Preservation of some invariants of modules by specialization

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 [r]

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

47

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(ExtjçM (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

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.

First o f all, we will recall the definition o f th e G orenstein m odule Lot ( A n ) he a

Noetherian local ring of dimension (I. Let L be a Cohen-Macaulay 4-moclule L is called

a Gorc.m tcin Æ-module if climL = inj.dim L = rf, [9] Before proving the preservation of

G orenstciness o f m odule, we will show th a t th e in jective dim ension o f m od u le L is not change by specialization

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.

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 (liniơ'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 ọl 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.

~ ( ( / / 1 )»>»•••

:.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<i )n 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>• • • ;«/,/|L/?7i£).

^('(lte)r *(ï/ii)o|0/„ : (î/i)«) = e(y2î-.- * ĩ / ií |0 l : 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 ,

ÎA/M-Finally, we denote by N o m( L ) th e non Cohen-Macaulay locus o f L, i.e.

N (’m(L) := Ị P g S u p p L I L p is not C ohen-M acaulay}.

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

= (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

ịi(L) = e( L) + I(L).

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)

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 =■ { (î/ 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(ÿ«-f- 2 y<i)L : Î/H1/(»*■+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

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 < i ) a

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

and therefore the equalities e' = i = 0 , ,p(L), follow from Theorem 1.6

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 Ỉ 11 KỈ 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

10 K Yaumgishi Recent aspect o f the theory o f B uch sbaum m odules College of Liberal Arts Himeji Dokkyo University.

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 Toán 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 ç R là iđèan:

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 hoá c h o m ôdun 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 hoá môđun Gorenstein (Buchsbaum tuyến tính lỏi dại) vẩn là Gorenstein (Buchsbaum luyến lính tối đại)