Let K be an algebraically closed field of characteristic zero and let I = (f 1 , . . . , fn) be a homogeneous R + -primary ideal in R := K[X, Y, Z]. If the corresponding syzygy bundle Syz(f 1 , . . . , fn) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n = 3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n = 4) does not necessarily have the Weak Lefschetz property, answering negatively a question of Migliore and Miró-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable. (2000): primary: 13D02, 14J60, secondary: 13C13, 13C40, 14F05.
Mathematical Subject Classification
Abstract. We show that the Hilbert-Kunz multiplicity is a rational number for an R + −primary homogeneous ideal I = (f 1 , . . . , f n ) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. More specific, we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f 1 , . . . , f n ) on the projective curve Y = Proj R.
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersectionThis answers a question of Migliore, Miró-Roig and Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X d , Y d , Z d ) on P 2 satisfies the Grauert-Mülich theorem. As a corollary we obtain that for p = 2 the algebra A has the weak Lefschetz property if and only if d = 2 t +1 3 for some positive integer t. This was recently conjectured by Li and Zanello.
With an appendix by Georg Hein: Semistability of the general syzygy bundle.Abstract. We study (slope-)stability properties of syzygy bundles on a projective space P N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring. (2000): 13A35; 13D02; 14D20; 14H60; 14J60; I * := {f ∈ R : ∃c = 0 such that cf q ∈ I [q] for all q = p e } .
Mathematical Subject Classificationℓ λ=1 (|I λ |−1) = r and such that, setting V λ = v i , i ∈ I λ , the following holds: the set {v i : i ∈ I λ } is linearly dependent modulo the subspace V 1 +. . .+V λ−1 , but all strict subsets are independent. Then for all subsets J = J 1 ⊎ . . . ⊎ J ℓ , J λ ⊂ I λ , |J λ | = |I λ | − 1 the vectors v i , i ∈ J, are linearly independent and so the determinantial coefficients (for these J) are = 0. Hence deg(hcf( k∈I−J f k , |J| = r, det((a ji ) 1≤j≤r,i∈J ) = 0)) ≤ deg(hcf( k∈I−J f k , J as above)).
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