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We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefschetz property. (C) 2007 Elsevier Inc. All tights reserved.

DOI： 10.1016/j.jalgebra.2007.09.011

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Let A be a standard graded Artinian K-algebra, with char K = 0. We prove the following.
1. A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr((z)) (A) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of it. 2. If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A, z) have the Strong Lefschetz Property.
As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property. (C) 2006 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2006.10.016

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Let A be a standard graded Artinian K-algebra, with char K = 0. We prove the following.
1. A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr((z)) (A) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of it. 2. If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A, z) have the Strong Lefschetz Property.
As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property. (C) 2006 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2006.10.016

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Let A = circle plus(c)(i=0) A(i) be a graded Artinian K-algebra, where A(c) not equal (0) and char K = 0. (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g is an element of A(1) such that the multiplication x g(c-2i) : A(i) -> A(c-i) is bijective for every i = 0, 1, . . . , [c/2]. The main results obtained in this paper are as follows:
1. A has the strong Lefschetz property if and only if there is a linear form z is an element of A(1) such that Gr((z))(A) has the strong Lefschetz property.
2. If A is Gorenstein, then A has the strong Lefschetz property if and only if there is a linear form z is an element of A such that all central simple modules of (A, z) have the strong Lefischetz property.
3. A finite free extension of an Artinian K-algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does.
4. The complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. (c) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2007.01.019

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Let A = circle plus(c)(i=0) A(i) be a graded Artinian K-algebra, where A(c) not equal (0) and char K = 0. (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g is an element of A(1) such that the multiplication x g(c-2i) : A(i) -> A(c-i) is bijective for every i = 0, 1, . . . , [c/2]. The main results obtained in this paper are as follows:
1. A has the strong Lefschetz property if and only if there is a linear form z is an element of A(1) such that Gr((z))(A) has the strong Lefschetz property.
2. If A is Gorenstein, then A has the strong Lefschetz property if and only if there is a linear form z is an element of A such that all central simple modules of (A, z) have the strong Lefischetz property.
3. A finite free extension of an Artinian K-algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does.
4. The complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. (c) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2007.01.019

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DOI： 10.1016/S0021-8693(03)00118-2

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DOI： 10.1016/S0021-8693(03)00118-2

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Let A = circle plus(igreater than or equal to0) A(i) be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element iota of degree 1 such that the multiplication x iota: A(i) --> A(i+1) has maximal rank, for every i, and A has the Strong Lefschetz property if x iota(d) : A(i) --> A(i+d) has maximal rank for every i and d. The main results obtained in this paper are the following.
(1) Every height-three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank-two vector bundles on P-2 and the Grauert-Mulich theorem.)
(2) We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one!).
(3) We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties! Some Hilbert functions in fact force the algebra to have the maximal Betti numbers. (4) Every Artinian ideal in K[x, y] possesses the Strong Lefschetz property. This is false in higher codimension. (C) 2003 Elsevier Science (USA). All rights reserved.

DOI： 10.1016/S0021-8693(03)00038-3

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Let A = circle plus(igreater than or equal to0) A(i) be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element iota of degree 1 such that the multiplication x iota: A(i) --> A(i+1) has maximal rank, for every i, and A has the Strong Lefschetz property if x iota(d) : A(i) --> A(i+d) has maximal rank for every i and d. The main results obtained in this paper are the following.
(1) Every height-three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank-two vector bundles on P-2 and the Grauert-Mulich theorem.)
(2) We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one!).
(3) We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties! Some Hilbert functions in fact force the algebra to have the maximal Betti numbers. (4) Every Artinian ideal in K[x, y] possesses the Strong Lefschetz property. This is false in higher codimension. (C) 2003 Elsevier Science (USA). All rights reserved.

DOI： 10.1016/S0021-8693(03)00038-3

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Language：English   Publisher：CANADIAN MATHEMATICAL SOC  

Let H be the Hilbert function of some set of distinct points in P-n and let alpha = alpha (H) be the least degree of a hypersurface of P-n containing these points. Write alpha = d(s) + d(s-1) +...+ d(1) (where d(i) > 0). We canonically decompose H into s other Hilbert functions H <----> (H-s',...,H-1') and show how to find sets of distinct points Y-s,..., Y-1, lying on reduced hypersurfaces of degrees d(s),...,d(1) (respectively) such that the Hilbert function of Y-i is H-i' and the Hilbert function of Y = boolean OR (s)(i=1) Y-i is H. Some extremal properties of this canonical decomposition are also explored.

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Let H be the Hilbert function of some set of distinct points in P-n and let alpha = alpha (H) be the least degree of a hypersurface of P-n containing these points. Write alpha = d(s) + d(s-1) +...+ d(1) (where d(i) > 0). We canonically decompose H into s other Hilbert functions H <----> (H-s',...,H-1') and show how to find sets of distinct points Y-s,..., Y-1, lying on reduced hypersurfaces of degrees d(s),...,d(1) (respectively) such that the Hilbert function of Y-i is H-i' and the Hilbert function of Y = boolean OR (s)(i=1) Y-i is H. Some extremal properties of this canonical decomposition are also explored.

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In this paper, using a standard fact in linkage theory, we give a new construction of Artinian Gorenstein algebras achieving all possible sets of graded Betti numbers for codimension three. Furthermore, as an application, we give another proof of Stanley's well-known characterization theorem for the Hilbert functions of codimension three Artinian Gorenstein algebras. (C) 1999 Elsevier Science B.V. All rights reserved.

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We study h-sequences of certain Gorenstein standard G-algebras and, for a given SI-sequence ($) under bar h = (h(1),..., h(s)) with h(1) = 3, give a new construction of Gorenstein standard G-algebras with h-sequence ($) under bar h. Furthermore, using a similar idea, we give some examples of Gorenstein SI-sequenceswith h(1) = 4.

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