GIBSON, GAVIN J.

Gibson, Gavin J.

Osaka University Knowledge Archive

Seminormality and F-purity in local rings

Osaka City University Repository

Gibson, G.J.
Osaka J. Math.
26 (1989), 245-251
SEMINORMALITY AND F-PURITY IN LOCAL RINGS
GAVIN J. GIBSON
(Received February 25, 1988)
In this paper we consider the properties of seminormality and F-purity in
commutative Noetherian local rings, two properties which are closely related,
especially in the situation where the ring in question has Krull dimension
equal to 1. By making use of this relationship, we obtain a simplified proof of
a generalization of a result due to Goto and Watanabe [7], which describes the
structure of the completion of a 1-dimensional F-pure ring whose residue field
is algbraically closed. Finally, in our main result, we demonstrate that for a
1-dimensional local ring of prime characteristic whose Forbenius endomorphism
is finite, the properties of seminormality and F-purity are equivalent if the re-
sidue field of the ring is perfect.
We should like to express our thanks to Liam O'Carroll for his interest
and helpful suggestions regarding this research, and in particular for his assis-
tance with the proof of 2.6..
0. Notations and conventions
From this point forth, A will denote a reduced Noetherian local ring with
maximal ideal m and residue field k^A/rti. The classical ring of quotients of
A will be denoted by O(A) and A will denote the integral closure of A in Q(A).
The term dimension will always refer to the Krull dimension of A.
1. Preliminaries
DEFINITION 1.1 (see [6, 1.1]). A ring A is seminormal if it satisfies the
following equivalent conditions:
(i) if a&Q(A) and a2, c?^A, then a^.A\
(ii) if a^Q(A) and there exists a positive integer k such that a*^A when-
ever t>k, then a^A.
In. the particular case where A is 1-dimensional, we have a further characteriza-
tion of seminormality, namely:
(iii) A is seminormal if m=J(A), where J(A) denotes the Jacobson radi-
cal of A.
DEFINITION 1.2. Suppose that A is a ring of prime characteristic p and
246 G.J.GIBSON
let F: A-*A be the Forbenius endomorphism of A. Let AF denote A when
regarded as an -4-module via F. Then A is said to be F-pure if, for all A-
modules E, the map hE:E-*A
F®AE defined by hE(x) = l®Ax9 for all x^E, is
injective. We say that F is finite when AF is a finite ^4-module.
In addition, we shall require the following results which concern the two
properties defined above. The first of these appears in [7], and although we do
not impose the same restrictions on the ring A that are present in the original
statement, the proof that appears therein applies equally well to the more gen-
eral situation which we consider in this paper.
Proposition 1.3 ([7, (2.2)]). Let A be an F-pure ring, let Q=Q(A) and
ktfQ/A: Q/A-+Q/A be defined by the relation
/Q/A(X mod A) — xp mod A ,
for all x&Q(A). ThenfQ/A is injective.
Concerining the property of seminormality we shall make use of the fol-
lowing results which are stated without proof.
Lemma 1.4 ([11]). Let k be a field and let L be a reduced Noetherian k-
algebra of dimension 0. Then k-\-TL[T] is seminormal
 t where T is an indeter-
minate over L, and k-}-TL[T] is identified with a subring of L[T] in the natural
way.
Proposition 1.5 ([11, 3]). Let A be a \-dimensional ring of prime charac-
teristic p, whose Frobenίus endomorphism is finite. Then the following conditions
are equivalent:
(i) A is seminormal;
(it) A is seminormal;
(Hi) Gr(A) is k-isomorphίc to k+TK[T], where Gr(A) denotes the associated
graded ring of A with respect to m and K=A/J(A)]
(iv) Gr(A) is reduced and seminormal.
At this point we remark that if A is a 1 -dimensional ring of prime charac-
teristic p, whose Frobenius endomorphism is finite, then it follows from [10,
Th. 108 and Th. 78] that A is a finite ^4-module. We are therefore justified in
replacing the latter condition, which appears in the original statement of 1.5.,
with the former condition which is more appropriate to the work of this paper.
2. The results
The first result is an immediate consequence of 1.3.
Proposition 2.1. Let A be a ring of prime characteristic p. If A is F-pure
SEMINORMALITY AND F-PURITY 247
then A is seminormal.
Proof. Suppose that a^Q(A) and a2, a*^A. Then it is easily seen that
, so that, by 1.3., a^A and the result follows.
From this point on we focus our attention on the situation where A is a
1 -dimensional ring. Concerning this situation we have the folloaing result which
sheds light on the relationship between seminormality and the prime ideal
structure of a ring.
Theorem 2.2 ([5, 5.1.5., cf. 7, (1.1)]). Let Abe a seminormal ring of di-
mension 1, and let φlt ••-,£„ be the minimal primes of A. Then for each i, \<i
<n,
(i) m = frθ ΓΊ ty, and
(it) m = θ ( ή j > y ) .
ί=ι yφί
n
Proof. Since A is reduced, the natural map A-* ® -4/J),- is injective, so
n
that A can be regarded as a subring of © A/$iC:Q(A) under this map. Fix ί,
l<i<n, let #em, and consider (0, •••,#,•••, 0)e © Afoj. where x^Afoi is
1
the image of x. Now r(t>, ® Π£, )— m> so ^^ there exists a positive integer 5,
yφ»
such that m'c b.φnt); whenever £>s. It follows that for each t>s, there
exists we nt>/ such that #'— ί/e&. This implies that (0, •••,£, •• ,0)e^4 by
1.1. (ii). There therefore exists an element v€Ξ Π^ such that x — ^ Eϊί>t , so that
#efc® Π t>y, and (i) follows. /Φ<
yφί
To see that (ii) holds, observe that the above argument shows that tti^ φ tπt ,
1
where m, is the image of m in Afa, \<i<n. It follows from (i) that m^m/p,
« n^. , l<i<n, so that m= © Πp. as required.
ίΦί » =ιyΦί
We now make use of 2.1. and 2.2. to prove a structure theorem for a cer-
tain class of 1 -dimensional F-pure rings, originally proved by Goto and Watanabe
under more restrictive hypotheses.
Theorem 2.3 ([7, (1.1)]). Let A be a \-dimensional ring of prime character-
istic p. Suppose that the field k^A/m is algebraically closed. Then A is F-pure
if and only if
A <* k[[X19 -, XJ]/( , Xi X j , -)w ,
where A denotes the completion of A with respect to m.
248 G.J. GIBSON
Proof. We begin by noting that A is F-pure if and only if A is F-pure
by the argument on p. 466 of [4]. Now A satisfies the conditions of the state-
ment of the theorem, so that we can assume with no loss of generality that A
is complete. It is a straightforward consequence of [4, 1.12] that rings of the
form k\ίXl9 -, *„]]/(-> Xi X;, •••)*=, are F-pure.
Let us now suppose that A is F-pure and that \>19 ••-,£» are the minimal
primes of A. As in the proof of 2.2. we can consider A to be a sub ring of
θ-4/fc. Furthermore, by [2, Chap. V, 1.2], J= 0Z/ft so that we have
Since A is complete, it follows that A /fa is complete, l<z<w, so that each
is a local ring with maximal ideal tπ,-, say. By 2.1., A is seminormal so
that, by 1.1. (iii),
thus the natural image of ttt in A /fa is m,-. Moreover since k is algebraically
closed, it follows that (^7^)/tή
ί
=ft, \<i<n, so that Afip^Afa, and thus each
is integrally closed. It now follows from [1,9.2] that m, is a principal ideal,
so that by 2.2., there exist elements x^ Π t>, , !</<w, such that
/Φ
m = 0
1
Since A is complete, it is possible to define a surjective ^-algebra homomorphism
/: k[[Xl9 •••, Xn]]-*A, such that f(Xi)—xiy l<i<n. It can easily be verified
that Ker /=(•••, X{Xj, * )ίΦ/> so
^ = k[[Xl9
as required.
Now in [3], Davis considers the situation where A is a 1 -dimensional ring
such that A is a finite ^4 -module and k is algebraically closed. In § 3 therein he
shows that such a ring is seminormal if and only if A^k[[Xly •••, Xn]]l
(•• ,XiXj, •••)**./• In yiew °f Λis and the remarks which follow 1.5. we are
able to deduce the following corollary to 2.3..
Corollary 2.4. Let A be a l-dimensional ring of prime characteristic p
whose Frohenius endomorphίsm is finite and suppose that k^Ajm is algebraically
closed. Then A is seminormal if and only if A is F-pure.
In our main result of this paper, we demonstrate that the equivalence of
SEMINORMALITY AND F-PURITY 249
the two properties will continue to hold if we relax the condition that k be al-
gebraically closed and only insist that it be perfect. We shall require the fol-
lowing auxiliary result.
Lemma 2.5 (cf [8, 4.6]]). Let kdk' be fields of non-zero characteristic p,
and suppose that k' is separable over k. Let R be a k-algebra such that R®kk' is
F-pure. Then R is F-pure.
Proof. This follows in a straightforward way from [8, 4.6] and 1.2.
Theorem 2.6. Let A be a l-dίmensional ring of prime characteristic p, whose
Frobenius endomorphism is finite and whose residue field k is perfect. Then A is
F-pure if and only if A is seminormal.
Proof. If A is F-pure then A is seminormal, by 2.1. Let us now sup-
pose that A is seminormal so that, by 1.5.,
Gr(A)e*kk+TK[T],
where K=A/J(A). Moreover, by 1.5., we can assume that A is complete, so
that kC.A, by [10, (28) P]. Now, if £ denotes the algebraic closure of k, then
(k+TK[T])®
 k^(k® kk)®(TK[T}®kk)^®T(K®kk) [T] .
We claim that, as ^-algebras, %®T(K®kk) [T]^Gr(A® J).
Since k is integral and flat over k and since kdA®kky it easily follows that
k®km=m is the unique maximal ideal of A®kk. We have the exact sequence
of ^-vector spaces
O -> mn+1 -> m« -> mΛ/ntn+1 -> O ,
which on applying — ®*& yields the following exact sequence,
O -* %®k m
n+1
 -> %®k m" -> Έ®k tnΎm"-
1
 -> O .
It now follows that, as ^-modules,
k®k τn
n/mn+1 ^  (k®k m")/(%®k mn+1) ^  mn/mΛ+1 .
Therefore
k®T(K®kk) [T]
Now K is a finite direct product of fields, each of which is a finite separable
extension of k, since A is a finite A -module and since k is perfect. It follows
from [9, 3.3. (iv)] that the ring K®k k is reduced. In addition, K®k k is Noe-
therian and zero dimensional, so that by 1.4. and (*), Gr(A®kk) is reduced, as
is easily seen, and seminormal.
250 G.J.GIBSON
Now kd A so that, by a straightforward adaption of the proof of [10, p. 212,
Cor. 2], A is a finite &[[#]]-module, where x is an indeterminate over k. Hence
A®kk is a finite module over k[[x]]®kk. We now show that k[[x]]®kk is
Noetherian.
Let us first consider the domain &[[#]]. It is easily seen that the quotient
field of k[[x]] consists of elements of the form Σ &»#', where d^Z, and it is
now a simple matter to verify that k is algebraically closed in ζ?(& [[#]])• This
means that k is maximally algebraic in £?(&[[#]]), in the notation of [13, p. 196]
so that by [13, p. 198, Cor. 2], Q(k[x]])®k k is a domain. This in turn implies
that &[[#]]®fc£ is also a domain, k being flat over k. In addition; k is integral
over k and it follows that &[[#]]®/? is a 1-dimensional local domain whose
maximal ideal is generated by the single element x®k 1, a so that, by [1, p. 84,
Ex. 1], k[[x]]®k k is Noetherian. This implies that A®k k, a finite &[[#]]®* k-
module, is itself Noetherian. We now have that A®k k is a 1-dimensional local
ring of prime characteristic £, whose Frobenius endomorphism is easily seen to
be finite, so that we can deduce from 1.4. that A®kk is seminormal. Fur-
thermore, since the residue field of A®k k is algebraically closed, it follows from
2.4. that A®k k is F-pure. Now k is perfect, so that k is separable over &, and
we deduce from 2.5. that A itself is F-pure. This completes the proof.
It is clearly of interest to investigate whether the properties of seminor-
mality and F-purity are equivalent under more general hypothesis than are
present in the statement of 2.6. The following example, taken from [12],
demonstrates that the result of 2.6. is not necessarily true if we do not impose
the condition that the residue field of the ring be perfect.
EXAMPLE 2.7. [12] Let A=k[xjy]/(y2—xp—a), where k is a non-perfect
field of characteristic p, and a^k—kp, and let B=Q(A). Then A is integrally
closed in B, and hence seminormal, but k(alίp}®k A is not seminormal. By 2.1.,
k(a1/p)®kA cannot be F-pure and an appealing to [8, 4.6], we see that A itself
is not F-pure.
References
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dison-Wesley, Reading, Mass., 1969.
[2] N. Bourbaki: Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
[3] E. Davis: On the geometric interpretation of seminormality, Proc. Amer. Math.
Soc. 68 (1978), 1-5.
[4] R. Fedder: F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983),
461-480.
SEMINORMALITY AND F-PURITY 251
[5] GJ. Gibson: Modules of generalized fractions, direct systems of determinantal
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[6] R. Gimler and R. Heitmann: On Pic(R[X]) for R seminormal, J. Pure Appl.
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[8] M. Hochster and J.L. Roberts: The purity of the Frobenius and local cohomologyt
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[9] J-P. Jouanolou: Theoremes de Bertini et applications, Birkhauser, Boston,
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University of Edinburgh
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Seminormality and F -purity in local rings

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Seminormality and F -purity in local rings

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