Let A=K[x1, ..... ,xn] be a standard graded polynomial ring over a ﬁeld K, let M = (x_1,&nbsp;....&nbsp;, x_n) be the graded maximal ideal and I a graded ideal of A. For each i the Betti numbers b_i(I^k ) of I^k are polynomial functions for k&gt;&gt;0. We show that if I is M-primary, then these polynomial functions have the same degree for all i 

Failla, Gioia

La Barbiera, Monica

Staglianò, Paola L.

Let A=K[x1, ..... ,xn] be a standard graded polynomial ring over a ﬁeld K, let M = (x_1,&nbsp;....&nbsp;, x_n) be the graded maximal ideal and I a graded ideal of A. For each i the Betti numbers b_i(I^k ) of I^k are polynomial functions for k&gt;&gt;0. We show that if I is M-primary, then these polynomial functions have the same degree for all i .</p

Gioia Failla

Monica La Barbiera

Paola L. Staglianò

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Betti numbers of powers of ideals

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXIII (2008) – Fasc. II, pp. 191–195BETTI NUMBERS OF POWERS OF IDEALSGIOIA FAILLA - MONICA LA BARBIERAPAOLA L. STAGLIANO`Let A=K[x1, . . . ,xn] be a standard graded polynomial ring over a fieldK, letM = (x1, . . . ,xn) be the graded maximal ideal and I a graded idealof A. For each i the Betti numbers βi(Ik) of Ik are polynomial functionsfor k 0. We show that if I isM -primar, then these polynomial functionshave the same degree for all i .1. IntroductionLet A = K[x1, . . . ,xn] be a standard graded polynomial ring over a field K andI ⊂ A be a graded ideal. Many authors studied the resolution of the gradedideal Ik,k > 0. More precisely they are interested to the total Betti numbers, thegraded Betti numbers and the regularity of Ik as a function of k, see ([3]) for asurvey on these results. To study these invariants one considers the Rees algebraR(I) of I, since Ik is its kth graded component. Let S = K[x1, . . . ,xn,y1, . . . ,ym]be a bigraded polynomial ring over the field K with degxi = (1,0) for i =1, . . . ,n and degy j = (d j,1) for j = 1, . . . ,m. The Rees algebra degR(I) =⊕k≥0 Iktk ⊂ S[t] is a finitely generated bigraded module over S, andR(I)(∗,k) =⊕jR(I)( j,k) = Ik. In our paper, we are interested in the asymptotic behavior ofthe total Betti numbers of Ik. Kodiyalam [2] proved that there are polynomialsPi(t)with Pi(k) = βi(Ik) for k 0, and Singla [3] showed degPi+1(t)≤ degPi(t)Entrato in redazione: 2 gennaio 2009AMS 2000 Subject Classification: 13A30, 13F20, 13A99Keywords: Betti numbers, Rees Algebra, Primary ideals192 GIOIA FAILLA - MONICA LA BARBIERA - PAOLA L. STAGLIANO`for any i≥ 0. Now we prove that, if I is anM -primary graded ideal of A whereM = (x1, . . . ,xn) is the graded maximal ideal of A, then degPi+1(t) = degPi(t)for all i≥ 0.2. Notation and ResultsLet A = K[x1, . . . ,xn] be a standard graded polynomial ring over a field K, andlet M = (x1, . . . ,xn) ⊂ A be the graded maximal ideal of A. Let I ⊂ A be agraded ideal, minimally generated by the homogeneous elements f1, f2, . . . , fswith deg fi = di for i = 1, . . . ,s, and let R(I) =⊕k≥0 Iktk ⊂ S[t] be the Reesalgebra of I.Let S = K[x1, . . . ,xn,y1, . . . ,ym] be a bigraded polynomial ring over the fieldK with degxi = (1,0) for i = 1, . . . ,n and degy j = (d j,1) for j = 1, . . . ,m. Thenthe K-algebra homomorphism S→ R(I) induced by xi 7→ xi and y j 7→ f jt isa surjective homomorphism of bigraded K-algebras (provided we assign to anelement f tk ∈R(I) the natural bidegree (deg f , k)). Thus R(I) may be viewedas bigraded S-module.Let W by a graded A-module. The numbers βi, j(W ) = dimK Tor i(K,W ) jare called the graded Betti numbers of the module W , and the numbersβi(W ) =∑jβi j(W ) = dimK Tor i(K,W )are called the total Betti number of W .Now let N be any finitely generated bigraded S-module. We setN(k) =⊕iN(i,k).Then for each k, N(k) is a graded A-module. In the special case mN =R(I), wehaveR(I)(k) = Ik.We quote the following results from [2] and [3, Theorem 2.2.4]:Theorem 2.1. With the assumptions and notation introduced one has:(a) (Kodiyalam) There exist polynomials PNi such that PNi (k) = βi(N(k)) forall k 0.(b) (Singla) degPNi+1 ≤ degPNi for all i≥ 0.In this note we showTheorem 2.2. Let I ⊂ A = K[x1, . . . ,xn] be aM -primary ideal. ThendegPI0 = degPI1 = · · ·= degPIn−1 = n−1.For the proof of the theorem we need the following simpleBETTI NUMBERS OF POWERS OF IDEALS 193Lemma 2.3. Let (R,N ) be a Noetherian local domain of dimension 1, and letI ⊂N be a nonzero ideal. Then R : I 6= R.Proof. The assumptions imply that R/I is a local ring of dimension 0. There-fore, there exists an integer k > 0 such that (N /I)k = (0). This implies thatN k ⊂ I. Hence R :N ⊂ R :N k ⊂ R : I, and it is enough to prove that R is aproper subset of R :N . In order to see this, consider the exact sequence:0→N → R→ R/N → 0.This sequence yields the exact sequence0→ Hom R(R/N ,R)→ HomR(R,R)→ HomR(N ,R)→ Ext1R(R/N ,R)→ 0.Since Hom R(R/N ,R) = 0 and Hom R(R,R) = R, it follows that (R :N )/R∼=Ext 1(R/N ,R), and since Ext 1(R/N ,R) 6= 0, we conclude that R 6= R :N , asdesired.Now we are ready to prove our main result.Proof of 2.2. We observe that β0(Ik) = µ(Ik) = dim(Ik/M Ik), where µ(Ik) isthe minimal number of generators of Ik. Thus β0(Ik) is the Hilbert functionof R(I) =R(I)/MR(I), and hence β0(Ik) is a polynomial function for k 0(which we denoted by PI0), whose degree is dimR(I)−1. Since I isM -primary,one has, according to [1, 4.6.13], that dimR(I) = n, so that degPI0 = n−1.By Theorem 2.1, degPI0 ≥ degPI1 ≥ ·· · ≥ degPIn−1. Thus it remains to provethat degPIn−1 = n−1.Note thatβn−1(Ik) = dimK Tor n−1(x1, . . . ,xn; Ik) = dimK Tor n(x1, . . . ,xn;A/Ik)= dimK Hn(x1, . . . ,xn;A/Ik) = dimK(Ik :M )/Ik).Thus PIn−1 is the Hilbert polynomial of the gradedR(I)-module(R(I) :MR(I))/R(I),and hence degPIn−1 = d−1, where d = dim(R(I) :MR(I))/R(I).In order to complete the proof of the theorem, we have to show that d = n.It is clear that d ≤ n. Suppose d < n; then there exists a prime idealQ of R(I)with dimR(I)/Q= dimR(I)= n, and such that (R(I) :MR(I))/R(I))Q = 0.LetP ∈ Spec (R(I)) be the preimage of Q under the canonical epimorphismR(I)→R(I). ThenP is a minimal prime ideal ofMR(I) of height 1, sinceheightP = dimR(I)−dimR(I)/P =194 GIOIA FAILLA - MONICA LA BARBIERA - PAOLA L. STAGLIANO`= (n+1)−dimR(I)/Q = (n+1)−n = 1.It follows that dimR(I)P = 1, and (R(I) :MR(I))/R(I))Q = 0 implies thatR(I)P :MR(I)P =R(I)P , contradicting Lemma 2.3.AcknowledgementsWe would like to thank Professor J. Herzog of University of Essen for the sug-gestion of the present research during the Pragmatic course in Catania and forthe useful discussions, comments and advices.BETTI NUMBERS OF POWERS OF IDEALS 195REFERENCES[1] W. Bruns - J. Herzog, CohenMacaulay rings, Cambridge Studies in AdvancedMathematics, 39. Cambridge University Press, Cambridge, 1993.[2] V. Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math.Soc. 118 n.3 (1993), 757–763.[3] P. Singla, Convex-geometric, homological and combinatorial properties of gradedideals, PHD Thesis, University of Duisburg-Essen 2007.GIOIA FAILLADepartment of MathematicsUniversity of Messina,C.da Papardo, salita Sperone, 3198166 Messina, Italye-mail: gfailla@dipmat.unime.itMONICA LA BARBIERADepartment of MathematicsUniversity of Messina,C.da Papardo, salita Sperone, 3198166 Messina, Italye-mail: monicalb@dipmat.unime.itPAOLA L. STAGLIANO`Department of MathematicsUniversity of Messina,C.da Papardo, salita Sperone, 3198166 Messina, Italye-mail: paolasta@dipmat.unime.it

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Betti numbers of powers of ideals

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