Jan 19, 1982 - Let H(t) denote the Hilbert-Samuel series of the graded ring SIPâ, i.e., H(t)= .... [ 2 ] E. G. Evans, Jr. and P. A. Griffith, Local cohomology modules for ...
J. Math. Kyoto Univ. (JMKYAZ)
23-2 (1983) 269-279
Every Noetherian uniformly coherent ring has dimension at most 2 By Shiro GOTO* ) (Communicated by Prof. Nagata, Jan. 19, 1982. Revised, Feb. 12, 1982)
1. Introduction
The purpose of this paper is to give a theorem stated in the title. Let A be a commutative ring and let v (M ) denote, for an A-module M, the least number of elements in systems of generators fo r M . L et N denote the set of positive integers. W e put A
f A (n) =
sup
heH om A (A .,A )
v (Ker h) A
for each n E N . Then f A (n) is possibly infinite, and the ring A is called uniformly coherent if the supremum f A (n) is finite for every n e N. The concept of uniform coherence was introduced by Soublin [14], and he showed that a given ring A is uniformly coherent if and only if the direct product A N is a coherent ring (Proposition 7). Quentel [12] succeeded Soublin and mentioned that if A is a Noetherian uniformly coherent ring, then s u p v (p ) f A (1) + f A (4) A
peSpecA
and, consequently, A must have finite dimension. This is a pretty application of Soublin's remark above and Gulliksen's theorem ([6]) that every prime ideal in a Noetherian ring belongs to some ideal generated by three elements. As was given in [12], it is not difficult to see that Noetherian semi-local rings of dimension at most one are uniformly coherent. Sally [13] has extended this result and proved that any two-dimensional Noetherian local ring is uniformly coherent (Ch. 3, 2.2 Theorem). In a subsequent joint paper [5] of the author and Suzuki, one may find a crucial use of her theorem, from which the motivation of the present research has started. Nevertheless, no one knows any examples of higher dimension and, because polynomial rings k[X i , X 2 ,..., X a ] and formal power series rings k [IX ,, X 2 ,..., X ,I] over a field k are not uniformly coherent for any 3 ( R e c a ll the famous examples of Macaulay [11], cf. [12], Proposition 3.1.), it seems to be rather reasonalbe to *) Partially supported by Grant-in-Aid for Co-operative Research.
270S
h
i
r
o
Goto
doubt about the existence of Noetherian uniformly coherent rings with dimension at least three. Now let us state explicitly our conclusion: Theorem 1 . 1 . L et A be a Noetherian r i n g . Then the following conditions are equivalent. (1) A is a uniform ly coherent ring. (2) dim A 2, and the suprem um f A(fl ) = RIP f A (0 M axA
m
is finite for every n e N. Unfortunately the latter part of the condition (2) in Theorem 1.1 is, in general, not superfluous, and such an example shall be given in Section 4. However with some suitable additional assumption on A , one may easily omit this part. Corollary 1 . 2 . Suppose that A is a Noetherian sem i-local ring, or that A is a f initely generated algebra ov er a field. T hen A is a uniform ly coherent rin g if and only if dim A 2. We will prove Theorem 1.1 and its corollary in Section 3. The next section is deveoted to some preliminaries which we need to prove Theorem 1.1. Throughout this paper, let A denote a commutative ring and N the set of positive integers.
2. Preliminaries In this section let A denote a Noetherian local ring with maximal ideal in. H V .) be local cohomology functors. We begin with the following
Let
Lemma 2 . 1 . Suppose that dim A = 2 and depth A ?.: 1. L et a, b be a system of param eters for A and assume that (a, b). II,En (A)-=(0). Then the follow ing equality (a", b"): (ab)" ' = [(a): b ] [ ( b ) : a] -
holds for every integer n 2. P ro o f . Let fe A and assume that (a b )'if= a "g +L oh where g, h e A . Then clearly. H ence, as (a"): to ' =(a"): b (cf. [4], (2.7)), we get bh e (a"): b that an 'b f= a x "± b h for some x e A . Similarly we may express abf=a x + b y with y e A . Now notice that x E (b): a (=(b): a ), and express ax=bz with z e A. Then abf=b(az)+b y, whence af=az +b y since b is a non-zerodivisor o f A (cf. [4], (2 .6 ) (1 )). Consequently f — z is in (b ): a , whence f e [(a): b ] [ ( b ) : a ] as z E (a): b by our choice. Thus we have proved the inclusion -
n -1
-
2
2
2
2
(a", b"): (ab)n The opposite one is obvious.
-1
C [ ( a ) : b] + [(b): a] .
2
N oetherian uniformly coherent ring
271
Proposition 2 . 2 . S uppose th at dim A = 3 an d depth 2. A ssum e th at the A -module 11 ( A ) is f initely generated. L et a, b, c be a system of param eters for A and assum e that (a, b )• Iii i ( A ) = ( 0 ) . Then the following equality -
(an, bn, c"): (abc)n ' =[( a, c ) : b ]+[( b , c ) : a] +(a, b): cn -
holds for every integer n
-1
]
2.
P ro o f . First of all, we prove the following C la im . (a"1 , b "2 ): c "3 =(a"1 , b "2 )+a"' 'bn2 ' • [( a , b ) : c " 3 ] fo r a ll n 1 , n 2 , n e N. Let f e A and assume th a t c"3f =a"ig + bn2h fo r som e g, h e A . T h e n h e (an', cn3): bn2 c le a rly . L e t ;4= A /(a"i, c " 3 ), a n d let d e n o t e th e reduction mod ((el, C " ) . Then as 5 is a parameter for the one-dimensional local ring A , it must be a non-zerodivisor mod the ideal 1 1 (A l(an l, c n 3 )). H ence w e get that h e H,VAI(ant, 03)), as h e (an', cn3): bn2 b y o u r c h o ic e . N o w re c a ll th a t a n ., cn3 is an A-regular sequence (cf. [4], (2.6)) and apply the functors H V . ) to the following exact sequences -
-
3
4 A — ■ Al(c"3)
0-
A
O,
A l(c" 3 )
0,
A l(cn3) — > A l(ce", cn3) ,
O.
Then we get two inclusions H 1 (A /(c n 3 ))c H U A )
and 1 /2 (A /(a" ,e 3 ))c H „,(A /(c " 3 )),
which yield that bh e ( a" , c"3), as b • 1-1 (A )=(0 ) by our standard assumption. Let us express b h =an ix +c n 3 y with x, y e A . T hen cn3f = anv + b " '( a" I x + cn3y ), whence f— b"2 'y e ( a n i) : c " 3 . A s an', cn3 i s a n A-regular sequence, this guarantees that f— b"2 ' y E (a ') and so w e have th a t f E (an9+ b"2 ' [(anl, b): c n 3 ]. (Notice that y e (an', b ) : c " 3 .) Thus -
-
n
-
(an', b"2): cn3
-
(a" , ) +bn2 - 1 . [( a" , b): c"3].
As the opposite inclusion is obvious, we have proved that (a"., bn2): c"3 =(a" , ) +b"2 - '. [(an tb ): c "3 ]
for all n , n , n e N. By virtue of this equality and the symmetry between a and b, we further get that 1
2
3
( a n t , b n 2 ) c "3
( a n ] ) b n 2 -1 .
[( a n' , b )
: C
n3]
= (a") + b"2 - 1 .[(b)+ a"1 - 1 . [(a, b ): c "3 ]] =
b n 2 )+
a
ni -i bnr
[(a, b): cn3].
Thus (a" ,
bn2 )
:c" = (a", 1 ) 1ant bn2 . [(a, b): cn3] 3
0 2
- -
-1
-1
Shiro Goto
272 as required.
Now return to the main proof. L et f e A and assume that (abc)" 'f =anx + b y+ cuz -
for some x, y, z e A . Then, as (ab)"lf — cz e (a", bn) : en-i, w e see b y th e above claim that (ab)11-1f— cz e(an, b")+(ab)"t [(a , b): c
n -1
].
Accordingly (ab)"l(f— g) e (a", b", c) f o r so m e g e (a, b): 0 . Let A =A /(c), and let — denote the reduction mod (c). Then, as (a, 5). II,I,(A)= (0) by our choice of a and b (recall that 11,(A )c H (A )), we see by (2.1) that 1--# is contained in [(a): 5] + [ ( b ) : a], i.e., -1
f — g e [(a, c): b]+[(b, c): a]. Hence f e [(a, c): b]+[(b, c): a]+[(a, b): and we have proved that (a n , b n , Ca ) . (abc)"
1
[ ( a , c): b]+[(b, c): a] + [(a, b):
As the opposite inclusion is clear, this completes the proof of (2.2). Corollary 2 .3 . In the sam e situation as (2.2), (abc)"
-1
(a", b", c")
f or ev ery ne N. Let us assume that dim A =3 and let a= a , a 2 , a 3 , a, be a system of elements in in such that dim A l(a)= O. Let H(a; A) denote the homology of the Koszul complex K (a; A ) generated by a over A . L et h: A —>A b e th e A-linear map defined by where ei a re th e canonical basis of the A-module A . h(e 1) =a 1 f o r a ll i, 1 i Then l
4
4
v (H i (a; A))5..v (Ker h ) v A
A
A
(11 1 (a; A ))+ 6
clearly. We put s(A )= sup v A (H i (a ; A)) , where a=a i , a 2 , a 3 , a, runs over systems of elements in ni such that dim A l(a)= O. Notice that if this supremum s(A) is infinite, then A is not a uniformly coherent ring. Proposition 2.4. ( 1 ) L e t A * denote the com pletion o f A . T hen s(A *)=s(A ). ( 2 ) Suppose that A contains a regular local ring R over which A is module-finite. Then s (A ) is infinite, if R is a direct sum m and of A.
273
Noetherian uniformly coherent ring
P ro o f . ( 1 ) Let b= b 1 , b 2 , b 3 , b 4 be elements of mA* with dim A *I(b)A * =O. Choose elements a=a 1 , 0 2 , a 3 , a 4 o f m so that (a)A *=(b)A *. Then, because K(b; A * O K ( a ; A ) as complexes of A*-modules, w e g et a n isomorphism A
Consequently v . (H (b; A*))= (a; A )), whence s (A *). s ( A ) . The opposite inequality is similarly proved. VA ( 2 ) Let a= a,, a , a , 0 be elements of R . Then, because K (a; R ) is a direct summand of K (a; A ) as a complex of R-modules, the R-module H 1 (a; R ) is also contained in H ,(a; A ) as a direct summand. Consequently, we get that A *O H ,(a; A ) of A*-modules.
1-11 (b;
A
i
A
i
2
3
4
v R (111(a; R ))5 R (I 1 (a; A)) v R(A) • v A(1- l i (a; A )).
v (A ). s (A), which tells us that it suffices to show that s (R ) is Therefore s infinite. Now let x, y , z be a system of generators for the maximal ideal n of R , and let 5 be an odd integer. Let H„ denote the n by n alternating matrix defined by [H a i = x
(i
odd a n d j= i + 1) even and j= i + 1)
=y ( i =z
( i+j=n +1 )
=0
otherwise
j . .n), and let I„ be the ideal of R generated by n - 1 by n —1 Pfaffians of the matrix H„ (cf. [1]. See also [13], C h . 5 .). Then as X e , y e , z e I„ (e=(n — 1)/2), the ideal I„ is certainly n-primary and therefore, by virtue of Theorem 2.1 in [1], we see that R II„ is an Artinian Gorenstein local ring with v , ( I „) = n . Let J„.= Then, according to Proposition 3.1 in [10], we get that v2, 1 ( x ,3 ) (xe, y e , ze): I„. y e, z e))=1 and that the dimension r (R 1J,) of the socle of the ring R IJ„ is (T„I(xe, equal to n - 3 . Thus v (J„)= 4. Let J„=(x e, z e , y„) for some element i)„ of n. Then, as r(R IJ„)=v ,(E x tl(R IJ„, R ) ) (cf. [7], 6.10), we find that the R-module R
R IJ„ has a minimal free resolution of the following form:
R"-3
0
R" — > R4
,v
R
0.
Therefore \T (H (x e, y e , z e, v„; R ) ) n —6, whence we know that s (R ) is infinite. R
3.
i
Proof of Theorem 1.1
First of all, we note the following two results due to Quentel [12]. (A detailed proof of Lemma 3.2 may be found also in [13].) Lemma 3 . 1 . L et S be a m ultiplicativ e system i n A . Then f s - A(fl) 5 f A(fl) f or ev ery ne N . H ence 5 - 4 A is a uniform ly coherent ring if so is A.
274
Shiro Goto
Lemma 3 . 2 . L et f: A -4B be a homomorphism of commutative rings making B a finitely presented A -m o d u le . T hen B is a uniform ly coherent rin g if so is A. In case Kerf is a finitely presented and nilpotent ideal of A , the converse is also true.
Proof of Theorem 1.1. (2) (1 ) (c f. [1 2 ], Corollaire)
L e t h: A"—>A b e a n A-linear m a p . Then h ) f (n) for every maximal ideal in of A . Consequently, by Satz 2 "Am (A ® A in [3 ], we see that vA (Ker h ) f (n)+ 2, whence f (n) q .- (n)+ 2 for every n e N. Thus A is a uniformly coherent ring by definition. (1)= .(2) By virtue of (3.1), it is enough to show that dim A 2. Assume the contrary and choose a prime ideal p of A so that dim A i, = 3 . Then, as the ring A by (3.1), again uniformly coherent, to produce a contradiction we may assume that A is a local ring of dimension 3. Suppose that A contains a field and let k be a coefficient field of the completion A * of A . Choose a system x, y, z of parameters for A * and put R =k [lx , y , z l]. Then R is a regular local ring, over which A * is module-finite. Moreover R is, by Theorem 2 in [8], a direct summand of A *, whence from (2.4) it follows that s(A) is infinite — this is a contradiction, and we find that A does not contain any field. Let In denote the maximal ideal of A and put p = ch A/m, the characteristic of the field A /in . T h e n 0 p E M . O n the other hand, as A lpA certainly contains a field and is, by (3.2), uniform ly coherent, w e get, by th e above argument, that dim A lpA . 2. Consequently, the element p of in may be extended to a system of parameters for A , say p, x , y. L et W denote a coefficient ring of A* and put R = W [ix , y l]. Then R is a regular local ring and A * is module-finite over R . Take a prime ideal P of A * such that dim A*IP = 3, and let B denote the normalization of the ring A * I P . Then B is a module-finite extension of A *IP and is again a local ring. Moreover depth B 2, and the B-module H ,( B ) is finitely generated. Now choose the elements x, y so that (x, y). H (B)= (0). Then, according to (2.3), we get that (pxy)n - 1 e(p", xn, y")B for every n e N . Consequently A
A
A
A
B
(PxY)" - 1
(P", x", Y ")A *
for all n e N , which guarantees, by Theorem 1 in [8], that R is a direct summand of A*. T h u s s (A ) must be infinite by (2.4) — this is the final contradiction, and we conclude that dim A 2. Corollary 3 . 3 . L et R be a regular r i n g . Then R is a uniform ly coherent ring 2. if and only if dim
P ro o f . The only if part is a direct consequence of Theorem 1.1, and the if part is due to [14] (cf. Théorème 3). Proof of Corollary 1.2. In case A is a Noetherian semi-local ring, our assertion immediately follows from (1.1). Assume that A is a finitely generated algebra over a field k. Let d =dim A, and choose elements X , X 2 ,..., X d of A so that A is module-finite over the poly1
Noetherian uniformly coherent ring
275
nomial ring R = k [X X 2 ,..., X d i Then by (3.2), we see that A is a uniformly coherent ring if and only if so is R . The latter statement is clearly equivalent to the condition that ( s e e (3.3)).
4.
Example
In this section, we shall construct a Noetherian integral domain A containg an algebraically closed field and satisfying the following conditions: (1) A contains cou ntably m any m ax im al ideals, say { 111 .}..N(2) dim A,„„ = 2 f or every n e N. ( 3 ) For each n e N , there is an A -linear m ap h„: A 2 -4A such that v A (Ker h„)=V A , ( A
rt
„OK er h„) =n+ I.
Of course, this ring A is not uniformly coherent, and one knows by this example that the latter part of the condition (2) in Theorem 1.1 is, in general, not superfluous. Let S = k [X 2 , X 2 , X 3 , X 4 ] be a polynomial ring over a n algebraically closed field k , and let M= S + , the irrelevant maximal ideal of S . Let E be a graded Sm o d u le . W e d e n o te b y E n (g E Z ) the graduation of E . L e t Him (E ) stand for local cohomology modules of E relative to M, which we regard as graded S-modules. For an integer p, we denote by E(p) the graded S-module whose underlying S-module coincides with that o f E a n d whose graduation is given by [E(p)] n =E p + q f o r all g e Z. We put k =S IM . Lemma 4.1. Let n e N . Then there exists a graded prim e ideal P„ of S such that dim SIP„= 2 and 111,1 (SIP„) k"(3— 5n) as graded S-modules. P ro o f . Let 0
0
F2 - > F 1 - - > F o
F4 --> F3
be a graded minimal free resolution of k . We put Z = hn (F 2 F
1 )
a n d E = Z"(2).
T hen, since depths m Em = 2 a n d E r i s a f r e e S r -m odule for every prim e ideal p of S (p M ), one may take, by virtue of Theorem B in [2], an exact sequence 0
S3"-'( — 1)
E
P(r)
0 (*)
of graded S-modules, where r is an integer and P„ is a graded prime ideal of S with dim SIP„= 2. Now consider the following exact sequence 0
—>S
SI P„
0 (**) .
Then we get, by both the exact sequences (*) and (**), that [14(E)] (— r),
276
Shiro G oto
which yields an isomorphism k "(2– r)
1/14 (S/P„)
of graded S-modules since E = Z"(2) and H ( Z ) k . Let us check that r =5n –1. C la im . T h e S-module P „ (r) h a s a graded m inim al fre e resolution of the following form :
0 —> Fft(2)
Fg(2)10S3"- I( – I)
P1(2)
P„(r) —> O.
P ro o f . Recall that E has a resolution
0
P1(2)
F1(2)
P4(2)
E --> O.
Choose a homomorphism g : 5 3 "- 1 ( - 1 ) – F 1 ( 2 ) o f graded S-modules making the following triangle
E
F1 (2 )
com m utative. Then it is a routine work to get that the mapping cone of the following homomorphism S3' - 1 ( - 1 )
•••
0
F'4.(2)
Fg(2)
o
F1(2)
of complexes of graded S-modules provides a graded minimal free resolution of P„(r). Let H (t) denote the Hilbert-Samuel series of the graded ring SIP„, i.e., H (t)= (4 ) [ S I P ] q ) - t . T h e n , since F = S " ' ( – i) and the Hilbert-Samuel series of S E d im k
q=0
we get, by the resolution of P ( r ) obtained by the above claim,
is given by (1
t )
'
that H (t)–
1. -6ntr + (7n– 1)tr+1 –mr+2
_04
On the other hand, as dim SIP,,= 2, we know that H (t)–
f _(1 ) for some polynomial )2
f (t) in t. Consequently, in the following equation 1 – 6 n tr ±
(7 n _
1 )tr+ 1
fl
tr + 2 = 0 ,
t =1 must be a multiple root, whence we get that r = 5n –1 as required.
We put R„ =SIP„ and choose a linear system x„, y„ of parameters for the ring
Noetherian uniformly coherent ring
277
R „. Let gn : R .--+R,, denote the R„-linear map defined by g „ ([ ab j) = a x„+ b y „ for each [ a l E R .
Lemma 4.2. v R . (Ker gn )= n+ 1.
Pro o f . Let IT= R u tx„R n , a n d let 7 - d e n o te th e reduction mod x „R „. Then, since x„ • 1111 (R n ) =(0), by the exact sequence -
0
R„(— 1)
o
R„
w e see that In i (R „) [HL (R n )] (— 1). Because H ( R ) k " ( 3 -5 n ) b y (4.1), this yields a n isom orphism H ( R kn(2— 5n). T ake hom ogeneous elem ents zi o f Rn with degree 5 n -2 so that f o r m a k-basis of 1-17 1 1 (K ) and, subsequently, express y„zi =-- —x„v i ( 1 . i n ) with homogeneous elements vi i n R „ of degree Sn —2. )
C la im . The R„-module Kerg„ is generated by the following elements
IC'
and
[
—
— } 15iSn
.Y,.
-
X,, _
Proof. Let a, b be elements of R„ such that ax,, + by„=O. Then, since y„ is a non-zerodivisor on the ring R„I M (X ), we get that 5 is in I n i (R n ), which allows us to express b = c i zi +ux„ w ith ci E k a n d u e R „. Therefore byn = c i (y„z i ) + (y„u)x„, w hence byn = t c 1(—x„v 1 ) +(y„ti)x„ because y„zi = —x„vi by our choice. i=i Accordingly, we get that a = ± c i vi +u(— y„), since ax„= —by„ a n d since x„ is a non-zerodivisor of R„.
Thus [ ab l = t c i [ v il+ u „ i= 1 z ix
n l, a n d w e conclude that
Ker g„ is generated by n+ 1 elements [ 1(1 n) „and [ — Y"1. zix Because [ v il and [ - -v ni are homogeneous elements of R„ with degree 5 n -2 z ix „ and 1 respectively, to prove that vk , (Ker g„)= n +1 it is enough to check that they form a m inim al basis o f th e graded R„-module Ker g,,, i.e.,[ vz i] (1.._i n ) and Y n are linearly independent mod M • (Ker g„) over the field k. This is routine x„1 and we omit it. Let Q„= [Ra, (= Tv 1/P,,) IP„) for each n e N and put B = C) R „, where the tensor i=1 product is taken over the ground field k. Then, as k is an algebraically closed field, the ring B must be a n integral d o m a in . Clearly Q„B is a prim e ideal of B for all ne N . We put A =7' - ' 8 , where T= B\
n=1
Q„B.
Proposition 4.3. ( 1 ) A is a Noetherian integral domain. (2) Max A = {Q„Aln e N} . (3) dim AQ „A = 2 for every n e N.
Shiro Goto
278 P ro o f. See Proposition 1, [9].
L et Iv A -- A denote, for each n E N , the A-linear map induced by th e R„linear map g„: Ri—>R„, i.e., h„= A ® g„. Recall the following commutative diagram 2
(R
n
)Q
A Q.A
of rings, where all homomorphisms are flat. ( T h e bottom one is, especially, faithfully flat.) Then we find that v „ (Ker g „ )_ v (Kier h,,) R
A
„>vA Q u A (Ker (A Q „A 0 g„)) = v ( R „) o ,, (Ker ((R„) (2 „
g,,)).
Because Q„ is the irrelevant maximal ideal of R„ and Kerg„ is a graded R„-module, we further get v „ „ (Ker ( ( R „) Q , 0 g „ ))= v ( R
)Q
'
R„
„) Q „ ((R n ) Q , 0 K er g„) ' R„
( R
=V R (Ker g„). .
Thus v (Ker h„)=v A
A Q A
( Ker (A Q „A h „ ) )
=v „(Ker g,,), R
whence v (Ker h„) =vA,2,,A (A(2A0Ker h,,)=n +1 b y ( 4 .2 ) . T h is completes our A
construction, and the ring A is a required example. DEPARTMENT OF MATHEMATICS NIHON UNIVERSITY
R e fe re n c e s
[I ] D. Buchsbaum and D. E isenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math., 99 (1977), 447-485. [ 2 ] E. G. Evans, Jr. and P. A. Griffith, Local cohomology modules for normal domains, J. London Math. Soc., 19 (1979), 277-284. [ 3 ] O. Forster, Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z., 84 (1964), 80-87. [ 4 ] S. G o to , Blowing-up of Buchsbaum rings, London Mathematical Society Lecture Note Series 72 (Commutative Algebra: Durham 1981), 140-162, Cambridge Univ. Press, 1983. [ 5 ] S. Goto and N. S u zu ki, Index of reducibility of parameter ideals in a local ring, to appear.
Noetherian uniformly coherent ring
279
[ 6 ] T. H. G ulliksen, Tout idéal premier d'un anneau noethérien est associé d un idéal engendré par trois élém ents, C. R. Acad. Sc. Paris, 271 (1970), 1206-1207. [ 7 ] J. Herzog and E. Kunz et al., Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics 2 3 8 , Springer Verlag, 1971. [ 8 ] M. H ochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J., 51 (1973), 25-43. , Non-openness of loci in Noetherian rin g s, Duke Math. J., 40 (1973), 215-219. [9] [10] E. Kunz, Almost complete intersections are not Gorenstein rin g s, J. Alg., 28 (1974), I I I 115.
[11] F. S. Macaulay, The Algebraic Theory of M odular System s, Cambridge University, 1916. [12] Y. Q uen tel, Sur l'uniforme cohérence des anneaux noethériens, C . R . Acad. Sc. Paris, 275 (1972), 753-755.
[13] J. Sally, N u m b e rs of Generators of Ideals in Local Rings, Lecture Notes in Pure and Applied Mathematics 35, Marcel Dekker, Inc., 1978. [1 4 ] J. -P. S o u b lin , Anneaux et modules cohérents, J. Alg., 15 (1970), 455-472.
