Though specialization is a claiisical method in A lgebraic Geometry, there is no system atic theory for what can be “specialized”.. Wc show ed that the Cohen-M acaulayness, the Gorenste
Trang 1VNLJ Journal o f S c ie n c e , M ath em atics - P h ysics 25 (2 0 0 9 ) 3 9 -4 5
The total specialization o f m odules over a local ring
Dao Ngoc Minh*, Dam Van Nhi
D e p a r tm e n t o f M a th e m a tic s, H a n o i N a tio n a l U n iv e rsity o f E d u ca tio n
1 3 6 X u a n Thuy R o a d , H a n o i, Vietnam
Received 23 March 2009
Abstract, ỉn this paper we introduce the total specialization o f an fínỉtcly generated module
over local ring This total specialization preserves the Cohen-Macaulayness, the Gorensteiness
and Buchsbaumness o f a module The length and multiplicity of a mcxiule are studied.
1 In tr o d u ctio n
G iven an object defined for a fam ily o f parameters u ~ ( u i , , U r n ) w e can often substitute u
by a fam ily a ( a j , , a ,n ) o f elem ents o f an infinite field K to obtain a similar object w hich is
called a specialization The new objcct usually behaves like the given object for almost all a , that is,
for all a cxccpt perhaps those lying on a proper algebraic subvariety o f K ^ Though specialization is
a claiisical method in A lgebraic Geometry, there is no system atic theory for what can be “specialized”.
The first step toward an algcbraic theory o f specialization w as the introduction o f the special- ualiun ut an ideal by w Krull m |1 | (Jivcn an ideal 7 in a polynom ial n n g i i — /c(u)Ịx|, where k IS
a subficld o f K , he defined the specialization o f / as the ideal
= { f ( a , x ) \ f ( u , X ) e I n A:|ti,a:|}
o f the polynom ial ring R o c k { a ) \ x \ For almost all a € K ^ , l a inherits most o f the basic properties
o f / Let pu be a separable prime ideal o f R In [2], w e introduced and studied the specializations
o f finitely generated m odules over a local ring Rp^ at an arbitrary associated prime ideal o f pa (For
specialization o f m odules, sceT |3 |) N ow , w c w ill introduce the notation about the total specializations
o f m odules Wc show ed that the Cohen-M acaulayness, the Gorensteincss and Buchsbaumncss of a module arc preserved b y the total specializations.
2 S p ecia liza tio n s o f p rim e sep a ra b le ideals
Let pu be an arbitrary prime ideal o f R The first obstacle in defining the specialization o f Rp^
s
is that the specialization pa o f pu need not to be a prime ideal B y [1], pa ~ n pi is an unmixed
i J \
ideal o f
/?a-* C orresponding author E-m ail: m inhdn@ hnue.edu vn
39
Trang 2A ssum e that d im p u — d and (^) is a generic point o f pu over k Without loss o f gcncralit>',
w e may suppose that this is normalised so that ^0 ~ 1- Denote by (ti) - { vi j ) w ith / “ 0 , 1 , ., d,
j ~ 1 , , n , a system o f {d + l ) n new indeterminates Vij, w hich arc algebraically independent over
k{u^ Cl) • • • J 6 i ) ‘ We enlarge k{ u) by adjoining (z;) We form d + 1 linear forms
n
Vi ~ ~ ^ ^ '^ijX j, 2 = 0, 1 , , d.
Then ti)[x| n /c ( u , ^;)[ỉ/Ị = ( / ( u , ti; yO) M 2/ư)) is a principal ideal We put Aj = with
i “ 0 , 1 , , d Then A o , , satisfies / ( u , t;; Ao, J Ad) = 0 and is callcd the g ro u rid -fo n n o f pu* The prime ideal pu is called a separable prime ideal if it’s ground-form is a separable polvnom ial
We have the follow in g lemma;
L em m a 2.1.[1, S a tz 14Ị A sp ecia liza tio n o f a p rim e se p a ra b le id e a l is an in tersection o f a fin ite p r im e
id ea ls f o r a lm o st a ll a
Let the prime ideal pu be separable A ssum e that Pa = n pi set T = n \ pi)-
L em m a 2.2 F or a lm o st a ll a , w e h ave { R o ,)t '■y sem i-lo ca l ring.
Proof Note that T is a m ultiplicative subset o f R a- We show that { R o ) t is a sem i-local ring Indeed,
let m be a maximal ideal o f { R o )t - Then, there is a prime ideal q o f R a such that m - (\{R cx )t -
Suppose th atm D p i(/ỉa )T ,iT i P i( /? « ) t - We have q D p i , q pi - Since m = q ( /? „ ) r is a maximal
ideal, q n T = 0 Hence q c u pi- Therefore, it exists j such that q c p j Then pi c p j, contradiction.
i = l Hence m = p i(/? a )r
-The natural candidate for the total specialization o f Rp^ is the sem i-local ring { R „ ) r -
D efín ition We call ( /? „ ) r a to ta l sp ecia liza tio n o f Rfi, w ith resp ect to a For short vvc will put
5 = Rp^, S a = { Ra ) p and S t = { R a ) T , where p is one o f the pj Then there is ( S r i p r
-4 0 D.N M inh, D v N h i / VNƯ Jou rn a l o f Science, M ath em a tics - P h ysics 25 (2 0 0 9 ) 3 9-4 5
3 T he total sp ecia liza tio n o f /?p„-tnoduIes
Let / be an arbitrary elem ent o f R We may write / = p{ u, x ) / q { u ) , p{ u, x ) € Ả:Ị?Í, a;|, q{u) £ fc[u] \ { 0 } For any Q such that q { a ) / 0 w e define f a := p ( a , x ) / q { a ) It is easy to chcck that this elem ent does not depend on the choice o f p (u , x ) and q{ u) for alm ost all a N ow , for every fraction
« — / / ổ ) f ĩ 9 ^ R, ổ 7^ 0, w e define Ua := f a / 9 a i f 9 a Ỷ 0 Then Cq is uniquely determined for
almost all a
The follow ing lem m a show s that the above definition o f S t reflects the intrinsic substitution
u —> a o f elem ents o f R.
L em m a 3.1 L et a b e an a rb itra ry elem ent o f s Then a„ € S t f o r a lm o st a ll a
Proof Since pu is a separable prime ideal o f R , pa / R a for alm ost all a Let a — Ị / g with Ị , g £ R ,
g ị p„ Since p is prime, pu : 5 = p„ B y ị l , S a tz 3 |, p„ = (pu : g)oc = pa : 5a- Hence e T Then
Oq G S a for almost all a
First w e want to recall ứie definition o f specialization o f finitely generated 5-m oduIc by |2
Let F, G be finitely generated free 5-m odules Let (/>: F —> G be an arbitrary hom om oiphism o f free 5-m od ules o f finite ranks With fixed bases o f F and G , Ộ is given b y a matrix A = (ttý ), aịj € s
Trang 3D.N M inh, D v Nhi / VNƯ Jou rnal o f Science, M ath em a tics - P h ysics 2 5 (2009) 39-45 41
By Lcm m u 3.1, the matrix Acc ((a ij)tt) has all its entries in {R a )p for alm ost all a Let Fa and
b e fn;c (/?a)p-nioduIcs o f the same rank as F and G , respectively.
D efin itio n |2ị For fixed b ases o f Fa and G a , th e h om om orp h ism 4>a ' p'a G a given by th e
tnatri X is ca lled th e sp e c ia liza tio n o f Ộ w ith resp ect to a
The definition o f ộtỵ does not depend on the choice o f the bases o f F, G in the sense that if B
is the matrix o f Ộ w ith respcct to other bases o f F, G , then there arc bases o f G a such that B a is
the m.atrix o f ộct w ith rcspcct to these bases.
D e f ì a ì t ỉ o n 1 2 | L e t L b e a f i n i t e l y g e n e r a t e d 5 - m o d u l e a n d F\ ^ Fo —* L 0 3, f i n i t e f r e e
preseiTitation o f L T h e (/? « )p -n io d u le La '.—C o k evộ a is called a sp e c ia liza tio n OĨ L (w ith respect
to Ộ )
Then, w c have the follow in g results.
L em m a 3.2 |2, T h eo rem 2 2 | Let O ^ L - > M - ^ N —* O b e a n exact sequence o f fin ite ly
L em m a 3.3 |2, T h eo rem 2 6 | Let L b e a fin ite ly g en era ted S -m odule, Then, f o r a lm o st a ll a , we
have
(ii) (Ann L )a ^A nn (Lrt).
!i i) (iirn L — dim Lf^.
L em m a 3.4 [2, T licoren i 3 1 1 Lei L b e a fin ite ly g en era ted S -m odule Then, f o r a lm o st a ll a , we
have
(j) projL,^ F>rojL.
(ii) depthL,^ d e p th L
Now w e w ill define the total specialization o f an arbitrary finitely generated 5-m od ule as follow s A s above, the matrix ((ajj),^) has all its entries in S t for alm ost all a Let F t and G t be free
5 r-n i odulcs o f the same rank as F and G, respectively, and is the matrix o f ỘT with respect to
thcsi bases.
Dcfiiiiition Let L be a fin itely generated 5 -rn o d u le and Fi Fo —> L —> 0 a finite free
Ị)rc\<cĩìlation o f L T h e 5 7’-n io d u le L r C ok er^ x is called a to ta l sp e cia liza tio n o f L (w ith re.spCK:t to Ộ), The module L t depends on the chosen presentation o f L, but L f is uniquely determined
up to isom()q-)hisms Mcncc the finite free presentation o f L w ill be choscn in the form
L - 0.
L e n m a 3.5 Let L b e a fin ite ly gen era ted S-m odule S u ppose that p = p \ JTien {Lr)piT^ — L<xfor
alm )isi a ll a
^ 0 be a finite free presentation o f L , There exists an exact sequence
\4 >r\vr
aim js i a il a
P ro)f Let iS'* > L ^ 0 be a finite free presentation o f L , There exists an
( lĩc) Ỷ ^ ^ ^ 0 This w ill induces also an exact sequence |(/? a )r |p
Lq\p>.j- —^ 0 B y an easy computation Afx = ((a jj)a ) = ịị follow s that
( ổ t j ) a / l Since |(/? a )r lp r ” “ *5«} w e have a commutative diagram
{Rah
{< ỉ > t ) pt
r
P t
ệa
-<ỉ>a
P t { ỉ ĩ a h r P t \ L t P t
q s V'a
Trang 4where to rows are finite free presentations o f and and an isomorphism ^
Hence {L t ) p ' t ~ I^OI for alm ost all a
P roposition 3.6 L et L b e a fin ite ly g en era ted S -m odule F o r a lm o st a ll a , we have
(i) { k x m L ) o , ^ k x m { L T ) p T ’
(ii) dim L = dim
Lx-Proof, (i) Since { L t ) pt - L, ol b y Lemma 3.5, there is A m \{L T )p T = A n n ((L T )p 7-) ^ Ann(Lc^) Since A n n (L )a = A n n (L a ) b y Lemma 3.3, therefore A rm (L )a — A n n (L r )p 7^ for alm ost all a (ii) We have d im L = d im La by Lemma 3.3 Then d im L = dim (L T )p ^ Semilarly, ( h m L ~
d im (L x )p 7* f o r i — 1 , , s Hence dim L = dim L t for alm ost all a
T heorem 3.7 L ei O —^ L —^ M ^ N —^ O b e a n exact sequ en ce o f f in ite ly g en era ted S -m odides
Then 0 —> L r M t N t 0 is exact f o r a lm o st a ll a
Proof Since 0 —> L —► M —^ A T ^ O i s a n exact sequence, the sequence 0 —► La M a —» 0
is also exact by Lemma 3.2, or the sequence 0 —> {L t ) pt^ —> { N t ) pt 0 exact for
every maximal ideal pT- H ence 0 —> L r —► M t N t —> 0 is exact for alm ost all a
Proposition 3.8 L et L b e a fin ite ly g en era ted S -m odule F o r a lm o st a ll a , w e have
(i) projL = prcjL T ,
(ii) d ep th L = d ep th L x
-Proof, (i) Since p rojL = p rojL a for almost all a b y Lem m a 3.4, there is p ro jL r — su p
tn^supi^x) {p ro j(L a )m } = p rojL „ = p rojL
(ii) B y [4, L em m a 1 8 Ij, there is a maximal ideal m o f S t such that d e p th L r = d e p th (L 7-)„, =
à im { L r )p ^ Then d e p th L x = d ep th L a = d ep th L by Lem m a 3.4.
P roposition 3.9 L et L b e a S -m o d u le o f fin ite length Then L t is a S r -m o d u le o f fin ite length f o r alm ost a ll a M oreover, i ị L r ) = s t ( L )
Proof Since t { L a ) = i { L ) b y [2 , P ro p o sitio n 2.8] and í { L t ) = ^ ((L r)m ) by |5, 3.
T h eorem 12|, there is £ { L t ) = s £{ L)
Proposition 3.10 L et L b e a fin ite ly g en era ted S -m o d u le o f dim ension d an d q = ( a i , , Ufi)S a
p a ra m eter id e a l on L Then, w e h ave e{ q x , Lt ) = se(q , L) f o r a lm o st a ll a , where e { q r , W )
e(q, L) a re the m u ltip licities o f L t and L with respect to (\T o n d q, respectively.
Proof First, w e w ill to show that e(q a, L a ) = e(q, L ) Indeed, Since a i , , a j € p 5 , for almost all
a there are ( a i ) a , (a^)a G p a S a - B y Lemma 3.2 and b y Lemma 3.3, d im L a / ( ( a i ) „ , («d)a)
L a = di m L / ( a i , , a j ) L = 0 Then ( a i ) c , , (a<i)a is a system o f parameters on La- The m ulti
plicity sym bol o f a j , , a j w ith respect to L w ill be denoted b y e ( a i , , OfilL), and the m ultiplicity sym bol o f ( a i ) a , , { ad) a w ith respect to L a by e ( ( a i ) a , , (a<i)a|L a) Then w e have
e(qa;L„) = e((ai)a, , ( a d ) a | L a )
e (q ;L ) = e { a x , , a d \ L )
We need only show tìiat e ( a i , , a d \ L ) = e ( ( o i ) a , { a d )a \L o ) This claim w ill be proved by induction on d For d = 0 , b y applym g [2, P ro p o sitio n 2 8 ], there is
e ( 0 |L „) = £ ( L „ ) - £ ( L ) - e ( 0|L ).
4 2 D.N M inh, D v N h i / VNƯ Jou rn al o f Science, M a th e m a tics - P h ysics 25 (2009) 3 9 -4 5
Trang 5N o w w e assum e that d > Ỉ, and the claim is true for all 5 -m od u les with the dim ension < d — 1 By
[2, L em m a 2 3 and L em m a 2.5Ị, there are
Lo c / { a i ) a L a ~ { L / a i L ) c and Oi,„ : ( a i) a = (O i : a i ) „
S in ce the dim ensions o f these m odules < d — 1, therefore
• -i {(^d)oi Lct/{<xi)fỵLa) ” e ( a 2, • , L /a iL )
e ( ( t t 2 ) a , • • • , ( « d ) a | 0 £ „ , : ( a i ) a ) = e ( c 2 , , a d | O i ; U i )
The statment fo llo w s from the definition o f the m ultiplicity.
N o w w e prove tíie result e (q x , L t ) = se (q , L) Sm ce
meECfir)
by Ị5, 7.8 T h eo rem 15], there is e{qTi L t ) = se(q , L) for alm ost all a
D.N M inh, D v Nhi / VNU J o u rn a l o f Science, M a th em a tics - P h ysics 25 (2009) 3 9 -4 5 43
4 P reservation o f so m e p ro p ertíes o f m od u les
B y virtue o f Proposition 3 10 one can show that preservartion o f Cohen-M acaulayness by total specializations.
T h eorem 4 1 L et L b e a fin ite ly g en era ted S -m odule F o r a lm o st a ll a , w e h ave
(i) L t is a C uh en -M acaulay S r -m o d u le i f L is a C ohen-M acaulay S-m odule.
(ii) L r is a m axim al C vh en -M acau lay S r -m o d u le i f L is a m axim al Cohen-M acaulay S-m odule.
P roof Wc need only show that { L t ) p t is a (maximal) Cohen-M acaulay (5r)pj7.-module if L is a (iiiux.inial) Cuhcn-M acaulay iS m odule.
(i) Assum e that L is a Cohcn-M acaulay 5 -m o d u le Therefore d im L = d ep th L Since dim L = dim Lcfc
by Lemma 3.3 and ciepthL — d ep th L a b y Lemma 3.4, w e get d im La ~ depthLrt* Hence Loc is also a Cohen-M acaulay 5 a -m od u le for alm ost all a Since L a = (L x )p ^ , it follow s that L t is a Cohen-M acaulay 5r-naodule for alm ost all a
(ii) Assum e that L is a m axim al Cohcn-M acaulay 5-m od ule Therefore d i r n L = dim 5 Since
(limLfj = c l i mL and d im 5 a = d im 5 , it follow s that dim La = dim 5 a - Hence L r is a maxi
mal Cohen-M acaulay 5 7 ^-module.
The iih B a ss and ith B e tti n u m b ers o f L, w hich are denoted b y respectively, are defined as follows:
ị i ị ( L ) = d im 5/^ E xt*5 (5 / m , L ) , A ( ^ ) = d im 5/ „ T o r f ( 5 /m , L ), V i > 0.
Lem m a 4.2 L et L b e fin ite ly g en era ted S -m odu les Then, f o r a lm o st a ll a , w e have
= A ( L ) , V i > 0.
Proof Since L and L a are the fin itely generated m odules, all integers f i g i L) and ( La ) are finite
We have
Trang 6t ì y Ị2, P ro p o sitio n 3 3 |, there IS E x t^ ( S c / m a , L c ) - E x t s { S / x n , L) c Sincc pa IS a radical ildcal,
from |2, P ro p o sitio n 2 8 | it follow s that
^ ( E x t ^ ^ ( 5 „ / m « , L „ ) ) = i { E x t s { S / m , L ) a ) = ^ ( E x t 5 ( 5 / m , L ) )
Hencc fẨ.^{L) = ỊẤg (L ,i) Similar, w e obtain Pi { L) ~ Pi { La )
Before invoking Lemma 4.2 to reprove Corollary 3.8 in |2], w e w ill define a quasi'Buchsbaum
module A finitely generated module over a Noetherian commuUitivc ring is said to be a (fuasz-
B uchsbaum module if its localization at every maximal ideal is a suijective Buchsbaum.
C orollary 4.3 Let L b e fin ite ly gen era ted S-m odules Then, f o r a llm o st a ll a , w e h ave
(i) I f L is a su rjective Buchsbaum S-moduIe, then L a d lso a su rjective Buchsbaum Sf^-module.
(ii) I f L is a quasi-B nchsbatim S-m odule, then L t is a lso a quasi-B uchsbaum S r-m o d u le.
Proof, (i) Put d — dim L B y Lemma 3.3, dim L c — d Since S' is a regular ring, by [6, Chapt€^r 2
T h eorem 4.2| w e known that L is a surjective 5-m od ule i f and only if
i f/s (L ) = ỵ ^ ạ ,^ j { S /m ) e { I P J L ) ) ,i ^
j =0
Since Í { H Ì { L ) ) < oo, therefore í { Hị n^{ La) ) = í { Hị n { L) ) by [2, T h eo rem 3.G| N o w the p ro o f is
immedialtely from Lemma 4.2.
(ii) It is easily seen that the localization o f L t at every maximal ideal is a surjcctive Buchsbaum,
Hence L t is also a quasi-Buchsbaum 5r-m od u le.
We w ill now recall the definition o f the Gorcnstcin m odule A non-zero and finitely generated
L is said to be a G o re n ste in module if and only i f the cousin com plex for L provides a injcctivc
resolution for L , see |7] Before proving the preservation o f Gorensteiness o f modulo, w c w ill show that the injective dim ension o f module L is not change by specialization.
I emma 4.4 ỉ.et ỈJ h e fin ite ly gen era ted S -m n d u lea Then, f o r a lm o st a ll fv, we h a ve
in j.d im (L a ) = in j.dim ( L).
In particular, i f L is an in jective module, then La is a lso an in jective m odule.
Proof Since s and 5 a have finite global dim ensions, therefore in j.d in iL and inj.ciiniLtt arc finite
From |8, T h eorem 3.1.17] w e obtain the follow ing relations
in j.d im L ,, = d e p th 5 c = d c p t h 5 = in j.diniL
If L is an injectivc m odule, then inj.dirnL = 0 Hence in j.d im L a = 0, and therefore La is also an
injective module.
T heorem 4.5 Let L b e fin ite ly gen era ted S-m odules I f L is a G orenstein S -m odule, then { L t ) pt again a Gorenstein { ST) pr - f ^odul e f o r alm ost a ll a
Proof Assum e that L is a Gorenstein 5-m odule o f dim ension d Then L is a Cohen-M acuulay S -
module and dim 5 = in j.d im L = d by |7, T h eorem 3 1 1 | Since dim L a = dim L = li by Lemma
3.3 and in j.d im (L a ) = in j,d im (L ) by Lemma 4.2, therefore d im 5 a = in j.diniL ^ = d i i n L „ IIciicc
{ L t )pt is again a Gorenstein (5r)pT^-module for alm ost all a
C orollary 4.6 Let I b e an id e a l o f s I f S / 1 is a G orenstein ring, then S t / I t again a Gorenstein ring f o r a lm o st a ll a
Proof We first w ill recall the definition about the Gorenstein ring A Noctherian ring is a Gorcnstein
ring if its localization at every maximal ideal is a Gorenstein local ring Sincc the localization o f
4 4 D.N M inh, D v N hi / VNƯ Jou rn al o f Science, M a th em a tics - P h ysics 25 (20 09 ) 3 9 -4 5
Trang 7S t / I t every maximal ideal is also a Gorenstein ring b y Theorem 4.5, therefore S t / I t is again a Gorenistcin ring for alm ost all a
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[3] D V Nhi, N Y Trung, specialization o f modules, C om m A lgebra 27 (1999) 2959.
[4] D Eiscnbud, C o m m u ta tiv e algebra w ith a v ie w tow ard a lg eb m ic g e o m etry , springer-Verlag, 1995.
[5] D O, Northcott, L esso n s o n I~ings, m odules a n d m u liip lic itie s, Cambridge at the University Press 1968.
[6| K Yamagishi, R e se n t a sp ec t o f th e th eory o f B u cksbau m m odĩães, College o f Liberal Arts Himeji Dokkyo ưni-
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[71 R,Y Sharp, Gorenstcin Modules, M ath z 115 (1970) 117.
[8] w Bruns, J Herzog, C o h en -M a ca u la y rin gs, Cambridge University Press, 1993.
D.N M inh, D v Nhi / VNƯ Jo u rn a l o f Science, M a th em a tics - P h ysics 25 (2009) 39-4 5 45