THÉORIE DES NOMBRES BESANÇON
Années 1996/97-1997/98
O n t h e g l o b a l a n d w e a k d i m e n s i o n s o f p u l l b a c k s o f n o n - c o m m u t a t i v e r i n g s
O n t h e g l o b a l a n d w e a k d i m e n s i o n s o f p u l l b a c k s o f n o n - c o m m u t a t i v e r i n g s
N i k o l a i K o s m a t o v
Ail rings will be assumed to have identity elements preserved by ring ho- m o m o r p h i s m s , and ail modules, unless specified otherwise, will be left m o d - ules. For a ring S, lgld S and lwd S will dénoté t h e left global dimension of S and t h e left weak global dimension of S, respectively. For an 5 - m o d u l e X and a right S - m o d u l e F, t h e projective dimension of X , t h e injective dimen- sion of X , t h e flat dimension of X , and t h e flat dimension of Y are denoted by p d5 X , ids X , f d s X and r f d s Y, respectively.
A c o m m u t a t i v e square of rings and ring h o m o m o r p h i s m s R — ^ Ra
«2 ii (1)
R2 R '
is said to b e a pullback (or a cartesian square, or a fibre p r o d u c t ) if given r i E R \ , r% G Ra with Ji(r'i) = 3 2 ^ 2 ) t h e r e is a unique element r Ç R such t h a t î'i(r) = r j and i2( r ) = r2 (note t h a t if j2 is a surjection t h e n so is b u t not conversely). T h e ring R, is called t h e fibre p r o d u c t (or pullback) of Ri a n d R,2 over R!.
Assuming t h a t j2 is surjective, Milnor [5, C h a p t e r 2] characterized pro- jective modules over such a ring R. Facchini and V â m o s [2] established ana- logues of Milnor's t h e o r e m s for injective and flat modules. K i r k m a n a n d K u z m a n o v i c h [3, T h e o r e m 2] showerl t h a t if (1) is a pullback square with j2
surjective, t h e n
lgld R ^ max{lgld Rk + r f dR (2)
For c o m m u t a t i v e rings, Scrivanti [6] s h a r p e n e d this u p p e r b o u n d on lgld i?
a n d o b t a i n e d an u p p e r b o u n d on lwd R,. Moreover, she gave examples which 1
showed t h a t , in a certain case, those results were best possible. Besides, Cowley showed t h a t it could b e bénéficiai to work only on one side of t h e rings in question [1, E x a m p l e 3.4]. He obtained t h e following "one-sided"
b o u n d [1, T h e o r e m 3.1]: if (1) is a pullback square with j2 surjective, t h e n
lgld R ^ max{lgld Rk -f p df i Rk} . (3)
k= 1,2
Our T h e o r e m s 9, 10 and Corollaries 12, 13 are t h e generalizations t o n o n - c o m m u t a t i v e rings of Scrivanti's results. In T h e o r e m 8, we provide a "one-sided" b o u n d on t h e left global dimension of a pullback ring. In Propositions 5, 6 and 7, we give sufficient conditions for an /?-module M t o have injective, projective and fla.t dimensions ^ n. In Corollaries 11 and 12, we deduce t h e u p p e r b o u n d s (3) and (2) as i m m e d i a t e conséquences of our Propositions 5 and 6. Here we relax t h e conditions and only require to be surjective.
W e begin with t h e following conséquence of [2, T h e o r e m 2].
T h e o r e m 1. Let (1) be a pullback diagram with i\ surjective. Then an R-module M is injective (projective, flat) if and only if H o m ^ - R i , M ) a n d H o m r ( R .2, M ) (R i ® r M and R2® r M ) are R.y - and R,2-injective (projective, f l a t ) modules respectively.
P r o o f . Set R " — j2( R2) . Since ii is a surjection, we o b t a i n j i ( R i ) C j2(R-2) = R"- T h u s we have a n o t h e r c o m m u t a t i v e square of rings a n d ring h o m o m o r p h i s m s
R - R !
>2 ii
R2 - n
It is clear t h a t this d i a g r a m is a pullback square with j2 : R.2 —>• R " surjective.
So t h e desired resuit follows f r o m [2, T h e o r e m 2],
P r o p o s i t i o n 2. Let M be an R-module, n be a positive integer, a n d let 0 — > M A / o A / , J î + I2— + . . .
be an injective resolution of M . Let Kt dénoté im(/<+i), t ^ 0. Suppose that idRk (Extl R(R,k, M ) ) ^ n — l f o r l = 0, 1, . . . , n and k = 1,2. Then id#fc(Hom/?(./?A;, Kt)) ^ n — t — 1 f o r t = 0, 1, . . . , n and k = 1,2.
P r o o f . For any n ^ 1 and k = 1 , 2 , t h e proof is by induction on t.
For t = 0, if we apply t h e f u n c t o r Ext*R(Rk, —) to t h e short exact sequence of fl-modules 0 —> M —> I0 I<0 — y 0, we obtain an exact sequence of iîfc-modules
0 — • H o mR{ Rk, M ) — • E o mR{ Rk, I0) h .
^ E o mR( Rk, K0) — > E x tl R( Rk, M ) (4) a n d isomorphisms of i?^-modules
E x tl R( Rk, K0) ~ E x t g ^ i f c , M ) , l > 1. (5) Set Akio = im / i * and break u p (4) into two short exact sequences of
i?fc-modules:
0 —> E o mR( Rk, M ) —> H o mR( Rk, I0) H Ak f i — • 0, (6) 0 —> Ak,o M- H o mR( Rk, Ko) —> E x t J ^ , M ) — > 0. (7) Since IQ is an injective i?-module, it can easily be checked t h a t R.k-module
HomJR(/t'fc, /o) is injective. Since k \R k( E o mR( Rk l M ) ) ^ n , we o b t a i n f r o m (6) t h a t idR k(Ak to) ^ w — 1. At t h e s a m e t i m e idnk(Ext}j(Z?.fc, M ) ) ^ n — 1.
Therefore, using (7), we get i d f>k ( Hom R ( R.k, K0 ) ) ^ n — 1.
For t ^ 1, we apply t h e f u n c t o r Ext*R(R.k, — ) to t h e short exact sequence of /^-modules 0 —> Kt~ \ M- It Kt —> 0. We get an exact sequence of /«^-modules
0 —> HomR(R.k, Kt- i ) —> H o mR( Rk, It) ^ H o m R( Rk, Kt) — • Ext^(jRfc, A'f_i) —> 0
and isomorphisms of / ^ - m o d u l e s
Extl R(R.k, Kt) ~ Extl+*{Rk, Kt- \ ) , l > 1. (9) P u t Ak,t = im (ft+i)* and break u p (8) into two short exact sequences of
/ 4 - m o d u l e s :
0 — • H o mR( Rk, A'4_i) —> H o mR( Rk, It) Ak,0 — • 0, (10) 0 —> Ak,t H o mR( Rk, Kt) —> E x t A ' t _ i ) —+ 0. (11)
By t h e inductive hypothesis, we have idp,k{H.omjt(Rk, Kt- i ) ) ^ n — t.
Since /< is an injective R-module, H o m / ^ / i ^ , It) is an injective / ^ - m o d u l e . Therefore, froin (10), idnk(Ak,t) ^ n — t — 1. C o m b i n i n g (5) a n d (9), we have E x t ^ / ? ^ A V i ) ^ . . . ~ E x tl R( Rk, K0) ~ E x t ^ i f c , M ) . Hence
i df l f c( E x tl n( Rk, K ^ ) ) = M ) ) ^ n - t - 1. Finally, f r o m (11),
we obtain idflfc(HomFi(Rk, Kt) ) ^ n — t — 1, as required.
T h e proofs of t h e following Propositions 3 and 4 are similar t o t h a t of Proposition 2 if we apply t h e f u n c t o r s T o r ^ ( R k , — ) t o t h e given resolutions.
P r o p o s i t i o n 3. Let M be an R.-module, n be a positive integer, a n d let
• • • —> P'i A Pi A Po A M —> 0
be a projective resolution of M . Let I \t dénoté ker ft, t ^ 0. Suppose that p df l f c( T o r p ( J 2 * , M ) ) ^ n - l f o r l = 0, 1, . . . , n and A: = 1,2. Then p d nk( R k A't) ^ n — t — 1 / o r f = 0, 1, . . . , « and k = 1,2.
P r o p o s i t i o n 4. Lei M 6e «n R,-module, n be a positive integer, a n d let
be a fla,t resolution of M . Let Kt dénoté ker ft, t ^ 0. Suppose that
f df t f c( T o r A f ) ) « - l f o r l = 0, 1, . . . , n «ml k = 1,2. Then
î dR k( R k Kt) ^ n - t - 1 f o r t - 0, 1, . . . , n and k = 1,2.
P r o p o s i t i o n 5. L d (%) be a pullback diagrarn with %i surjective, M be. an R.-module, and let n be a non-negative integer. Suppose that idK j c(Ext l R( Rk, M ) ) ^ n - l f o r l = 0, 1, . . . , n a n d k = 1,2. T h e n \ dR M ^ n.
P r o o f . For t h e case n = 0, t h e resuit follows f r o m T h e o r e m 1. For n ^ 1, consider an injective resolution of /?,-module M
By définition, p u t Kt = i m ( /< +i ) for t ^ 0. From Proposition 1, we get t h a t / 4 - m o d u l e Homp_(Rk, Kn- i ) is injective (k = 1,2). Hence, by T h e o r e m 1, Kn-X is an injective i?-module. This m e a n s t h a t i d r M ^ n.
Arguing as above, t h e reader will easily prove the following analogues of Proposition 5 for projective and flat dimensions.
P r o p o s i t i o n 6. Let (1) be a pullback diagram with ii surjective, M be an R,-module, a n d let n be a non-negative integer. Suppose that
p df l f c( T o r ? ( Rk, M ) ) ^ n - l f o r l = 0, 1, . . . , n and k = 1,2. Then
p drM < n.
P r o p o s i t i o n 7. Let (1) be a pullback diagram with ii surjective, M be an R-module, and let n be a non-negative integer. Suppose that f d ^ ( T o r f ( Rk, M ) ) ^ n - l f o r l = 0 , l , . . . , n a n d k = 1,2. T h e n î àRM ^ n .
Proposition 5 clearly implies t h e following t h e o r e m .
T h e o r e m 8 . Let (1) be a pullback diagram with i\ surjective, and let n be a non-negative integer. Suppose that f o r any R-module M we have that
idf l f c(Extl R{Rk, M ) ) ^ n - l f o r l = 0, 1, . . . , n a n d k = 1 , 2 . Then lgld R ^ n.
T h e o r e m 9 . Let (1) be a pullback diagram with i\ surjective, and let n be a non-negative integer. Suppose that f o r any left idéal J of R, we have that
p d ^ T o r P ( Rk, R / J ) ) ^ n - l f o r l = 0, 1, . . . , n a n d k = 1 , 2 . Then lgld R ^ n.
P r o o f . Let J b e a left idéal of R. Proposition 6 shows i m m e d i a t e l y t h a t p dr( R / J ) ^ n. Therefore, using Auslander's t h e o r e m , we have lgld R = s u p { p dr{ R . / J ) | J is a left idéal of R } ^ n.
T h e o r e m 1 0 . Let (1) be a pullback diagram with i\ surjective, and let n be a non-negative integer. Suppose that for any finitely generated left idéal J of R we have that
f d ^ ( T o r f [ Rk, R / J ) ) ^ n - l f o r 1 = 0 , 1 , . . . , n a n d k = 1 , 2 . Then lwd R ^ n.
P r o o f . Let J be a finitely generated left idéal of R. Propo- sition 7 evidently implies t h a t i dR( R / J ) ^ n. Therefore, since lwd R, = s u p { p dR( R / J ) | , / is a finitely generated left idéal of R}, we have lgld R ^ n.
C o r o l l a r y 1 1 . If (1) is a pullback diagram with i\ surjective, then lgld R ^ max{lgld Rk + p dH Rk} .
P r o o f . Set nk = lgldR.k, rnk = p df i R.k, Nk = nk + mk (k = 1 , 2 ) and N = max{iVi, N2}. It can be assumed t h a t mk, nk< 00. Let M be an i?,-module and k G {1, 2}. Silice p df l Rk = mk, we have E x t 'R( Rk, M ) = 0 for ail l ^ mk + 1 . At t h e s a m e t i m e , since l g l d i î ^ = nk, we get i dR k( E x tl R( Rk, M ) ) ^ nk = Nk - mk ^ Nk - l ^ N - l for any l = 0, 1, . . . , mk. Therefore, by Proposition 5, p dRM ^ N . This m e a n s t h a t lgld R ^ N.
Similarly, Propositions 6 and 7 allow us to prove Corollaries 12 and 13.
C o r o l l a r y 1 2 . If (1) is a pullback diagram with i\ surjective, then lgld R < max{lgld Rk + r f df l Rk} .
C o r o l l a r y 13. If (1) is a pullback diagram with i 1 surjective, then lwd R sC m a x { l w d Rk + r f dH Rk} .
A c k n o w l e d g e m e n t . T h e a u t h o u r would like to t h a n k Professor A. I. Generalov for lus help and guidance.
R e f e r e n c e s
[1] K . M . CoWLEY. One-sided bounds and the vanishing of E x t . J . Alge- bra, 1 9 0 ( 1 9 9 7 ) , 361-371.
[2] A . FaCCHINI, P . V a m o s . Injective modules over pullbacks. J . London M a t h . Soc., 3 1 (1985), 425-438.
[3] E . IClRKMAN, J . KuZMANOVICH. On the global dimension of fibre prod- uctif. Pacific J. M a t h . , 1 3 4 ( 1 9 8 8 ) , 121-132.
[4] S. MACLANE. Homology. Springer-Verlag, New York, 1963.
[5] J . MlLNOR. Introduction to Algebraic K-theory. Annals M a t h . Studies, vol. 72, Princeton University Press, Princeton, 1971.
[6] S. SCRIVANTI. Homological dimension of pullbacks. M a t h . Scand., 7 1 (1992), 5-15.
E q u i p e de M a t h é m a t i q u e s U . M . R . 0623 du C.N.R.S.
16, r o u t e de Gray
25030 Besançon Cedex France
D e p a r t m e n t of Mathernatics and Mec.ha.uics S a i n t - P e t e r s b u r g S t a t e University
Bibliotechnaya pl. 2
Saint-Petersburg, 198904, Russia E-Mail: koko@nkl442.spb.edu
