Complete Intersections with the SLP
The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Le
- PDF / 229,277 Bytes
- 16 Pages / 439.36 x 666.15 pts Page_size
- 20 Downloads / 190 Views
Complete Intersections with the SLP
The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Lefschetz property.
4.1 Central Simple Modules In order to prove the main theorem, we introduce the notion of central simple modules for a graded Artinian K-algebra, and review the definition of the SLP for modules and a characterization of the SLP in terms of central simple modules. Central simple modules are a useful tool to study the SLP. For details we refer the reader to [52, 53] and [54]. Definition 4.1. Let A be a graded Artinian K-algebra, z a linear form of A and mf mf m P .z/ D f1 1 ˚ f2 2 ˚ ˚ fs fs the Jordan decomposition of z 2 End.A/, where f1 > f2 > > fs . The graded A-module Ui D
.0 W zfi / C .z/ .0 W zfi C1 / C .z/
is called the i -th central simple module of .A; z/, with 1 i s and the convention fsC1 D 0. Note that these are defined for a pair consisting of the algebra A and a linear form z 2 A. By the definition, it is easy to see that the modules U1 ; U2 ; : : : ; Us are the nonzero terms of the successive quotients of the descending chain of ideals A D .0 W zf1 / C .z/ .0 W zf1 1 / C .z/ .0 W z/ C .z/ .z/:
T. Harima et al., The Lefschetz Properties, Lecture Notes in Mathematics 2080, DOI 10.1007/978-3-642-38206-2 4, © Springer-Verlag Berlin Heidelberg 2013
141
142
4 Complete Intersections with the SLP
Remark 4.2. We indicate a property of central simple modules of graded Artinian K-algebras with the SLP. Let A D ˚ciD0 Ai be a graded Artinian K-algebra with the SLP and z a strong Lefschetz element of A. Set 0
A=.z/ D ˚ci D0 Ai ; where Ai D Ai =zAi 1 and c 0 is the largest integer such that .A=.z//c 0 6D 0. Since A has the SLP, one sees easily that .A; z/ has c 0 C 1 central simple modules U1 ; : : : ; Uc 0 C1 and that Ui Š Ai 1 as graded K-vector spaces. This shows that Ui has only one non-trivial graded piece concentrated at the degree i 1. Hence Ui has the SLP for trivial reasons. In analogy to a graded module over a graded L ring, we define the Hilbert series for a graded vector space as follows. Let V D biDa Vi be a graded vector space over a field K, where Va ¤ 0, Vb ¤ 0 and Vi D 0 if i < a or i > b. We call a the initial degree of V and b the end degree of V . Then the Hilbert function of V is the map i 7! dimK Vi , which we denote by h.V; i /, and its Hilbert series is the polynomial Hilb.V; t/ D
b X
.dimK Vi /t i :
i Da
We define the Sperner number of V by Sperner.V / D max f dimK Va ; dimK VaC1 ; : : : ; dimK Vb g and CoSperner number CoSperner.V / D
b1 X
min f dimK Vi ; dimK Vi C1 g :
i Da
L The Hilbert function h.V; i / of V D biDa Vi (where Va ¤ .0/ and Vb ¤ .0/) is symmetric if dimK VaCi D dimK Vbi for all i D 0; 1; : : : ; Œ.b a/=2, and we call the half integer .a C b/=2 the reflecting degree of h.V; i /. The Hilbert function of V is unim
Data Loading...
