THÉORIE DES NOMBRES BESANÇON
Armées 1998/2001
O n t h e h o m o l o g i c a l d i m e n s i o n s o f p u l l b a c k s . I I
N . K O S M A T O V
O n t h e h o m o l o g i c a l d i m e n s i o n s o f p u l l b a c k s . I I
N i k o l a i K o s m a t o v
In this article, ail rings are assumed t o have identity elements preserved by ring h o m o m o r p h i s m s , a n d ail modules are left modules. For a ring À, let lgld A a n d w d A d é n o t é t h e left global dimension of A and t h e weak dimension of A, respectively. For a A-module X , we d é n o t é t h e injective, projective a n d flat dimensions of X by idA X , p dA X a n d fdA X , respectively.
T h e left finitistic injective, projective a n d flat dimensions of A are denoted a n d deflned as follows:
1FID A = sup{idA M | M is a A-module w i t h idA M < oo}, 1FPD A = s u p { p dA M | M is a A-module with p dA M < oo},
1FFD A = s u p { f dA M | M is a A-module with fdA M < oo}.
Consider a 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 —ii-»- Rx
h ii (1)
R2 R',
where R is t h e pullback (also called fibre p r o d u c t ) of R i a n d R2 over R', t h a t is, given r} G R i a n d r2 G R2 w i t h j i ( r i ) = j2( r2) , there is a unique element r G R such t h a t i \ ( r ) = rx and i2( r ) = r2. W e assume t h a t is a surjection.
In t h e present p a p e r we continue our s t u d y of homological dimensions of pullbacks s t a r t e d in [1]. O u r p u r p o s e is t o give u p p e r b o u n d s for t h e finitistic dimensions of R ( T h e o r e m s 1, 2 a n d 3). W e also provide two simple examples of pullbacks where we use these results t o calculate homological dimensions, a n d show t h a t our conditions are essential. In t h e first example, a pullback of two h e r e d i t a r y rings has finite finitistic dimensions t h o u g h its global and weak dimensions are infinité. Therefore, it is impossible t o e s t i m a t e t h e global a n d weak dimensions of a pullback if only t h a t of t h e c o m p o n e n t rings are given. T h e second example d e m o n s t r a t e s t h a t our e s t i m â t e s would not be t r u e if we d r o p p e d t h e a s s u m p t i o n t h a t i\ is surjective.
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T h e o r e m 1. Let n be a non-negative integer. Suppose that f o r every R.- module M of finite injective dimension we have that
i d / ^ E x t l R(Rk, M ) ) ^ n - l f o r l = 0, 1, . . . , n a n d k = 1 , 2 . Then 1FID R ^ n .
P r o o f . Let M b e an i2-module of finite injective dimension.
F r o m [1, Proposition 5] it follows t h a t i dRM ^ n. Therefore 1FID R = s u p { i df l M \ i dH M < oo} ^ n.
Similarly, [1, Propositions 6 a n d 7] allow us t o prove analogous b o u n d s for finitistic projective a n d fiât dimensions.
T h e o r e m 2 . Let n be a non-negative integer. Suppose that f o r every R - module M of finite projective dimension we have that
p dH f c( T o r f ( Rk, M ) ) ^ n - l f o r 1 = 0, 1, . . . , n a n d k = 1 , 2 . Then 1FPD R ^ n .
T h e o r e m 3 . Let n be a non-negative integer. Suppose that f o r every R - module M of finite f l a t dimension we have that
f d ^ ( T o r f ( Rk, M ) ) ^ n - l f o r l = 0, 1, . . . , n a n d k = l , 2 . Then 1FFD R < n .
E x a m p l e 1. Let s ^ 2, R ' = Z / s Z , R i = R2 = Z, R = { ( m i , m2) e R i x R2\ m i = m2 (mod s) }. T h e n in t h e c o m m u t a t i v e square (1) w i t h canonical surjections ik a n d jk t h e ring R is t h e pullback of R \ a n d R2 over R'. T h e r e exist t h e periodic free resolutions of t h e f?-modules Rk
. . . H R m R H R - ^ R x - ï 0, (2)
(a,0) nR ^ R ^ R R (0,3) 0 fs,o; 2 0, (3)
where t h e syzygies are t h e s u b m o d u l e s s Z x 0 ~ R i a n d 0 x s Z ~ R2. It is easily seen t h a t t h e short exact sequences
0 — • 0 x s Z <-> R R i — • 0,
do not split. Hence t h e Zï-modules Rk are not projective. B y [2, T h e o r e m 3.2.7], t h e y are not fiât either. It follows t h a t p dR Rk = f d # Rk = oo and l g l d i ? = w d i ? = oo. At t h e same time, l g l d Rk = wd Rk — 1. W e see t h a t it is impossible t o e s t i m a t e lgld R a n d wd R with only lgld Rk a n d wd Rk given.
Let M b e a n /^-module of finite projective dimension w i t h a projective resolution
. . . — > 0 — • 0 — » Pn — * . . . — • P0 — • M — » 0. (4) Since l g l d Rk = 1, we have pdf l f c(Toro ( Rk, M ) ) < 1. A p p l y i n g [2, Exercise
2.4.3] t o t h e projective resolutions (2), (3) a n d (4), we o b t a i n for a sufficiently large t
T o r f ( Rk, M ) ~ T o v f+ 2 t( Rk, M ) ~ T o r f ( i ?f c, 0 ) = 0.
Consequently, p df l f c( T o r f ( Rk, M ) ) = pdHfc 0 = 0. T h e o r e m 2 now yields t h a t 1FPD R ^ 1. In t h e s a m e m a n n e r we can use T h e o r e m s 1 a n d 3 t o show t h a t 1FID R ^ l a n d l F F D i ? ^ 1.
Consider t h e following projective resolution of t h e -R-module R / ( s , s ) R : 0 — • R R - A R / ( s , s ) R — 0 .
Since this short exact sequence does not split, we have p dr( R / ( s , s ) R ) = f dr( R / ( s , s ) R ) = 1. T h i s clearly forces 1FPD R = 1FFD R = 1.
T h e s u b g r o u p R of t h e free Abelian g r o u p Z x Z is a free Abelian group also, therefore, applying t h e f u n c t o r H o m z (R, — ) t o t h e short exact sequence 0 —>• Z Q —> Q / Z 0, we o b t a i n a short exact sequence of i?-modules
0 — • H o mz( i ? , Z ) —> H o mz(JR , Q ) —•> H o mz( i ? , Q / Z ) —> 0.
T h i s sequence does not split and, by [2, Corollary 2.3.11], it is an injective resolution of t h e i î - m o d u l e H o m z ( i î , Z). It follows t h a t id/î(Homz(.R, Z)) = 1, a n d hence t h a t 1FID R = 1.
E x a m p l e 2. Let F be a field. Define R ' = F ( x , y ) , R i = F(x)[y), R2 = F(y)[x]i R — R i H R2 = F [ x , y ] , T h e n t h e ring R is t h e pullback of R i a n d R2 over R ' in t h e c o m m u t a t i v e square (1) with inclusions ik a n d jk none of which is surjective. W e claim t h a t in this case our results are not true.
By [2, P r o p o s i t i o n 4.1.5, Corollary 4.3.8], we have w d i ? = lgld i î = 2 a n d wd Rk — lgld Rk = 1. Since these dimensions are finite, we have t h a t
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l F F D i t ! = l F P D i ? = 1 F I D R = 2 a n d 1FFD7?* = l F P D i E * = l F I D i îf c = 1.
It is easy t o check t h a t t h e / ï - m o d u l e s Rk are flat. So t h e a s s u m p t i o n s of T h e o r e m s 2 a n d 3 hold for n = 1, b u t their conclusions are false. T h e s a m e observation c a n b e m a d e a b o u t t h e estimâtes [1, P r o p o s i t i o n 5, 6 and 7, T h e o r e m s 9 a n d 10, Corollaries 12 a n d 13], which are not t r u e in this case either.
It can b e explained by t h e fact t h a t t h e surjectivity condition cannot be d r o p p e d in t h e basic resuit [1, T h e o r e m 1]. Indeed, we see a t once t h a t t h e .R-module M — R / ( x R + y R ) is neither projective nor flat, whilst t h e Rk- modules Rk <S>r M = 0 are projective a n d flat. From [2, Proposition 3.2.4]
we conclude t h a t t h e R - m o d u l e X = H o mz( M , Q / Z ) is not injective, t h o u g h t h e i?f c-modules H o mf i( / 4 , X ) ~ Homz(i?f e A f , Q / Z ) = 0 are injective.
A c k n o w l e d g e m e n t . T h e a u t h o r would like t o t h a n k Professor A. I. Generalov for suggesting t h e problem and guidance.
R e f e r e n c e s
[1] N . KOSMATOV. On the global a n d weak dimensions of pullbacks of non-commutative rings. Publications M a t h é m a t i q u e s de l ' U F R Sciences et Techniques de Besançon, Théorie des Nombres, Années 1 9 9 6 / 9 7 - 1997/98, 1 - 7 (Univ. de Franche-Comté, Besançon, 1999).
[2] C . A . W e i b e l . An introduction to homological algebra. Cambridge, C a m b r i d g e Univ. Press, 1994.
E q u i p e de M a t h é m a t i q u e s Université de F r a n c h e - C o m t é 16, r o u t e de G r a y
25030 Besançon Cedex France
D e p a r t m e n t of M a t h e m a t i c s a n d Mechanics S a i n t - P e t e r s b u r g S t a t e University
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