Multi-graded Hilbert functions, mixed multiplicities

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1 ulti-graded Hilbert function, mixed multiplicitie Irena Swanon* ultiplicitie of ideal are ueful invariant, which in good ring determine the ideal up to integral cloure. ixed multiplicitie are a collection of invariant of everal ideal, generalizing multiplicitie, and capturing ome information on the interaction among ideal. Teiier and Riler [Tei73] were the firt to develop mixed multiplicitie, in connection with ilnor number of iolated hyperurface ingularitie: the equence of ilnor number obtained by interecting with general i-plane arie a mixed multiplicitie of the ideal generated by the partial derivative of the defining power erie with the ideal correponding to the point ee Theorem 2.5. Ree connected mixed multiplicitie to joint reduction ee Theorem 3.. Thi paper i meant to be an introduction to the topic of multi-graded Hilbert function, mixed multiplicitie, and joint reduction. There i much that i omitted, and a partial lit of known reult i given at the end. Familiarity with ordinary multiplicitie and reduction i aumed. Throughout, R i a Noetherian ring, i a poitive integer, I,..., I are ideal in R, and i a finitely generated R-module. We will denote -tuple of non-negative integer a n,..., n or a n. All comparion among -tuple are componentwie. We will ay that n i ufficiently large if there exit an -tuple e uch that n e; of coure, ufficent largene depend on the context. By I n we denote I n.. Preliminarie The technique ued to handle everal ideal at the ame time are imilar to the technique for handling ingle ideal. We need a multi-ideal form of the Artin Ree Lemma, and exitence of pecial, ufficiently general, element with repect to the given ideal. Theorem.: Generalization of the Artin Ree Lemma Aume that and N are R-module contained in a larger R-module T. Then there exit c uch that for all n c, I n N = I n c I n c I c Ic N. Proof: Let X,..., X be variable over R, and let A be R[I X,..., I X ], the o-called the multi-ideal Ree ring, namely the ubring of R[X,..., X ] generated by element a i X i, a a i varie over element of I i. Let G be the A-ubmodule T [I X,..., I X ] of T [X,..., X ]. Set H = n I n X n N G. Then A i an N -graded Noetherian ring, G * The author acknowledge upport from NSF grant DS

2 i a finitely generated N -graded A-module, and H i a graded A-ubmodule of G. Thu H i a finitely generated A-module. Let h,..., h t form a homogeneou generating et of H over A. Define c = max{deg h i, i =,..., t}. The theorem i of coure true if n = c. Now let n c, with n i > c i for at leat one i. Fix one uch i. Let m I n N. Then mx n Xn i a homogeneou element of H, and o it can be expreed a j m jh j, where each m j i multi-homogeneou in A, and where for each j =,..., t, deg m j + deg h j = n. A n i > c i, for each j =,..., t we may write m j = k a kjb kj, with each a kj homogeneou of degree e i = 0,..., 0,, 0,..., 0 in the ith place, and each b kj homogeneou of degree deg m j e i. Hence m = k,j a kj b kj h j I i I deg m j e i h j I i I deg m j e i +deg h j N = I i I n e i N, which i in I i I n c e i I c Ic N by induction on n. Thi prove that I n N I n c I c N. The other incluion hold trivially. Lemma.2: Aume that R i local with maximal ideal m, that the reidue field R/m of R i infinite, and that I i not contained in prime ideal P,..., P r. Then there exit an integer c > 0 and a finite union V of proper R/m-vector ubpace of I /mi uch that for each a I whoe image in I /mi i not in V, the following hold: a i not in i P i, 2 and for all i, n c + i, and n 2,..., n, I n : a i I c I n 2 Such a i ufficiently general. = I n i. Proof: The proof include the cae r = 0. If I i nilpotent, then necearily r = 0, and the lemma hold for all a I and for any c uch that I c = 0. So we may aume that I i not nilpotent. Then I /I 2 0. Let A be the Noetherian N-graded ring A = n 0 I n I n + where I ha weight, and I 2,..., I have weight 0. Then G = I n I n + n 0, i a finitely generated N-graded A-module. For each prime ideal in A that i aociated to G but doe not contain A, it interection with I /I 2 i necearily proper, hence by Nakayama Lemma, the image of thi interection in I /mi i a proper ubpace. Let 2,

3 W,..., W t be all the ubpace of I /mi obtained in thi way. Similarly, the image P i of the P i in I /mi are proper ubpace. Define V to be the union of the W i and P i in I /mi. Since I /mi i a finite-dimenional vector pace over the infinite field R/m, V i a proper ubet. We will prove that any element a I with a + mi V atifie the lemma. Since G i a Noetherian module, we can decompoe it zero ubmodule irredundantly into primary component 0 = i i. Each ideal j : A G i an aociated prime ideal of G. If A i not an element of j : A G, then ince j i a primary module, j : G a i = j for all i. If, however, a lie in of j : A G, then by the definition of V, j : A G contain A, and hence alo c A c. In particular, there exit an integer c uch that A c lie in all uch j : A G. Hence II c n 2 /I c+ lie in all correponding j. Therefore 0 : G a i Ic I n 2 I c+ = j : G a i a j : A G a j : A G j a j : A G j : G a i Ic I n 2 I c+ j : G a i a j : A G a j : A G j j j = 0. In other word, the lemma hold. 2. Hilbert-Samuel polynomial From now on, R i a Noetherian local ring with maximal ideal m, and I,..., I are m-primary ideal. It i elementary to prove that for any a I, I n : a 0 I n I n i a hort exact equence. Thu I n I n 2 = In For pecial a I, tronger reult are obtained: a I n + a I n I n + a 0 I n : a I n. Lemma 2.: Let a I. Aume that a i not contained in any prime ideal minimal over ann, and that there exit an integer c uch that for all n c and all ufficiently large n 2,..., n, I n : a I c I n 2 3 = I n.

4 I Then for all ufficiently large n, n : a I n I n 2 2 In = 0 : a, and o I n I n 2 = In I n 0 : a. + a Proof: Necearily dim > 0. By the generalization of the Artin Ree Lemma Theorem., there exit e N uch that for all n e, I n a ai n e. Thu I n : a I n e + 0 : a. A all ideal are m-primary, for all n ufficiently large, I n e II c e 2 2 Ie, o that I n : a I n e + 0 : a II c n : a. Hence if n i ufficiently large, by the aumption on c, But I n : a I n = In : a + II c n 2 I cin 2 = 0 : a + II c n 2 I cin 2 0 : a = 0 : a I cin 2. 0 : a II c n 2 n I n : a II c n 2 n I n = 0, o that I n : a I n I n 2 2 In = 0 : a. The ret follow from Equation *. Theorem 2.2: For all poitive integer n,..., n, /I n ha finite length. oreover, there exit a polynomial p Q[X,..., X ] uch that for all ufficiently large n,..., n, I n = pn In,..., n. The degree of the polynomial i dim where the degree of the contant zero polynomial i zero. Proof: The cae = wa proved by Samuel in [Sa5]. In general, clearly /I n ha finite length. The proof of the ret proceed by induction on dim. If dim = 0, then for all n = n,..., n with n ufficiently large, I n = 0. Thu for all n ufficiently large, = ha finite length, o p i a contant polynomial. I n In 4

5 Now let dim > 0. Firt a technicality. Let X be a variable over R, and R = R[X] mr[x]. Then R i a faithfully flat extenion of R, dim R = dim R, and the reidue field of R i infinite. For any finitely generated R-module N, dim N = dim N R R. In particular, if N ha finite length, then N R R ha finite length. Furthermore, in that cae, R N = R N R R. Thu it uffice to prove that there exit a polynomial p Q[X,..., X ] uch that for all ufficiently large n,..., n, R R I n R R = I n R R = pn,..., n. But I n R R = I n R R R R = I n R R R. Thu by replacing R by R, by R R, and I i by I i R, without lo of generality R ha an infinite reidue field. Let P,..., P r be the minimal prime ideal over ann. A I i m-primary and dim > 0, necearily I i not contained in any P i. Thu by Lemma.2, there exit c N and a I \ i P i uch that for all n > c, and all n 2,..., n N, I n : a I c I n 2 = I n. Then by Lemma 2., for all ufficiently large n, I n I n 2 = In I n 0 : a. + a A by the choice of a, dim /a < dim, by induction there exit a polynomial q Q[X,..., X ] of degree dim /a = dim uch that for all n ufficiently large, I n +a = qn. Then the right ide of the equation i a polynomial of degree exactly dim. The polynomial p can be built in the tandard way by recurion: p ha degree exactly one more than the degree of q, namely p ha degree dim. The homogeneou part of degree d = dim of the polynomial p Q[X,..., X ] a in Theorem 2.2 can be written a d! d! ei[d ] ; X d Xd 2, d + +d =d where ei [d ] ; denote a rational number. Thi number i called the mixed multiplicity of of type d,..., d with repect to I,..., I. If =, thi i the uual multiplicity of with repect to I. We prove next that ei [d ] ; i actually an integer: Theorem 2.3: Ue the et-up a above. Then ei [d ] ; i an integer. If dim = 0, then ei [0],..., I[0] ; =. If dim > 0, R/m i an infinite field, and d i > 0, then there exit a i I i uch that ei [d ] ; equal,..., I [d i ] i, I [d i ] i, I [d i+] i+ ; /a i. ei [d ] 5

6 In particular, there exit d element of I, d 2 element of I 2, etc., d element of I, labelled a,..., a d, uch that ei [d ],..., I [d a,..., a ] d : a d ; =. a,..., a d a,..., a d Proof: Clearly d = dim. The theorem i trivial if d = 0. Thu we aume that d > 0. Let X be an indeterminate over R, and R = R[X]mR[X]. A for all n, = R R I n, it follow that ei [d ] ; = ei [d ] R,..., I [d ] R ; R R. Thu we may aume that R ha an infinite reidue field. By poibly firt permuting the I i, without lo of generality d > 0, and then chooe a I a in Lemma.2, uch that a avoid each prime ideal minimal over ann. Then by Lemma 2., for all ufficiently large n, I n I n 2 = In I n 0 : a, + a and the two ide are polynomial of degree d. If d =, then we read off that ei [], I[0] 2,..., I[0] ; = ei [0], I[0] 2,..., I[0] ; /a 0 : a = /a 0 : a. Whenever d >, the homogeneou part of the polynomial of degree d give that I n R R d + +d =d,d >0 d! d! ei[d ] ; d n d n d 2 2 nd 2 = d + +d =d d! d! ei[d ] ; /a n d nd 2, for all ufficiently large n, o that ei [d ] ; = ei [d ] 2,..., I [d ] ; /a. Then by repeating the contruction of a, there exit d element in I, d 2 element in I 2, etc., d element in I, labelled conecutively a a 2,..., a d, uch that ei [d ] 2,..., I [d ] ; /a = a,..., a d Now combine the lat two diplay. a,..., a d : a d a,..., a d It i not immediately clear that all thee integer ei [d ] 2,..., I [d ] ; are poitive. Lemma 2.4: Let d = dim > 0, and a,..., a d m uch that J = a,..., a d +ann i m-primary. For all i =,...,, et i = /a,..., a i. If for all i, either a i i a non-zerodivior on i or 0 : i a i m n i = 0 for n ufficiently large, then the multiplicity of with repect to J equal a,...,a d a,...,a d : a d a,...,a d. Proof: The choice of a i guarantee that for all n ufficiently large, 0 : i a i J n i = 0. By Equation *, /J n /J n = /J n + a J n : a /J n. 6.

7 There exit a polynomial p Q[X] of degree d uch that for all ufficiently large n, pn = /J n. Alo, there exit a polynomial q Q[X] of degree d uch that for all ufficiently large n, qn = /J n + a. Set rx = qx px + px. By the diplay above, r i a polynomial in Q[X] of degree at mot d uch that for ufficiently large n, J n : a /J n = rn. If d =, then px = ex + f for ome rational number actually integer e and f. The leading coefficient, e, i the multiplicity of with repect to J. Hence the diplay ay that for n large enough, e = pn pn = /a a n : a /a n. But for poibly even larger n, a n : a a n = a n + 0 : a a n = 0 : a 0 : a a n = 0 : a, which prove the cae d =. Now let d >, and et I = a 2,..., a d. By the Artin Ree Lemma, there exit an integer c uch that for all n c, I n : a 0 : a + I n c. Then J n : a J n J n + I n : a J n + 0 : a + I n c = J n J n J n + 0 : a J n + I n c J n + J n. A J n +I n c J n i a module over R/J c +ann, it length i bounded by µµi n c time the length of R/J c + ann, which i a polynomial of degree at mot d 2. By the aumption on a, J n +0: a J n i iomorphic to 0 : a for large n. Thu rx i a polynomial of degree at mot d 2, and in Equation *, the reading of the polynomial in degree d yield that the multiplicity of with repect to J equal the multiplicity of /a with repect to I. Then by induction on d, the multiplicity of J on equal a,...,a d a,...,a d : a d a,...,a d. Now we combine Theorem 2.3 and Lemma 2.4: Theorem 2.5: Teiier-Riler [Tei73], Ree [Re84] With the et-up a above, the number ei [d ] ; i a poitive integer. Whenever R/m i an infinite field, thi number equal the mutliplicity of an ideal J in R, where J i generated by d ufficiently general element of I, d 2 ufficiently general element of I 2, etc., d ufficiently general element of I. Proof: Set d = dim. Then d = d + + d. The theorem i trivial if d = 0. Thu we aume that d > 0. Let X be an indeterminate over R, and S = R[X] mr[x]. Then for any R-module N of finite length, NS i an S-module of the ame length, o in particular ei [d ] ; = 7

8 ei S [d],..., I S [d] ; R S. The ring S ha an infinite reidue field. By Theorem 2.3, there exit d ufficiently general element in I S, etc., d element in I S, call them a,..., a d, uch that ei S [d],..., I S [d] S ; R S equal a,...,a d S. Set i = S/a,..., a i S. The contruction in the proof a,...,a d S: S a d a,...,a d S of Theorem 2.3 require that each a i avoid the minimal prime over ann i and atify the property that for ome c all n i c and all n,..., n i, n i+,..., n, I n i : i a i I n In i i Ic i I n i+ i+ i = I n In i i In i i I n i+ i+ i. By the generalized Artin Ree Lemma Theorem., for all n ufficiently large, the module I n i : i a i i contained in I n In i i Ic i In i+ i+ i. Thu for all ufficiently large n, 0 : i a i I n i = 0. Thu a,..., a d atify the condition of Lemma 2.4, which prove the theorem. An important ingredient in the proof above i the paage from R to a faithfully flat extenion S = R[X] mr[x] with an infinite reidue field that preerve length. Thi technique i alo the main tool in the proof of the following and the proof are left to the reader: It i clear that whenever I,..., I, J,..., J are m-primary ideal uch that for all i =,...,, J i I i, then /J n /I n. But even more: whenever d + + d = dim, then ej [d ],..., J [d ] ; ei [d ] ;. Hint: It i enough to prove thi in the cae J 2 = I 2,..., J = I. 2 For any poitive integer l,..., l, ei l [d],..., I l [d] ; = l d ld ei [d ] ;. 3 Aociativity formula for mixed multiplicitie ei [d ] ; = P RP P ei [d ] ; /P, where P varie over the prime ideal in R containing ann for which dim R/P = dim. 4 ixed multiplicitie behave well on hort exact equence. Explicitly, if 0 N K 0 i a hort exact equence of R-module, then i If dim = dim N = dim K, ei [d ] ; = ei [d ] ; N + ei [d ] ; K; 8

9 ii If dim = dim N > dim K, ei [d ] ; = ei [d ] ; N; iii If dim = dim K > dim N, ei [d ] ; = ei [d ] ; K. 5 Let R, m R, m be a module-finite extenion of local domain of dimenion d. Then ei R [d],..., I R [d] ; R = ei[d ] ; Rrk R S [R /m. : R/m] 3. Joint reduction There i interaction between mixed multiplicitie alo via the concept of joint reduction. Joint reduction were firt defined by Ree in [Re84]. We et-up ome notation. Let d be a poitive integer. Then a d-tuple a,..., a d of element of R i aid to be a joint reduction of the d-tuple J,..., J d with repect to, if for each i =,..., d, a i J i, and J = d i= a ij J i J i+ J d i a reduction of the ideal I = J J with repect to. In other word, there exit an integer l uch that JI l = I l+. If each J i i one of the I j, and if each I j appear d j time, then we ay that a,..., a d i a joint reduction of I,..., I of type d,..., d with repect to. The reader can rework the proof of Theorem 2.5 to ee that when the reidue field i infinite, there exit a joint reduction a,..., a d of I,..., I of type d,..., d with repect to uch that ei [d ] ; equal the multiplicity of with repect to a,..., a d. The cae d = i the cae of ordinary reduction, and the ret i proved by induction on d. Theorem 3.: Let d = dim. If a,..., a d i a joint reduction of I,..., I of type d,..., d with repect to, then ei [d ] ; equal the multiplicity of with repect to a,..., a d. Proof: Necearily a,..., a d + ann i m-primary. The theorem i clear if d = 0. Now aume that d =. Then by aumption there exit an integer l uch that a I I l = I I l+. Thu a R i a reduction of I with repect to = I 2 I l. There exit an integer c uch that for all n c, I n a n c R + ann. Then for all n, /a n /I n /a n c. Each of the three quantitie eventually equal a polynomial in n of degree, the firt and the third polynomial have the ame leading coefficient, namely the multiplicity of a R on, o necearily the middle polynomial ha the ame leading coefficient. But /I n = /I n / = /I n /, / i a contant, o the leading coefficient of the polynomial /I n equal the leading coefficient of the three polynomial above, which prove the cae d =. Now let d >. By the aociativity formula for multiplicitie and mixed multiplicitie, without lo of generality ann = 0 i a prime ideal, and = R. For implicity of notation we may aume that d > 0. A tep of Lemma 2.4 prove that the multiplicity 9

10 of a,..., a d on R i the multiplicity of a 2,..., a d on R/a R, and that the mixed multiplicity ei [d ] ; R equal ei [d ] 2,..., I [d ] ; R/a R. By paage to R/a R, a 2,..., a d i a joint reduction of I 2,..., I of type d, d 2,..., d with repect to IR/a c R for ome integer c. By induction on dimenion, ei [d ] 2,..., I [d ] ; IR/a c R equal the multiplicity of IR/a c R with repect to a 2,..., a d. But the hort exact equence 0 Ic +a R a R R a R R I 0 how that c dim R/a Rdim IR/a c R = dim R > dim R/I c = 0, o that by Property 4, ei [d ] 2,..., I [d ] ; IR/a c R = ei [d ] 2,..., I [d ] ; R/a R, and the multiplicity of IR/a c R with repect to a 2,..., a d i the ame a the multiplicity of IR/a c R with repect to a 2,..., a d. Now combine all the equalitie. The convere hold ometime. Namely, let a,..., a d be d element of I, etc., d element of I, uch that ei [d ] ; R equal the multiplicity of R with repect to a,..., a d. Then if R i formally equidimenional, a,..., a d i a joint reduction of the d-tuple I,..., I, I 2,..., I 2,..., I,..., I with repect to R, where each I i i lited d i time. The proof of thi i in [Sw93], and it require technique from [B69]. 4. Other topic We finih with liting a few other topic related to mixed multiplicitie which an intereted reader may wih to read. Teiier [Tei78], Ree and Sharp [ReSh78] proved variou inkowki-type inequalitie. Katz and Verma [KaVe89] generalized mixed multiplicitie to ideal not all of which are zero-dimenional. Verma [Ve88, Ve92] connected mixed multiplicitie to multiplicitie of multi-graded Ree algebra. Cohen acaulayne of multi-ideal Ree algebra wa tudied by Verma [Ve88], [Ve90], [Ve9]; Tang [Ta99]; Herrmann, Hyry, Ribbe [HHR93]; and Herrmann, Hyry, Ribbe, Tang [HHRT97]. Trung [Tr0] generalized mixed multiplicitie to bigraded algebra, uing filter-regular equence. Trung and Verma [TrVe06] interpreted mixed multiplicitie of monomial ideal in term of mixed volume of polytope. Reference [B69] E. Böger, Eine Verallgemeinerung eine ultiplizitätenatze von D. Ree, J. Algebra 2 969, [HHR93]. Herrmann, E. Hyry and J. Ribbe, On the Cohen acaulay and Gorentein propertie of multi Ree algebra. anucripta ath , [HHRT97]. Herrmann, E. Hyry, J. Ribbe and Z. Tang, Reduction number and multiplicitie of multigraded tructure. J. Algebra , 3 34 [KaVe89] D. Katz and J. Verma, Extended Ree algebra and mixed multiplicitie, ath. Z , -28. [Re84] D. Ree, Generalization of reduction and mixed multiplicitie, J. London ath. Soc , [ReSh78] D. Ree and R. Y. Sharp, On a theorem of B. Teiier on multiplicitie of ideal in local ring. J. London ath. Soc ,

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