A Generalization of Lefschetz Elements
In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (
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A Generalization of Lefschetz Elements
In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin.I /, etc) are needed at a few points in Chap. 5.
5.1 Weak Rees Elements Theorem 5.1. Let .A; m/ be a local ring with the residue field K D A=m, y1 ; y2 ; : : : ; yt 2 m and y D x1 y1 C x2 y2 C C xt yt with x1 ; x2 ; : : : ; xt 2 A. Let X1 ; X2 ; : : : ; Xt be indeterminates over A, Y D X1 y1 C X2 y2 C C Xt yt 2 AŒX1 ; X2 ; : : : ; Xt and A.X / D A.X1 ; X2 ; : : : ; Xt / the polynomial ring AŒX1 ; X2 ; : : : ; Xt localized at mAŒX1 ; X2 ; : : : ; Xt . Finally let a be an m-primary ideal of A. 1. lengthA.X / .A.X /=aA.X / C YA.X // lengthA .A=a C yA/. 2. Further, suppose that K is infinite. Then there exists a non-zero radical ideal b of KŒX1 ; X2 ; : : : ; Xt such that the above inequality becomes an equality if the ideal .X1 x1 ; X2 x2 ; : : : ; Xt xt / does not contain b, where xi denotes the image of xi in K. We first prove the following proposition. Proposition 5.2. Let .B; n; K/ be an Artinian local ring, X an indeterminate over B, and B.X / D BŒX nŒX . Then lengthB.X / .B.X /=.Xz1 C z2 /B.X // lengthB .B=.xz1 C z2 /B/ for any x 2 B and z1 ; z2 2 n.
T. Harima et al., The Lefschetz Properties, Lecture Notes in Mathematics 2080, DOI 10.1007/978-3-642-38206-2 5, © Springer-Verlag Berlin Heidelberg 2013
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5 A Generalization of Lefschetz Elements
Proof. Set n D lengthB.X / .B.X /=.X z1 C z2 /B.X // and let B.X / D J0 ¥ J1 ¥ ¥ Jn D .Xz1 C z2 /B.X / ¥ ¥ Jn0 D .0/ be a composition series of B.X /. Consider the natural homomorphisms '1 W BŒX ! B.X / and '2 W BŒX ! BŒX =.X x/ Š B. Let Ik D '11 .Jk / for all 0 k n and let Ik be the image of Ik in B. Note that In .Xz1 C z2 /BŒX and In .xz1 C z2 /B. To prove the desired inequality, it suffices to show that I0 ¥ I1 ¥ ¥ In . Step 1. Between two ideals Ik and IkC1 , we can take a chain of ideals Ik D a0 ¥ a1 ¥ ¥ ask D IkC1 such that, for any j D 0; 1; : : : ; sk 1, aj =aj C1 Š BŒX =pj for some pj 2 Spec.BŒX /. We show that aj =aj C1 Š BŒX =nŒX Š KŒX for some j . Note that nŒX is the unique minimal prime ideal of BŒX ; in fact, nŒX is nilpotent and BŒX =nŒX Š KŒX . We show this by way of contradiction, for suppose that nŒX ¤ pj for some j . Then, B.X /=pj B.X / D .0/, and hence aj B.X /=aj C1 B.X / Š aj =aj C1 ˝BŒX B.X / Š BŒX =pj ˝BŒX B.X / Š B.X /=pj B.X / D .0/: This implies that IkB .X / D IkC1 B.X /. Therefore we have that Jk D JkC1 , so this is a contradiction. Step 2. We would like to prove that Ik ¥ IkC1 under the assumption that Ik =IkC1 Š BŒX =nŒX Š KŒX . First we show that .Ik W .X x// D Ik for X x f 2 IkB .X /. Noting that .X x/ all k. Let f 2 .Ik W .X x//. Then 1 1 f 2 Jk , and hence f 2 Ik . is a unit in B.
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