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The weak and strong Lefschetz properties for Artinian \(K\)-algebras. (English) Zbl 1018.13001
Summary: Let \(A=\bigoplus_{i\geq 0}A_i\) be a standard graded Artinian \(K\)-algebra, where \(\text{char} K=0\). Then \(A\) has the “weak Lefschetz property” if there is an element \(\ell\) of degree 1 such that the multiplication \(\times\ell: A_i\to A_{i+1}\) has maximal rank, for every \(i\), and \(A\) has the “strong Lefschetz property” if \(\times\ell^d :A_i\to A_{i+d}\) 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 \(\mathbb{P}^2\) and the Grauert-Mülich 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.

MSC:
13A02 Graded rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Software:
Macaulay2
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[1] D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp
[2] Bigatti, A., Upper bounds for the Betti numbers of a given Hilbert function, Comm. algebra, 21, 7, 2317-2334, (1993) · Zbl 0817.13007
[3] Bigatti, A.; Geramita, A.V.; Migliore, J., Geometric consequences of extremal behavior in a theorem of Macaulay, Trans. amer. math. soc., 346, 203-235, (1994) · Zbl 0820.13019
[4] Boij, M., Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function, Pacific J. math., 187, 1-11, (1999) · Zbl 0940.13009
[5] Buchsbaum, D.; Eisenbud, D., Algebra structures for finite free resolution, and some structure theorems for ideals of codimension 3, Amer. J. math., 99, 447-485, (1977) · Zbl 0373.13006
[6] Chiantini, L.; Orecchia, F., Plane sections of arithmetically normal curves in \(P\^{}\{3\}\), (), 32-42
[7] Diesel, S., Irreducibility and dimension theorems for families of height 3 Gorenstein algebras, Pacific J. math., 172, 2, 365-397, (1996) · Zbl 0882.13021
[8] Ein, L., Stable vector bundles on projective spaces in char p>0, Math. ann., 254, 53-72, (1980) · Zbl 0431.14003
[9] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Grad. texts math., 150, (1995), Springer-Verlag · Zbl 0819.13001
[10] Eliahou, S.; Kervaire, M., Minimal resolutions of some monomial ideals, J. algebra, 129, 1-25, (1990) · Zbl 0701.13006
[11] Geramita, A.V.; Harima, T.; Shin, Y.S., Extremal point sets and Gorenstein ideals, Adv. math., 152, 1, 78-119, (2000) · Zbl 0965.13011
[12] Gordan, P.; Noether, M., Ueber die algebraischen formen, deren hessesche determinante idensisch veschwindet, Math. ann., 10, (1878)
[13] Harima, T., Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property, Proc. amer. math. soc., 123, 3631-3638, (1995) · Zbl 0857.13013
[14] Hulett, H., Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. algebra, 21, 7, 2335-2350, (1993) · Zbl 0817.13006
[15] Iarrobino, A., Associated graded algebra of a Gorenstein Artin algebra, Mem. amer. math. soc., 107, (1994) · Zbl 0793.13010
[16] Ikeda, H., Results on dilworth and Rees numbers of Artinian local rings, Japan. J. math., 22, 147-158, (1996) · Zbl 0857.13014
[17] Kreuzer, M.; Migliore, J.; Nagel, U.; Peterson, C., Determinantal schemes and buchsbaum – rim sheaves, J. pure appl. algebra, 150, 155-174, (2000) · Zbl 0999.14014
[18] Migliore, J.; Peterson, C., A construction of codimension three arithmetically Gorenstein subschemes of projective space, Trans. amer. math. soc., 349, 3803-3821, (1997) · Zbl 0885.14022
[19] J. Migliore, U. Nagel, Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math., to appear · Zbl 1053.13006
[20] Migliore, J.; Nagel, U.; Peterson, C., Buchsbaum – rim sheaves and their multiple sections, J. algebra, 219, 378-420, (1999) · Zbl 0961.14031
[21] Nagel, U., Arithmetically Buchsbaum divisors on varieties of minimal degree, Trans. amer. math. soc., 351, 4381-4409, (1999) · Zbl 0941.14016
[22] Pardue, K., Deformation classes of graded modules and maximal Betti numbers, Illinois J. math., 40, 564-585, (1996) · Zbl 0903.13004
[23] Okonek, C.; Schneider, M.; Spindler, H., Vector bundles on complex projective space, Progr. math., 3, (1988), Birkhäuser
[24] Stanley, R., The number of faces of a simplicial convex polytope, Adv. math., 35, 236-238, (1980) · Zbl 0427.52006
[25] Stanley, R., Weyl groups, the hard Lefschetz theorem, and the sperner property, SIAM J. algebraic discrete methods, 1, 168-184, (1980) · Zbl 0502.05004
[26] Watanabe, J., The dilworth number of Artin Gorenstein rings, Adv. math., 76, 194-199, (1989) · Zbl 0703.13019
[27] Watanabe, J., The dilworth number of Artinian rings and finite posets with rank function, (), 303-312
[28] Watanabe, J., A note on complete intersections of height three, Proc. amer. math. soc., 126, 3161-3168, (1998) · Zbl 0901.13019
[29] Watanabe, J., A remark on the Hessian of homogeneous polynomials, The curves seminar at Queen’s, XIII, (2000)
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