over K A,

D' := LflAh"

is a local etale extension of A.

Proof:
By the previous theorem, D’ is finite and unramified over A and has Henselization

(D’)’” = A’”. Since then A’” is faithfully flat over both A and D’, D’ is faithfully

flat over A. Thus D’ is a local etale extension of A by Remark 1.2. CI

Theorem 3.2.7 Let ( A, m, k ) be an excellent Henselian reduced local ring, and
let A Q D Q A’” be an intermediate ring which is module finite over A. Then the

following are equivalent:

i) D is etale over A and the minimal prime ideals of A’” = D’” are extended from

D.

ii) The residual field kg of D is separably closed in each residual field of the nor-

malization D of D.

iii) For each minimal prime ideal Q of A’” and p 2: Q (1 D the morphism

D —> D/P ‘-> k(P) fl (Aha/Q)

is residually trivial.

73

Proof:

(i => ii) : This is immediate from Theorem 2.1.3.

(ii => iii) : Set D’ := k(p) fl (AM/Q). By Theorem 3.2.1, D’ is finite over A and
hence D/p ¢—) D’ is also finite. Since D/p and D’ have the same quotient field k(p),

we must have
137;» = ’5. (3.61)

Since kg: is a finite separable field extension of kg and since by ii) kg is separably

closed in the residual field k 5/7; = k it follows that kg 2 kg.

57,
(iii => i) : Let F be the unique local etale extension of A with residual field kg.
Since D’ is unramified over A by Theorem 3.2.1, D’ is a homomorphic image of a
local etale extension of A which has residual field kg: = kg. But by Proposition 1.11,
all such local etale extensions of A are isomorphic to F. Thus D’ is a homomorphic
image of F via an A-morphism 1b which induces the identity map on residual fields.
Furthermore by Proposition 1.10, w is the unique local A-morphism from F to D’
which induces the identity on residual fields. Using Proposition 1.10 again, since D
is finite over A and hence Henselian, the identity map kg ——> kg lifts to a morphism
F ——) D which is injective since, for E some local etale extension of A containing D,

the composite

F —> D —> E (3.62)

74

is etale. But since the composite

F <—> D —> D/p 9 D’ (3.63)

induces the identity map on residual fields, this composite must be the surjection 2,!)

by uniqueness. Therefore we have isomorphisms

Fagnmza, am

where

m:=pflF=QflF awm

By Theorem 3.2.1 we have isomorphisms

(F/po)”3 "a (0’)“ 2—3- Ahs/Q. (3.66)

Then since F is local etale over A, we must have Q = poA’“.
Since this argument is independent of the choice of the minimal prime ideal Q
of A’”, it follows that the minimal prime ideals of A’” are all extended from F.

Furthermore, for any minimal prime ideal Q E Min(A’“) we have

F D
QnF QnD'

IIZ

 

 

(3.67)

Since there are injections

F c—> D c—> A’“, (3.68)

the rings F, D and A’” all have the same number of minimal prime ideals. By 3.67

75

it follows that considered as subrings of K Aha, F and D have the same total quotient

ring. So by faithful flatness of F —+ A’” we have

F : KF r) A’” 2 D. (3.69)

Therefore F ——> D is an isomorphism. [:1

Theorem 3.2.8 Let ( A, m, k ) be an excellent Henselian reduced local ring, and let
A Q D Q A’“ be an intermediate ring. Then D is Noetherian if and only if for every
residual field I of the normalization D of D, the separable closure of kg in l is a finite

field extension of kg.

Proof:

(=>) : This follows by the Mori-Nagata integral closure theorem (sect. 33 of [M]).

(<=) : Suppose that for every residual field I of D, the separable closure of kg in l is
a finite field extension of kg.

To show that D is Noetherian, it is enough to show that D/ P is Noetherian for
any P E Min(D). So fix P E Min(D). Replacing A by A/(Pfl A), we may assume A
is a domain.

Let Q be a minimal prime ideal of A’” which lies over P, and set

D’ := k(P) n (AM/Q) (3.70)

so that we have containments

A g D/P g D’ g Ahs/Q. (3.71)

76

We argue that D’ is excellent by exhibiting it as a homomorphic image of a direct
limit of local etale extensions of A. We then argue that D’ is finite over D / P, so that

D/ P is Noetherian by Eakin-Nagata (Thm. 3.7 of [M])

Write D as a direct limit of finite type A-subalgebras,

—— -lim D =U D. (3.72)

7GP 76F

Since A ——-> A’” is integral and local, the D7 are finite local extensions of A, hence
Henselian.
Let A, be the unique local etale extension of A with residual field [$137. By

Proposition 1.10, since D, is Henselian, there is an isomorphism
HomzocA(A.,, D7) '5’ Homk(kg7, [1307). (3.73)

Hence there is a morphism A7 ——> D, inducing the identity on residual fields. Since
the composite A7 —) D7 —> A"8 is faithfully flat,
A, H D, is injective.

Now for any 70, 71 E I‘ we have an isomorphism
Hom¢OCA(A.,0, A71) 9:" Homk(kgm, k0” ). (3.74)

So the A7 form a direct system with 70 g 71 if and only if D70 Q D7,, in which case

the structure morphism A.,0 —-) A7, is induced by kg70 —> k0,”. Set

F := 1131A, (3.75)

’YEI‘

Then F is a direct limit of local etale extensions of A and has residual field kg.

77

Furthermore, we have

F E“ lirn A, Q lirn D7 = D. (3.76)

76F 76F

Let N := P F) F so that there are containments
A g F/N g D/P g D’ g Ahs/Q. (3.77)

For each 7 E I‘, we let P, := P 0 D7 and set
D; := k(P,) n (AM/Q). (3.78)

Then by Theorem 3.2.1, the D; are unramified over A. Furthermore, we have

0’ = 1131 0;. (3.79)
761‘

Let A; be the unique local etale extension of A with residual field kgg. By Proposi-
tion 1.10, there is a morphism A; —-> D11. Since D; is unramified over A, D57 is a
homomorphic image of a local etale extension of A which has residual field kpgl. But
all such local etale extensions of A are isomorphic to A’7. It follows that A; ——-> D1,
is surjective.

Set

F' := lim A'. (3.80)

'7
’YEF

Taking the direct limit of the surjections

A; ——> D1,, (3.81)

78

we obtain a surjection
F’ —> D’ c_: Ahs/Q. (3.82)
Thus
D’ ’_—‘_’ F’/N’, (3.83)

where N’ :2: Q (1 F’, and therefore D’ is a homomorphic image of a direct limit of
local etale extensions of A. Hence D’ is excellent.

We now have containments
F/N g D/P g F’/N’ 2 D’ g Ahs/Q. (3.84)

Since D / P ——> D’ is integral and since these rings have the same quotient field k(P),
they must have the same normalization. Then by hypothesis the residual field kg, of

D’ must be finite and separable over kg. Since
HomzocA(F, F’) °_-’ Homk(kg, kg), (3.85)

the inclusion kg H kg: induces a morphism F ——> F’ and we have a commutative

diagram

F———> F’

l l

F/N —> F’/N’ = D’.

Then to show that F /N -—> F’ /N’ = D’ is finite and hence that D’ is finite over
D/ P, it suffices to show that F ——> F’ is finite.

Let c 6 ’60! be such that kg: = kg[c] and let G’ be the unique local etale extension

79

of A with residual field k[c]. By Proposition 1.10 we have a morphism G —> F’ which

induces k[c] ——> kg: on residual fields. This yields a residually trivial morphism

(F ®A G)U ——> F, (3.86)

where U is the preimage of the maximal ideal of F’. By Remark 1.12, since (F ®A G)U
and F’ are both direct limits of local etale extensions of A having isomorphic residual
fields, these rings are isomorphic. Since (F 8),; G)U is local etale over the Henselian

ring F, it is finite over F by Proposition 1.8. Hence F’ is finite over F. 1:]

As a simple application of the previous theorem, let A be the power series ring

over Q in n variables, X1, . .. ,Xn, modulo some ideal I,

QllXI a - - - a anl

A2: I

 

. (3.87)

Then for Q an algebraic closure of Q, the strict Henselization of A is given by A”3 ’5
A ®Q Q = A[Q]. Let a E A be a regular element and set D :2 A[Qa]. Since
A —+ D ——> A"3 is integral, D is a local ring. To see that D is nonNoetherian by
the previous theorem, note that D has residual field Q while Q is contained in any
residual field of D.

More generally, for A an infinite index set, {a A} )(e A a collection of regular elements
of A, and {egheA a collection of elements of Q which is contained in no finite field
extension of Q, then the ring D := A[{cAaghE/d is nonNoetherian since again D has

residual field Q whereas {c,\},\€/( Q D.

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