DSpace at VNU: Specializations of Rees rings and integral closures

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 Sharp, Local Cohomology: an algebraic introduction utk geometric applications, Cambridge University Press, 1998 V Krull, Pa.ra.meterspezialisierung in Polynomringen, Arch Math., 1(1948), 56-64 EV Nhi and N v TVung, Specialization of modules, Comm Algebra, 27(1999) 2*59-2978 E.V Nhi and N v Trung, Specialization of modules over local ring, / Pure Appl Agebra, 152(2000), 275-288 E V Nhi Specialization of graded modules, Proc Edinburgh Math Soc 45(2002) 41-506 OCV 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: Can lb Pil Soc., 50(1954), 145-158 8A Seidenberg, The hyperplane sections of normal varieties, Trans Arner Math Sc, 69(1950), 375-386 y j Sriickrad and w Vogel, Buchsbaum rings and applications, Springer Berlin ISC ()W / Vasconcelos, Computational methods in commutative algebra and algcbraic gorn.etry, Springer-Verlag Berlin Heidelberg 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