Preservation of some invariants of modules by specialization

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VNU. J OU R N A L OF SCIENCE. M a t h e m a t i c s - Physics t XVIII. n ° l - 2002

P R E S E R V A T IO N O F S O M E I N V A R I A N T S
O F M O D U L E S B Y S P E C I A L I Z A T I O N

Dam Van Nhi

Pedagogical College. T h a i B in h ' V ietn a m

Introduction

Throughout 11 ii> paper wo assume that k is an arbitrary perfect infinite field, and K
is an extension o f A*. We denote? R := A:(w.) [jt'3, and Rn :== A:(a)[.rj. w here It = (ỉ/ 1...Uut)
is a family of indeterminate and o = ( a i...(\,n) € I \ fn• III this paper, we shall say that
a property holds for almost all rt if it holds for all a except perhaps those lying on a proper
algebraic subvariety o f A'"\ For other n otations we refer the reader to [1Ị. For convenience 
we often ohmit. tile phrase “for almost all a " when we are working with specializations.

The theory of specialization of ideals was introduced by w . Krull [5j. [()]. Krull
defined the specialization of an ideal / of R with respect to the substitution u -4 a as th<‘
ideal /,, = { / ( osX) ị f ( i t . x ) € /nfcjtt,./:]}. The ideal I n inherits most of the basic properties
of /. Using specializations of finitely generated free modules and homoinorphisms between
them wo defined in (7j the specialization of finitely generated module. We showed that
the basic properties and operations oil modules are preserved by specializat ions. In [8] we
followed the1 samr approach to introduce and to study specializations of finitely generated
modules OVCT a local ring.

The aim of this paper is to show that some invariants of finitely generated modules
over a local ring are preserved by specializations. We will show that tilt1 specializations
of a Gorcnstein (linear maximal Budisbaum) module is again Gorenstein (linear maximal
Buchsbaum).

Preservation of some invariants of modules by specialization

Let p be an arbitrary prime ideal of R. By [6, Satz 14], the specialization p n of p 
is a radical nil mixed ideal. Let p he an arbitrary associated prime ideal of Pn . We consider
the specialization of finitely generated R p -module.

For short we will pu t s = R[> and S (i = ( R (i)p. D en ote P S and ps ,ầ by m and m0 . 
We start by recalling the definition of a specialization of a finitely generated 5-module.

Let L be a finitely generated S-m od u le. A ssum e th a t

s'* A s ' — > L — > 0

be a finite free presentation of L. As the definition of L a we obtain a finite free presentation

sil ^ s ' — > L n — > 0,

T y p e s e t by

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IK D a m Van N h i

where L n = Cok<T0a, see [8].

The /th B oss and /til B etti num bers of L. which are denoted by //Ç(L ) and 3 ,(L )

respectively. aro defined as follows

ụ?s(L) = d i m E x t < ; ( S / m , L), Vi > 0.
0 i ( L ) = dim.s/ni Torf (S/m. />), V/ > 0.

Proposition 1.1. Let L be a fin itely g en era ted S -n io d id c . Then, for nliiiost rill ( \ % we

fig (L0 ) = i i ‘s { L ) rind ( 3 i( L 0 ) = f t i ( L ) , V i > 0.
Proof. Since L is finitely generated, all integers ị i l$ { L ) are finite. We have

/£(!) = i(Exti(S/m,L)).

By [8. P roposition 3.3], there is

E x t s ( S J m0 , L „ ) a E x t5 (57m, £ )„ .
Since Pn is a radical ideal, from [8, Proposition 2.8] it follows that

f(ExtjỗM (S,v/m,r, L0 )) = i(E x t^ (5 /m ,L )rt) * *(Ext’s(S/m . L ) ) .
Mona*

M ^ (L 0) = 4 ( L ) 1 i > 0 .
Similar, we obtain

& { L „ ) = 0 i ( L ) , i > 0.
We invoke Proposition 1.1 to reprove Corollary 3.8 in [8].

C o r o lla r y 1 .2 . Let L be a finitely generated s-m odule . I f L is a Buchsbcium s - m o d u l e ' 
then L n is also a Buchsbaum S n -module for almost a/i a .

Proof. Put d = dim L. By [8. Theorem 2.7], dim La = d. Since s is a regular ring, by 10.
Chapter 2. Theorem 4.2] we known that L is a Buchsbaum 5-module if and only if

/4 =

(S/mK(tfi(£)), Î = (),... ,d -

.

j

= 0

Since f(//m (L)) < (£<>)) = V ( H m ( L ) ) by [8. Theorem 3.6]. Now the proof is
im m ediately from P roposition 1.1.

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P re se r v a tio n o f sortie in v a r ia n ts o f m o dules by sp e c ia liza tio n

Lemma 1.3. L('t L />('• a fin itely g c iic rn t e d S-mochilt'. T h e n, for almost CÌÌÌ Ụ* liỉìvo
inj. <Iim(Ln ) = inj. dim(L).

In Ị v ì r t ir u lỉir . i f I, is an in je c tiv f» m o d uli', then L n is also ỈÌÌÌ in jc c tiv e module.

Proof. Sinn* /? and R n are Gorenstt'in rings, s and S lt are Gorenstein rin*>s by 1. Proposi
tion ÌỈ.1.19Ì. By [8, Theorem 3.1], we have proj. dim L n = proj.dim L < oc. and therefore
inj. dim /v,, and inj.diniL are finite. From 1. Theorem 3.1.17] we obtain

inj. dim L (i = depth S n = depth 5 = inj. dim L .

If L is an injective 5-module, then inj.diniL = 0. Hence inj.ciiinL,, = 0. and therefore'
L n is injective. By using this lemma we have the following theorem.

Theorem 1.4. Let L be a fin ite ly gen era ted s -m o d u le . It L is a G o ìv iìs t e iìỉ S -iiio d u lc .

then L n is again a G oìvnsteiỉi S tt-ììio d uỉc fo r iihnost nil (X.

Proof. Assume th a t L is a Gorenstoin S-module o f dimension equal (L Then L is a Cohon-

Macaulay .S’-module hv [9. (3.11)] and dim 5 = inj.diniL = d. By [8. Tlu’oivm 2.71.
dim L n — <L By js. Corolllary 3.2]. L ix is also Cohen-Macaulay. Since (liniô'o = dim $ and

inj. dim L t% = inj. (lim L I)V Lmuna 1.3. wo havcMÌini S tị = ill), dim L iX = dim L n . Hence L ix

is also G orm stein.

Now \vr will show that the multiplicity of modulo is preserved by specialization. Wo
bogin with a following lemma.

Lemma 1.5- Let L be H fin itely g enera ted S -n io d u le o f d im en sio n (I. L e t (J\...iffi h r
a svstciỉi o f param eters oïl L . T h e n (vi)íYi-‘ - >{y<i)<* j*s â system o f p a ra m e te rs oil /,,, for 
almost till (I.

Proof. Since (J)...y,i 6 P S, there are (vi , {y(i)n £ P(*Six. By Ị8, Lemma ‘2.3 and
Theorem 2.7], there is

dim L n / ( { y \ )„ ...(y ti ) n ) L n = <lini(L/(//i...y , i ) L ) n = d \ u i L / ( t j \...y ,i)L - 0.
Tilt'll (y\ ),,...(ijti)tt i‘s a sy stem o f param eters Ü11 L n .

Theorem 1.6. L et L he a fin itely genvvnted S - in o d iile o f d im en sio n (I and lot
q = (y i...Vd)S

he a p a ra m e te r ideal on L . T h en , fo r almost aII ft, we have
c (q „:L rt) = e(q;L),

U’lii'iv c(q,t:£ 0) and <*(q:L) are the m u ltip lic itie s o f L a and L w ith respect to q(l and q
resp ectiv e ly.

Proof. By [8, Theorem 2.7], ( lim L ,, = d. By Lemma 1.5, the ideal

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:.o D a m Van N h i
is a parameter ideal oil L (V. The multiplicity symbol of , y (Ị with respect to L is

d e n o t e d b y e ( y \ , . . . y y (i \ L ) , a n d t h e m u l t i p l i c i t y s y m b o l o f ( y i ) t t , • • • , ( y w i t h r e s p e c t

to L0 is denoted by e { ( y , (</r/)0 |Lfl ). We have

“ ^((ỉ/l 1(1! • • • 1 { ỉ j t l ) c ì L n ) ,

e(q: L) = t-(;vi...,VdỊL).
wv* Iim l only show that

e( ( y i , ( » d ) a | ^ « ) = e(/yi.yfz|£).

Wo prove this claim by induction on d. For the case d — 0, bv applying [8. Proposition
2.8] w<‘ have

e (0 |L fV) = f ( L ti) = f ( L) = e(0|L).

Now we rtssunic tliat d > 1. and the claim is true for all S-module with dimension < (I 1.
By [H, Lrmma 2.3 and Lemma 2.5]. there are

L n / { y i ) tiL n = ( L / y \ L ) n and 0 L" : (yi)o s* (0/, : Ij\)n - 
Siiuv tin* dimension of these modules < d - 1, there are

f’( 0 / 2 i{y<i)<x\L(x/ ( y \ ) n La ) = e(y-2>ã ã ã .;ô/,/|L/?7iÊ).
^('(lte)r... *(ù/ii)o|0/. : (ợ/i)ô) = e(y2ợ-.- */iớ|0l : A/i ).

The statment follows from the définition of the multiplicity.

We now will show the preservation of some invariants of module which are given by
N. T. C uoug in [2j.

Recall that (-4. n) is a local ring and L is a finitely generated /l-module of dimension

d. L(‘t (J — {7/1...tj'i} he a system of parameters on L, and q = (y1, . . . . yfi) A . For positive
integers / 1...t,Ị we put / = { il , . . - ,£,/}. In [3] the difference

h.(t.:y) := C{L/(y[ ' , . . . ,y '/ ') L ) - i l .. .írfp(q; L)

is considered as a function in t. Assume that A has a dualizing complex. By [2. Theorem
I II. it is well-known that there are systems of parameters y oil L such that Ifu( t : y ) is a
polynomial in / for all positive integers J ( i and the degree of //.(/: ij) is independent

of tlu* choice of y. Tlie least degree of all polynomials in t hounding If d(t: y), which is
(Irnotod by p (L)< is independent of the choice of y. This nuinerical invariant p ( L ) of L is
called the p o lyn o m ia l type of L . We set

CI,(L) := A n n H ị ( L ) a n d a(L ) := a0( L ) . . . a,/_ i(L ),
whoro H n ( L ) is the zth local cohomology module of L respect to n, and

en d im 0(/,)(!/) := inf{z I H ị ị L , ) ( L ) is n ot finitely gen erated over A\<

see !2]. In [31. p-standard system of parameters on L is defined. We recall that a system
of parameters { //I... //,/} of L is calk'd a p-sta n d a rd system o f p a ra m eters if

r yfi € a(L),

\ |/7 € a ( L /( j/,+ i ...*/<*)£), i = 1, — , rf — 1,
where ĨỊ, is the image of y x in A / ( y , + 1__ ,

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Theorem 1.7. 12. Tlirom n 1.2| Suppose tJiHt A tins H fin alizin g com plex. Then
( i) fj(L) «tim A / ci(L),

(ii) if L is c(Ịiii(liĩĩị('i:sio n iìl th ru p ( L ) — <lim(NcM(L)) = (I - endima ịỉ ) { L ) .

\ \ v set

b, ( / „ , ) : Ann (/,,,) and b{Ln ) := bo(L0 ) . . . b#/- I(Ln,).

Proposition 1.8. Lrt L be H finitely xcnem tcd s-m o d u le o f dimension (Ì. Then, for

íìhìiitsỊ nil a. U7‘ hiìVi'
(i) M /-*. ) = ML).

( i i ) it L is <‘<Ịiii<IiiiH'ìi>i()ỉi;il th en <‘ n < l i n i i , Ị / ) ( / - ! ! ■ ) = e n đ i m aị / ị ( L ) . < l i m ( N < m ( L , , ) )

<lim( Ncwi(L)).

Proof, (i) By [8. Theorem 2.7;. we have cliinLo = (I. Because the rings s and S n have
dualizing eoliiplxcs. In using Theorem 1.7 we only need to show that

dim S n / b ( L n ) = dim S/ b( L) .

lmlrrd. since A n il/7,^ (£,») = A nn/7ró(L)o by [8. Lemma 3.5], wc have
b, ( Ln ) = Oj(L)fV.

Tlioroforo

l l ( L n ) = = ( « < » { £ ) ) , . . l ( £ ) ) a

= (at)(L). . . a</_i(L))o = a(L)n .

(ii) From AiniLo — (AuuL)o and dim L n = (lim L by [8t Theorem 2.7]. it follows that
L n is cquidimonsional if L is <*qui(limcnsioiml. By (i) and by Theorem 1.7 (ii). we ob
tain <’n<limh(/ )(£,♦) = enđima(/,)(£). and (liin(NcM(L0 )) = dim(NcM(£)). The following

Corollary follows immediately from Proposition 1.8 and [2].

C o ro llary 1.9. (8. Theorem 3-0] I f L is H gen era lized C o h en -M a cau lay. then L iS is iìlso AI
g v iic rn liz cd C o h e n -M n cn u la y for almost- nil a .

Lot (/i.m ) he a local ring with dini;4 = (L For the finitely generated /l-mo<lul(‘ L 
of dimension (I. and a paramctiT ideal q, we set

ỉ(q.L) = e(L/cịL)-c.(c\.L),

I ( L ) = siip {/(q .L ) q is a parameter ideal of L }.

I lie minimal nuinlxT of generators of L is denoted by /'(/>). cind 111 0 multiplicity of L wit!»
rc'spcïc't tu ail m-primary ideal is denoted by v ( L ) . In [11], the module L is called a lin e a r 
m a x im a l Duchsbaufil moduli* if

P r e s e r v a t i o n o f s o m e i n v a r i a n t s o f m o d u l e s b y s p e c i a l i z a t i o n '*!

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D a m Van N h i
Proposition 1.10. Let L be a fin itely generated s - m o d u l e o f d im e n s io n (I = dim 5. I f
I. is a lin rn r iiuLxinuil B u ch sb n u m m odule, then L a is also lin e a r m a x im a l B liclislm n iii 
m o d u li' for ỉìlmost all Or.

Proof. By [8. Theorem 2.6]. we have dim L n = ciimL. Since d in i5a = dimS, dim L {X ==
(limS,,. Tilt* equality //(L it) = f i ( L ) follows from the proof in [8. Theorem 3.1]. By
Thromn 1.6. we have <j( L n ) = r ( L ) . Since L is a linear maximal Buchsbaum module,
H L ) < X. rhereforo. L is a generalized Cohen-Macaulay module. By Corollary 1.9. L u is
also a generalized Cohen-Macaulay module. Since f ( H ị n { L n ) = £ ( H ^ ( L ) y 1 = 0...d — 1
by [8. Theorem 3.G), we obtain

/ ( L „ ) = ^ ( f/ 1 V ( ( L (>) = ( d 7 = I(L)

i—ii ' ' / — Ỉ) ^ '

from[4. Satz 3.7]. Hence n ( L n ) = (*(Ln ) -f /(£<-»)• So L a is a linear maximal Buchsbauni
module.

Proposition 1.11. L e t L be a fin ite ly g en era ted S - m o d u lc o f d im e n s io n d. I f a system
o f para m eters 1/ — { / / 1... /yf/ } o f L is a p -sta n d a rd system o f p a ra m e te rs o f L . then, for
ỉìhìiost nil n. =■ { (ỵ/ 1 , (f/,/)o} is aJso a p - s t a iid ã rd system o f p a ra m e te rs o f L<y and

h tt(t : Vn) = h ( t ; y )

-Proof. By Lemma 1.5, Ijti = {(//i)o __ , (ỉ/d)tt} is <l system of parameters of L ( i . Since
( m € a(L),

\ //7 € a(L/(y,+ i ...

y<ì)L)i

i = 1-... , r f - 1.

we have

f

e a(L)„ = b(Lo),

I (.{/,),, € a ( L / ( ÿ i + i ----,y<j)L)» = b(/W ((*/*+i)„ * • • • 1 (ỉta)a)£<fc)' i = ---rf —

1-Hence //„ = i(y i)a i • • • 1 Íỉ/í/)a} is a p-standard system of parameters of L f>. We set
**/ = <*(/71... .Vil(ô-f- 2...y<i)L : ẻ/H1/(ằ*+2i--- <y<i)L)>

^ t * ( (? / l ) o ' • • • 1 ( ] J i ) o I (($/?•+• ‘2 ) o 1 • • • * ( .tyfi ) o ) I J t \ ' { V i -f - 1 ) i ì / ( ( 2 )fv » • • • 1 ( )<» )^<» )•

From [2 it is well-known th a t

/>(£o ) />( /' )

ll.„(t-Un) = ^ 2

an(1

I l \ t - , y ) Y ^ t [ . .

1=0 £ = ()

By Proposition 1.8. p ( L it) = p ( L ) . To prove I t (J;j/n) = Ỉ Ị , ( t : y ) we only need to shows
that , i — 0— ,/>(£)• By [8, Lemma 2.3 and Lemma 2.5], we obtain

( ( .{ /» 4 - 2 ) a T • • • s ( y d ) i i ) ) 0 / ( ( j / |4-2 ) 0 1 • • • • ( y

( ( ,VH- 2 ẻ ã • • » ,Vr/ ) ^ • V i + l / { V i + 2 ) • • • 1 ) ^ ) a j

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P r e s e r v a t i o n o f s o m e i n v a r i a n t s o f m o d u l e s b y s p e c i a l i z a t i o n

Theorem 1.12. Let L hv H finitely gvnei'Htcd S-inodulc o f dimension fl Ỉ11KỈ let

!/ - {.VI... .'A/i

he a p-stiUH till'd system o f p a ra m e te rs o f L . F o r a d -tu p e l o f p o s itiv e integers t = {11...(,Ị}
wo set

.</' = {//|‘ ...nil }' = { ( . V i ...( ÿ d ) a ) 
-T h r u , for alm ost fill a. \vc Ini VC

f ( L „ / ( y lJ L a ) = e ( L / ( y t)L ).

Proof. By 1’ropo.siton 1.8. jf/o = {(ỉ/ 1 . ( y , i) n} is also a p-standarcl system of param
eters of Lt%. and

ỈL.At-y») = ỉiẢt:y)-

\\\' obtain tho result (’( L <i/ ( ( j tn )I,iS) =. f ( L / ( y t )L) from the following equalities

i ( L lt/(i/n ) L (>) = l Ltt{t\ya ) + ti .. . t ,/e(j/ „:Ln ).

( ( L / ( i / ) L ) = //,(*:?/) + * 1 . ,./.f/e(ĩ/:L).

References

1. \v. Bruns and J. Herzog. C o h e n -M a ca u la y rin g s. Cambridge University Press. 1993.
2. w T. Cuong. On the dimension of the non-Cohen-Macaulay locus of local rings

admitting dualizing complexes. Math. Proc. Cam p. Phil. Soc. 109( 1991). 479-
488.

3. N. T. Ouong. On the lea.st degree of polynomials bounding above' the differonc.es
Ix'twivn lengths and m ultiplicities of certain system s of p aram ete rs ill local rings.
NatỊoyn Math. ./. 1 2 5 (1 9 9 2 ). 105- 114.

4. X. T. Cuong. |>-staii(lard systems of parameters and p-standard ideals in local rings.
A r i a Math. V ietnam ien 20( 1995). 146- 161.

5. \v. Krull. Prtrameterspozialisiorung in Polynoinringen, Arch. Math. 1 ( 1948). 56-64.
(). D.v. Nlii and N.v. Trung. specialization of modules, C om m . Algebra 27 (1999).

2059-2978.

7. D.v. Nhi and N.v. Trung. Specialization of modules over local ring, ,/. P w v Aply.

comm. Algebra. 152(2000). 275-288.

8. R. Y. Sharp. G orenstein M odules, Math. z. 115(1970), 117-139.

9. YY. V. Vasconcelos. C o m p u ta t io n a l methods i n com m utative algebra and algebraic 
(jcomcfry. Springer-Verlag Berlin Heidelberg New York, 1998.

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Il K. Yosliida. Oil linear maximal Buchsbaum module*. C on n u. Alf/t'hni. 23(100'»).
10«."»-1130.

T AP CHI KHOA HOC DHQGHN. Toán • Lý. t XVIII, n ° l - 2002

s ự B Á O T O À N M ỘT s ố BAT BIÊN C Ủ A M Ô Đ U N KHI Đ Ạ C BIỆT HOÁ
Đàm Van Nhi

Kh o a Tốn. C a o dání> Sư pliạm T h á i Bình

Troim hài báo này trường k được giả thiết là hoàn háo. vỏ hạn. Trường A' là một
mớ lộiiũ của A1. KÝ hiệu l ì := k ( u ) ị x } . R n Ả-{n)[./•). với tập u — ( » 1 .II,,,) gồm m
tliani số và n = ( ííị...u,„) t K " ' .

Lý thuyct đặc hiệt hoá iđêan được W.Krull dưa ra tron a Ị 5 |. |6 |. Đ ặc biệt hóa cua 
idẽan I ỗ R l iốan:

L = { f ( n . . r ) \ f ( u . . r) € / n A-Ị||. J-)} c /?„.

w . Krull dã chúmẹ minh nhiều lính chất cua iđẽan / được háo toàn khi đ ặ t biệt hoá. Sừc . . .
dụng dặc hiệt hoú mỏtlun tự d o hữu hạn sinh và đồng cấu giữa ch ú n g , chúng tỏi xây 
dựng lý thuyết đặc biệt hố c h o m ơdun hữu hạn sinh trên vành đa thức và trẽn vành ctịa 
phương. Trong hài này, chúng tôi chi ra sự khổng thay đổi một số bất biến môđun khi

đạc hiệt hoậ. Chúng lôi chi ra dặc biệt hố mơđun Gorenstein (Buchsbaum tuyến tính
lỏi dại) vẩn là Gorenstein (Buchsbaum luyến lính tối đại).

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