PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 9, September 2009, Pages 2935–2942 S 0002-9939(09)09922-5 Article electronically published on May 4, 2009
DERIVATIONS PRESERVING A MONOMIAL IDEAL
YOHANNES TADESSE
(Communicated by Bernd Ulrich)
Abstract. Let I be a monomial ideal in a polynomial ring A = k[x1,...,xn] over a field k of characteristic 0, TA/k(I) be the module of I-preserving k- derivations on A and G be the n-dimensional algebraic torus on k. We compute the weight spaces of TA/k(I) considered as a representation of G.Usingthis, we show that TA/k(I) preserves the integral closure of I and the multiplier ideals of I.
1. Introduction
Throughout this paper TA/k denotes the module of k-linear derivations of the polynomial ring A = k[x1,...,xn]overafieldk of characteristic 0. Let G = k∗ ×···×k∗ be the n-dimensional algebraic torus on k. There is an action of θ θ1 ··· θn · θ θ1 ··· θn · G on a monomial X = x1 xn defined by (t1,...,tn) X =(t1 tn ) Xθ, and we say that Xθ has weight θ.ThismakesA arepresentationofG and induces an action of G on TA/k which will make TA/k arepresentationsuchthat θ ∈ − ⊆ X ∂xj TA/k has weight (θ1,...,θj 1,...θn). Let I A be a monomial ideal, i.e. a G-invariant ideal. Let TA/k(I)={δ ∈ TA/k | δ(I) ⊆ I}⊂TA/k be the submodule of I-preserving derivations, which is a G-subrepresentation of TA/k. Our first result is a description of the weight spaces of TA/k(I). This implies in particular [1, Theorem 2.2.1] by Brumatti and Simis. Note that our use of the G-action significantly clarifies the structure of TA/k(I) and simplifies the proof. The action of G also gives a simple argument to the fact that the integral clo- sure I¯ is a monomial ideal; one may compare it to the more complicated proof ¯ in [7, Proposition 7.3.4]. For any ideal I,itisknownthatTA/k(I) ⊆ TA/k(I) [3, Theorem 3.2.2]. Again using the action of G and directly employing the integral equation of elements of I¯, we prove this inclusion when I is a monomial ideal. Let J (r · I) be the multiplier ideal of I for the rational number r>0; see [5]. Using Howald’s description [2] of J (r · I)whenI is a monomial ideal, we prove: Theorem 1.1. Let I be a monomial ideal and J (r · I) be any of its multiplier ideals. Then TA/k(I) ⊆ TA/k(J (r · I)).
It is a useful fact that TA/k(I) preserves J (r · I), as this radically restricts its form. The proof of the inclusion for any ideal is indicated in [3].
Received by the editors November 25, 2008, and, in revised form, January 5, 2009. 2000 Mathematics Subject Classification. Primary 13A15, 13N15, 14Q99. Key words and phrases. Derivations, monomial ideals, multiplier ideals.
c 2009 American Mathematical Society Reverts to public domain 28 years from publication 2935
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2. The structure of TA/k(I) ∇ n ∇ ∇ The Lie algebra of the torus is G = j=1 k xj ,where xj = xj∂xj .The representation TA/k of G decomposes as
(2.1) TA/k = A∇G ⊕ s, n where s = j=1 Axj ∂xj and Axj = k[x1,...,xˆj,...,xn]. Hence the representation is semi-simple. Since I is a monomial ideal, TA/k(I)isaG-subrepresentation of TA/k, hence semi-simple, having the decomposition
(2.2) TA/k(I)=A∇G ⊕ s(I).
Since all the weight spaces of s(I)=TA/k(I) ∩ s ⊆ s are 1-dimensional, we easily get the following lemma. ⊆ n ∈ Lemma 2.1. Let I A be a monomial ideal and δ = j=1 fj∂xj TA/k be a ∈ derivation where fj = ij mij A and mij are distinct monomial terms. Then ∈ ∈ δ TA/k(I) if and only if mij∂xj TA/k(I) for all i and j =1,...,n.
{ | θ ∈ }⊆Zn Put exp(I)= θ X I ≥0 and let (ej) denote the standard basis of Rn.Let{Xα1 ,...,Xαt } be the unique minimal generating set of I,where αi αi1 ··· αin X = x1 xn . Theorem 2.2. Consider the condition CI ∈ Zn − ∈ j (β): β ≥0,βj =0, and β + αi ej exp(I)
for all i such that αij > 0. n We have s(I)= j=1 Ixj ∂xj ,where
β |CI ⊆ Ixj =(X j (β)) Axj .
β ∈ If βj =0andαij > 0 for all i =1, ..., t,thenX ∂xj / s(I). Indeed, for a − θ ∈ ≤ β θ β+θ ej ∈ monomial X I such that θj αij for all i, one has X ∂xj (X )=θjX / I. CI Hence j (β) never holds, and we then put Ixj =(0).
β Proof. By Lemma 2.1 it suffices to determine when X ∂xj belongs to s(I), so in
particular βj =0.SinceIxj =(0)ifαij > 0 for all i, we may assume there exists 1 ≤ k ≤ t such that αkj =0.Then
β ∈ ⇔ − ∈ X Ixj β + αi ej exp(I) for all i such that αij > 0 ⇔ β αi ∈ X ∂xj (X ) I for all i =1,...,t ⇔ β ∈ X ∂xj s(I). n By [1] we have TA/k(I)= j=1[I :[I : xj]]∂xj , but the argument there does
not profit on the torus action. We now show that [I :[I : xj]] = (xj)+Ixj .Note
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Xθ θ first that colon ideals of monomial ideals are monomial, and [I : xj]=( | X ∈ xj I and θj > 0). Therefore,
θ β β X θ [I :[I : xj]] = (X | X · ∈ I for all X ∈ I such that θj > 0) xj β =(X | β + θ − ej ∈ exp(I) for all θ ∈ exp(I) such that θj > 0) β =(xj)+(X | βj =0andβ + θ − ej ∈ exp(I)
for all θ ∈ exp(I) such that θj > 0) β =(xj)+(X | βj =0andβ + αi − ej ∈ exp(I)
for all i such that αij > 0)
=(xj)+Ixj . CI We assert that j (β) is equivalent to the following condition: ∈ Zn { − } ∩ Zn ⊆ (2.3) β ≥0, βj =0,and[ β ej +exp(I)] ≥0 exp(I). ⇒CI CI ⇒ It is easy to see that (2.3) j (β), so we prove only j (β) (2.3): If αij > 0for CI ≤ all i,then j (β)doesnotholdforanyβ. Now assume there exists 1 (2.4) 0 = α1j = ···= αk−1,j <αkj ≤···≤αtj , ∈ − ∈ Zn and let θ exp(I). It is clear that θ + β ej / ≥0 if θj = 0; thus assume that θj > 0. If θj <αkj, then there exists some i =1,...,k− 1 such that θ = αi + γ with γj = θj > 0. Hence θ + β − ej = αi + β + γ − ej ∈ exp(I)sinceαi + β ∈ exp(I) − ∈ Zn ≥ ≥ and γ ej ≥0.Ifθj αkj, then there exists l k such that θ = αl + γ for ∈ Zn CI − − ∈ some γ ≥0. j (β) implies θ + β ej = γ + αl + β ej exp(I). Therefore, { − } ∩ Zn ⊆ [ β ej +exp(I)] ≥0 exp(I). It is interesting to see the structure of TA/k(I)whenA = k[x, y]. First, if I is principal, then TA/k(I) is either of the form A∇G, A∇G ⊕ Ay∂y or A∇G ⊕ Ax∂x. a1 b1 at bt Now assume I =(x y ,...,x y ) is non-principal with ai−1 y p p p p p p p p p p p p Xr p p p p p p p p p p p p (a1,b1) XX XXXz p p p p p p p p w(I)x x p p ∂yp rp p p p p p p p p p p AK p p p p p p yw(I)y ∂ A p p p p xp Arp p p p p p p p p p p (ai,bi) p rp p p p r (at,bt) x License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2938 YOHANNES TADESSE Corollary 2.3. Assume that A = k[x, y] and I =(xa1 yb1 ,...,xat ybt ) is a non- principal monomial ideal such that ai−1 Proof. It suffices to prove the cases a1 =0andbt > 0 since the other cases are similar. l l a b a b y ∂x ∈ s(I) ⇔ y ∂x(x y ) ∈ I for all x y ∈ I ⇔ (0,l)+(a, b) − (1, 0) ∈ exp(I) for all (a, b) ∈ exp(I) such that a>0 ⇔ (ai − 1,l+ bi) ∈ exp(I) for all 1 ⇔ l ≥ bi−1 − bi for all 1 ⇔ l ≥ w(I)y. w(I)y Hence s(I)=k[y]y ∂x. 8 2 6 5 4 7 2 8 12 Example 2.4. Consider I =(y ,x y ,x y ,x y ,x y, x ) ⊆ k[x, y]. Then w(I)x 2 4 =4,w(I)y =2andTA/k(I)=A∇x ⊕ A∇y ⊕ y k[y]∂x ⊕ x k[x]∂y. If I is a monomial ideal of A = k[x, y]andl>0 is an integer, it is not obvious l l how w(I )x and w(I )y depend on l. But using Corollary 2.3 and the obvious fact l l l−1 that TA/k(I) ⊆ TA/k(I ) where equality holds if I =[I : I ] [3, Remark 3.2.6], we get the following result. l Corollary 2.5. If I ⊆ k[x, y] is a monomial ideal, then w(I)x ≥ w(I )x and l l l−1 w(I)y ≥ w(I )y, and equality holds when I =[I : I ]. We give a guideline on how to use Theorem 2.2 to compute s(I). Since Ixj =(0) if αij > 0 for all i, we assume that there exists 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DERIVATIONS PRESERVING A MONOMIAL IDEAL 2939 3. Preservation of the integral closure and multiplier ideals It is well known that TA/k(I) preserves many naturally defined ideals related to I, for any ideal I [3,4,6]. We will investigate this question in the case of monomial ideals in relation to the integral closure and the formation of multiplier ideals. Given a commutative Noetherian ring R and an ideal I of R,anelementf ∈ R is integral over I if f satisfies the equation d d−1 i (3.1) f + g1f + ···+ gd−1f + gd =0, where gi ∈ I and i =1, ..., d. The integral closure I¯ consists of all elements in R which are integral over I.The following lemma is a standard fact. Lemma 3.1. Let I be a monomial ideal in A.ThenI¯ is also a monomial ideal. Furthermore, a monomial Xθ is in I¯ if and only if (Xθ)l ∈ Il for some integer l>0. Proof. Let f be integral over I. Applying the action of the torus G on (3.1) we obtain d d−1 (3.2) (f (t · X)+g1(t · X)f (t · X)+···+ gd−1(t · X)f(t · X)+gd(t · X)=0), where t =(t1,...,tn) ∈ G, X =(x1,...,xn)andt · X =(t1x1,...,tnxn). Since Ii is a monomial ideal for all i =1,...,d, hence invariant under the action of G, i we have gi(t · X) ∈ I for all i =1,...,d. Thus (3.2) is the integral dependence equation for f(t · X) ∈ A. Therefore f(t · X) ∈ I¯; hence I¯ is invariant under the action of G and it is a monomial ideal. To prove the second statement, assume Xθ satisfies (3.1). Since each Ii is a monomial ideal, considering terms of weight dθ in (3.1), we obtain an equation of the form θ d θ1 θ d−1 θd (X ) + k1X (X ) + ···+ kdX =0 θi i for some X ∈ I , i =1,...,d,andk1,...,kd ∈ k.Somecoefficientkl must be θ d θ θ d−l θ l θ l non-zero; thus (X ) = k0X l (X ) ,whereX l ∈ I and k0 ∈ k,so(X ) = θl l k0X ∈ I . The converse is immediate. The inclusion ¯ (3.3) TA/k(I) ⊆ TA/k(I) is proved in [3, Theorem 3.2.2] using the blow-up of I, for any ideal I. But it is difficult to see how this directly follows from equation (3.1). Here is a direct proof of (3.3) when I is a monomial ideal: By Lemma 2.1, it suffices to prove this for βj ∈ θ ∈ ¯ derivations of the form δ = X ∂xj TA/k(I). Let X I such that θj > 0. Then θ l l l (X ) ∈ I for some l>0. Since TA/k(I) ⊆ TA/k(I ), we have l θ l θ l−1 θ l δ ∈ TA/k(I ) ⇒ δ((X ) )=l(X ) δ(X ) ∈ I . Clearly δ((Xθ)l) is a monomial. We have that δ2((Xθ)l)=δ[l(Xθ)l−1δ(Xθ)] = l(l − 1)(Xθ)l−2(δ(Xθ))2 + l(Xθ)l−1δ2(Xθ) ∈ Il is also a monomial, split into the sum of two monomials of the same weight. It follows that (Xθ)l−2(δ(Xθ))2 ∈ Il. Similarly, δ[(Xθ)l−2(δ(Xθ))2]=(l − 2)(Xθ)l−3(δ(Xθ)3)+2δ(Xθ)δ2(Xθ) ∈ Il is a monomial; hence (Xθ)l−3(δ(Xθ))3 ∈ Il. Continuing in this way, we eventually get (δ(Xθ))l ∈ Il. Hence by Lemma 3.1 δ(Xθ) ∈ I¯. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2940 YOHANNES TADESSE ⊆ Rn Let conv(I) ≥0 be the convex hull of exp(I). The following lemma immedi- ately follows from (2.3). ∈ Zn − Lemma 3.2. Let I be a monomial ideal and β ≥0 such that βj =0.Ifθ + β ∈ ∈ { − } ∩ Rn ⊆ ej exp(I) for all θ exp(I) such that θj > 0,then[ β ej +conv(I)] ≥0 conv(I). ¯ ∩ Zn We have exp(I)=conv(I) ≥0 [7, Proposition 7.3.4]. Using this we can easily { − } ∩Rn ⊆ ⇒ prove (3.3) using (2.3) and Lemma 3.2, since [ β ej +conv(I)] ≥0 conv(I) { − } ¯ ∩ Zn ⊆ ¯ [ β ej +exp(I)] ≥0 exp(I). Let X =SpecA. By a log resolution of an ideal I ⊂ A we mean a proper birational map f : Y → X with the property that Y is smooth and f −1(I)= OY (−E), where E is an effective Cartier divisor and E + exc(f) has a normal crossing support. Let r>0 be a rational number. We define the multiplier ideal of I with coefficient r as the ideal J (r · I)=f∗OY (KY/X − rE). ∗ Here KY/X = KY − f KX is the relative canonical bundle and − is the round down for rational divisors. See [5] for a discussion of J (r·I). In general it is difficult to compute J (r · I), but for monomial ideals we have a complete description due to Howald. Theorem 3.3 ([2]). Let I ⊂ A be a monomial ideal and r>0 be a rational number. Then J (r · I) is a monomial ideal generated by the following set of monomials: { θ | ∈ · ∩ Zn } X θ +(1, 1,...,1) Int(r conv(I)) ≥0 , where Int(C) denotes the topological interior of a convex set C ⊆ Rn. Example 3.4. The figure below contains a graphical description of the ideals I =(y6,x2y3,x5y, x8)andJ (r·I)=(y8,xy7,x2y5,x3y4,x5y3,x6y2,x8y, x10)when 31 r = 18 . y p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p s p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p J (r · I) p p p p p p p p p sp p I p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p sp p p p p p p p p p p s x Proof of Theorem 1.1. Consider the monomial ideal J o(r · I) generated by the fol- lowing set of monomials: n { θ ∈ Zn | ∈ · ∈B · ∩ } X ≥0 θ Int(r conv(I)) or θ (r conv(I)) ( Hj) , j=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DERIVATIONS PRESERVING A MONOMIAL IDEAL 2941 B ⊆ Rn where (C) is the boundary of a convex set C ≥0 and Hj is the coordinate n θ θ o hyperplane xj =0inR . Clearly X ∈J(r·I) if and only if x1 ···xn·X ∈J (r·I). It suffices to show that s(I) ⊆ s(J o(r · I)) ⊆ s(J (r · I)). J o · ⊆ J · If αij > 0 for all i,then(0)=Ixj =( (r I))xj ( (r I))xj . Therefore assume that there exists 1 (3.4) θ + rβ − rej ∈ Int(r · conv(I)). Putting r = 1 in (3.4) and using the inclusion Int(conv(I)) ⊆ Int(r · conv(I)), − ∈ · CJ o(r·I) we obtain θ + β ej Int(r conv(I)). This implies j (β). ∈ · 1 ∈ r>1: If θ Int(r conv(I)), then r θ Int(conv(I)); so by Lemma 3.2, it follows 1 − ∈ 1 that r θ + β ej Int(conv(I)), since r θj > 0. By convexity all the points on the 1 1 − line segment joining r θ and r θ + β ej belong to Int(conv(I)). In particular 1 (θ + β − e ) ∈ Int(conv(I)). r j − ∈ · CJ o(r·I) Hence θ + β ej Int(r conv(I)), implying j (β). Acknowledgments The author wishes to express his warmest gratitude to his advisor, Rolf K¨allstr¨om, for introducing him to this subject and for his continuing support. This work is financially supported by International Science Programme, Uppsala University. References [1] Paulo Brumatti and Aron Simis, The module of derivations of a Stanley-Reisner ring,Proc. Amer. Math. Soc. 123 (1995), no. 5, 1309–1318. MR1243162 (95f:13014) [2]J.A.Howald,Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2665–2671 (electronic). MR1828466 (2002b:14061) [3] Rolf K¨allstr¨om, Liftable derivations for generically separably algebraic morphisms of schemes, Trans. Amer. Math. Soc. 361 (2009), no. 1, 495–523. MR2439414 [4] , Preservation of defect sub-schemes by the action of the tangent sheaf, J. Pure and Applied Algebra 156 (2005), no. 2, 286–319. MR2176658 (2006g:14002) [5] Robert Lazarsfeld, Positivity in algebraic geometry. II, Positivity for vector bundles, and multi- plier ideals, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 49, Springer-Verlag, Berlin, 2004. MR2095472 (2005k:14001b) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2942 YOHANNES TADESSE [6] G¨unter Scheja and Uwe Storch, Fortsetzung von Derivationen,J.Algebra54 (1978), no. 2, 353–365 (German). MR514074 (80a:13004) [7] Rafael H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR1800904 (2002c:13001) Department of Mathematics, Addis Ababa University, P. O. Box 1176, Addis Ababa, Ethiopia E-mail address: [email protected] Current address: Department of Mathematics, Stockholm University, SE 106-91, Stockholm, Sweden E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use