VN U J O U R N A L O F S C I E N C E , M athem atics - Physics T x x , N q - 2004 S P E C IA L IZ A T IO N S AND OF REES IN T E G R A L R IN G S CLOSURES D a m Van N hi Pedagogical University Ha Noi, Vietnam P h u n g T h i Y en The Upper Secondary School Dong Anil, Hrt Noi, Vietnam A b s t r a c t The paper presents the specializations of Rees rings, associated graded rings and of integral closure of ideals T h e preservation of some invariants of lings by special izations will also be concerned In tro d u c tio n Let k be an infinite field of arbitrary characteristic Denote by K ail extension field of k Let u = ( u i , , U t n ) be a family of indet.ennina.tes and o = ( a i , , O',,,.) a family of elements of K We denote the polynomial rings in n variables £ , , x n over k(u) and k(ot) by R = k(u)[x] and by R Q = k(ct)[x], respectively The first step toward ail algebraic theory of specialization was the introduction of the specialization of ail ideal by w Krull [2] Let I be an ideal of R The specialization of with respect to the substitution u — > a € k m is the ideal If* = { / ( a ,x ) | /(w,x) G /nfe[u,:r]} c fc[x] Following [2] the specialization of / wit h respect to the substitution u — > a G A defined as the ideal I Q of 7?a generated by elements of the set is {/(a ,a ;)|/K x )G /n A ;[^ x ]} A Seidenberg [7] used specializations of ideals to prove th at hyperplane sections of nor mal varieties are normal again under certain conditions Using specializations of finitely generated free modules and of homomorphisms between them, we defined in [4] the special ization of a finitely generated module, and we showed th at basic properties and operations on m od u les are preserved by s p ecia liza tio n s Ill [3] we followed th e sam e approach to in troduce and to s t u d y s p e c ia liz a tio n s o f finitely gen erated m od u les over a local lin g [4] and of graded modules over graded ring [5] We will give the definitions of specializations of Rees rings and associated graded rings, which are not finitely generated as /?-mođules and we want also to study specializations of integral closures of ideals In this paper, wc shall say th a t a property holds for almost all a if it holds for all points of a Zariski-open non-empty subset of K ni For convenience wo shall often omit the phrase "tor almost all a ” ill the proofs of the results 1T h e a u th ors are p a r tia lly su p p o rted by th e N ation al B asic Research Program T y p eset by ^Ạa/Í-5-T^X 25 26 D a m Van N h i , P h u n g Thi Yen S o m e r e s u l t s a b o u t s p e c i a li z a t io n o f g r a d e d m o d u l e s Let k be an infinite field of arbitrary characteristic Denote by K an extension field of k Let u = ( u , Urn) be a family of indeterminates and a = ( « ! , ,Qm) a family of elements of K Let m and ma be the maximal graded ideals of R and i?Q, respectively The specialization of ideals can be generalized to modules First, each element a(u, x) of R can be written in the form p( u, x) a( u, x) = —y " q{u) with p( u, x) G k[u,x] and q(u) £ k[u] \ {0} For any a such th a t q(a) Ỷ we define p(a,x) a{a, x) = > ■ q(a) Let F be a free 7?-nio(lulo of finite rank The specialization Fa of F is a free 7?tt-modulo of the same rank Let Ộ : F — > G be a homomorphism of free 7?-modules We can represent Ộ by a matrix A = (di j (u, x)) with respect to fixed bases of F and G Set Aa = (.atj ( a , x )) Then A a is well-defined for almost all a The specialization ộct • Fa — > Ga of Ộ is given by the matrix A a provided th a t A q is well-defined We note that the definition of (f)a depends on the chosen bases of Fa arid G a D e fin itio n [3] Let L be ail i?-mocỉule Lot Fi - A F() — > L — > be a finite free presentation of L Let Q : (F i) a — > (F0)a be a specialization of Ộ We call L a := Coker a a specialization of L (with respect to ợí>) If w e ch oo se a different finite free p resen ta tion — > Fq — > L — > we m ay get a different specialization L'a of L, but L a arid L[y are canonically isomorphic [4, Proposition 2.2] Hence L a is Iliiicjilcly deterniinecl up to isomorphisms T he following lemmas show that the operations and the dimension of modules are preserved by specialization L e m m a [3, Proposition 3.2 and 3.6] Let L be finitely generated R-inocIule ãiìd M , N submodules o f L, and Ỉ an ideal of /? Then, for almost all O', (i) ( L / M ) a = L a / M a , (ii) ( M n N ) a = M n n Na, (iii) (M 4- N)a — Ma -f Na, (iv) {IL)a = I a L a Let L be a finitely generated R -module The dimension and depth of L are denoted by (lim L and depth L, respectively L e m m a [3] Let L be a finitely generated R-nioclule T h e n , for almost a 11 a, we have (i) A n n L a = (Ann L)a , (ii) dim L a = dim L, (iii) depth = depth L We recall now some facts from [5] which we shall need later First we note that R is naturally graded For a graded /?-inođule L, we denote by Lị the homogeneous component S p e c ia liza tio n s o f R e e s r in g s a n d in te g l clo su res 27 of L of degree t For an integer h we let L(h) be the same module as L with grading shifted by //., that is, we set L(1i)t = L/H-* Let F = © s=1 R ( —hj) be a free graded i?-rnodule We make the specialization Fn of F a free graded 7?a -moclule by setting Fa — = i R o t(-h j) Let s1 50 j=l j=1 be a graded homomorphism of degree given by a homogeneous m atrix A = (dij(u,x)) Since d e g (a ii(u ,x )) + hoi = = deg (a iSo(u,x)) + h 0so = h u , A a = (a,; (a, j ) ) is a homogeneous m atrix with , x)) + hoi = = deg (aiso( a, x) ) + hQso = h UTherefore, the homomorphism ộct : R a (-h lj) j=1 » R a ( —hoj) j=1 given by the matrix i4a is a graded homomorphism of degree L e m m a 1.3 [5, Lemma 2.3] Let L be a finitely generated graded R-niodule Then La is a graded R a-inoduie for almost all a Let F — ■» Ft Fg-I — ■ > • • • — > Fi F() — > L — > be a minimal graded free resolution of L, where each free module Fi may be written in the form (Ị) R{ —j)- jlJ, and all graded homomorphisms have degree The integers Pi j Ỷ axe called th e graded B e t t i n u m b e r s o f L T h e follow ing lem m a sh ow s th a t th e graded B etti numbers are preserved by specializations L e m m a 1.4 [5, Theorem 3.1] Let F # be a minimal graded free resolution of L Then the complex ( F ) „ : — > (F ,)„ — > > (F ,)« (Fo)« — >— is a minimal graded free resolution o f L a with the same graded B etti numbers for almost ni l a S p e c ia liz a tio n o f R e e s r i n g s a n d a s s o c i a t e d g r a d e d rin g s Let 1/ , , Vs be a sequence of distinct indeterminat.es The polynomial ring of 2/1 , ■, i/s with coefficients in 7? is denoted by i?[y] Let L be a finitely generated i?-module Then besides considering the polynomial ring R{y\ we may also consider polynomials ill Ỉ/1 ? • • • iVs with coefficients belong to L The set L[y} of all this polynomials has a natural structure as a module over R[y] It is easily seen th at L[y] = L min{r, dim Bp} for all p £ Spec(-R) W ith out loss of generality we can assume th a t A = k(u)[x , ,Xd] is a Noether normalization of B In this case B is a finitely generated graded A-module Using the above proposition we are now in a position to prove the following result, see [6, Lemma 4.3] C o ro lla ry 3.2 I f B satifies Serre’s condition (Sr), so is Da for almost all a Proof We consider D as a finitely generated graded A-module Suppose that F : — > A dt ^ A d' - ' — > > A di ^ A d° — > D — > is a minimal graded free resolution of D Denote by I j ( B ) the ideal I, — rank(/?j By [10 , Proposition 7.1.3], we know th a t D satifies (Sr) if and only if ht I j ( B) > j + r , j > By Proposition 3.1, A a = Ả;(a)[xi, ,Xd] is a Noether normalization of Bex and F q : — A ị' A ị ' - — > ■> A ị l A ị° — * Da — ►0 is a minimal graded free resolution of Da by Lemma 1.4 Since n k (ipj)ct — n k ipj and lit I j ( B„) = ht I j ( B) f()r all j > by Lemma 1.2, therefore B cỵ satifies Serre’s condition (Sr) by [10 Proposition 7.1.3] The proplern of concern is now the preservation of the reduction number of D by specializations First, let us recall the definition of reduction num ber of a graded algebra Assume that B = ©t>o-ơf is a finitely generated, positively graded algebra over a field D q — k and z , , Zd G k \ [ D\ ] su ch th a t A = k \ [ z \ , , Zd] is a N ot her n orm alization of B Let , v s be a minimal set of homogeneous generators of D as an A-module s D = A v j , deg Vj = m j j= i The reduction n u m b e r t a ( B ) o f D w ith resp ec to is t h e su p r em u in o f all rrij P r o p o s it i o n 3.3 Let A be a Noether normalization o f B Then almost all a v a {B) = VAfX(Ba) for Proof As above, without loss of generality we can assume th a t A — /c(u)[a:i, , X(i\ is a Noether normalization of B Let V i , , v s be a minimal set of homogeneous generators of B as ail Ẩ-rnodule s B = A v j , d e g Vj = rrij 3=1 S p e c ia liz a tio n s o f R e e s r in g s a n d in te g l c lo s u re s 31 We have dim B a = d by Lemma 1.2 T hen A a = k ( a ) [ x i , \ , x d] is a Noether normaliza tion of Dry by Proposition 3.1 and Dn = Ỵ^S j =i.A a(vj)a, deg(Vj)a = degVj by definition of specialization Hence TAa {B,y) = sup{deg(uj)Q} = sup{degVj} = To study the specialization of integral closures of ideals we will recall the notion of reduction of ail ideal, an object first isolated by N orthcott and Rees, see [1] Let Q and b he ideals of D a is said to be a reduction of b if a c b and abr = br+1 for some nonnegative integer r and th e least in teg er V w ith th is p rop erty is called th e r e d u c tio n num ber of b with respect to a This number is denoted by r a(b), and it is the largest non-vanishing degree of b An element e B is integral over a if there is ail equation z'n + a i z ' n~ l + ■• • + a , „ = 0, a , e a* Denote the set of all elements of D, which are integral over a, by n ã is called the integral closure of ideal a Note th a t z £ 13 is integral over a if and only if z t € B[t] is integral over B[at} The set of all ideals of D which have n as a reduction has a unique maximal member T hat is Õ by [1, Corollary 18.1.6] An ideal a is said to be integrally closed if a — Õ To study specializations of integral closures we need the following L e m m a 3.4 Let a and (.1 be ideals u ỉ B (i) I f a c li, then n c b (ii) If a is a reduction of b , then b c Õ (iii) I f a is a reduction o f b, then ã = b Proof, (i) Assume th at a c b Suppose th at G ã There is an equation z'n + d \z'n ^ + ■ ■■ + ( l/n — , (lị E Cl Since a' c IV', therefore G b Hence ã c b (ii) Assume that a is a reduction of b, then each element of b is integral over a by [1 , Proposition 18.1.5] T hus b c a (iii) Assume that, n is a reduction of b T hen a c b Thus ã c b by (i) Because is a reduction of [i, therefore b c ã by (ii) Thus b c (ã) We need prove (ã) = ã Since a is a reduction of Õ and a is a reduction of (a) by [1 Corollary 18.1.6], a is a reduction of (a) It implies n = (ã) from the maximality of integral closure of a L e m m a 3.5 Let a be an ideal o f D Then (a)a c aQ and (a)„ = a,, for almost nil a Proof Note th a t if b is an ideal of D and a is a reduction of b, then there is an positive integer r such th at all'- = IV+1 Hence a« c bQ and a„b';, = b ra+1 by Lemma 1.1 (iv) Also, a,, is a reduction of bu Since a is a reduction of a, therefore aQ is a reduction of (o)o by above note Hence (0)o c 0^ and = (ã)o follows from Lemma 3.4 (iii) T h e o r e m Let a be an ideal o f B The integral closure o f the B ees ring Z3[a„/] is the integral closure o f a specialization o f the integral closure o f the Rees ring D[at], Proof We know th at the inegral closure of D[at] is the graded subring T = above definition, the specialization of T is the graded subring T a — © j> o(& )atJ ■By the D a m V a n N h i , P h u n g T h i Y en iiegalc.osure of T a is © j>o(aJ )QíJ Because (aJ)a = nJ Q by Lemma 3.5 and a and j „ y >econmmtc, i.e (a J)q = ( aa )j = aJc , therefore ® j > o a 3a P is t h e integral closure o f a, sjcili-tf'ion ®j>o(aj )a tJ for almost all a Fopiition 3.7 L e t q be a paramer ideal o f D Then e(q^; D a ) = e(q: B) for almost, all 1)0 ] K well-known that e(q 77: B a ) = e(qa ] B a ) and e(q; D ) = e(q: B ) by [7], The proof icjuidiite from the equation e(qa, B a ) = e(q; B) by [6 Theorem 1.6] p.fteices A- P Brodm ann and R Y 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Now York, 1998 ... Nhi, Preservation of some invariants of modules by specialization ,/ of Sc.iex'f, VNU t XVIIL Math.-Phys 1(2002), 47-54 DC N orthcott and D .Rees, Reductions of ideals in local rings, Math Plot:... closures of ideals we will recall the notion of reduction of ail ideal, an object first isolated by N orthcott and Rees, see [1] Let Q and b he ideals of D a is said to be a reduction of b if a c b and. .. closure o f the Rees ring D [at] , Proof We know th at the inegral closure of D [at] is the graded subring T = above definition, the specialization of T is the graded subring T a — © j> o(& )atJ ■By the