HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS

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1 HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS FRANCESCO RUSSO 1. WEAK AND STRONG LEFSCHETZ PROPERTIES FOR STANDARD ARTINIAN GORENSTEIN GRADED ALGEBRAS Let X P N C be a smooth irreducible complex projective variety of dimension n = dim(x) endowed with the euclidean topology. The cohomology groups with coefficient in the field C will be indicated by H i (X) := H i (X; C). As it is well known dim C (H i (X)) < for every i 0 and H i (X) = 0 for i > 2n. Let [H] H 2 (X) be the class of a hyperplane section of X. For every integer k 1 the cap product defines a C-linear map: [H] k : H i (X) H i+2k (X) [Y ] [H] k [Y ] We recall the following fundamental result of S. Lefschetz Theorem. (Hard Lefschetz Theorem) Let X P N C be a smooth irreducible complex projective variety of dimension n 1. Then q = 1,..., n (1.1) [H] q : H n q (X) H n+q (X) is an isomorphism. The following consequence of the Hard Lefschetz Theorem inspired the algebraic notions we shall introduce in a moment Corollary. Let notation be as above. Then: (1.2) [H] k : H i (X) H i+2k (X) is injective for i n k and surjective for i n k. Proof. Suppose i n k. The composition H i (X) [H]k H i+2k (X) [H]n k i H 2n i (X). is an isomorphism by Theorem 1.1 so that the first map is injective. 1

2 2 FRANCESCO RUSSO Analogously if i n k, the composition H 2n i 2k (X) [H]i n+k H i (X) [H]k H i+2k (X). is an isomorphism by Theorem 1.1 so that the second map is surjective. Let us recall that Poincarè Duality Theorem assures that H i (X) H 2n i (X) for every i = 0,..., n. There are important algebraic notions characterized by this property Definition. Let K be a field and let d i=0 be an artinian associative and commutative graded K-algebra with A 0 = K and A d 0. Let : A i A d i A d (α, β) α β be the restriction of the multiplication in A. We say that A satisfies the Poincarè Duality Property if: (i) dim K (A d ) = 1; (ii) : A i A d i A d K is non-degenerate for every i = 0,..., [ d 2 ]. The algebra A is said to be standard if A i A K[x 0,..., x N ], I as graded algebras, with I K[x 0,..., x N ] a homogeneous ideal. Let us remark that this implies I = (x 0,..., x N ) because (x 0,..., x N ) l I for l d + 1. To each artinian graded K algebra d i=0a i as above, letting h i = dim K A i, we can associate its h-vector h = (1, h 1,..., h d ). For algebras satisfying the Poincarè Duality Property we have h d = 1 and h d i = h i for every i = 1,..., [ d 2 ]. The h-vector of A is said to be unimodal if there exists and integer t 1 such that 1 h 1... h t h t+1... h d 1 1.

3 HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS3 The notion of Poincarè Duality Property was inspired by the remark that the even part of the cohomology ring with coefficient in a characteristic zero field of a compact orientable manifold of even real dimension satisfies the previous property due to Poincarè Duality Theorem. We now introduce the definition of Gorenstein ring which despite its apparent abstractness is a property shared by the even part of a the cohomology ring of compact orientable manifolds as we shall see below Definition. Let (R, m, K) be a local ring. Then R is called a local Gorenstein ring if it has finite injective dimension as an R-module. A commutative ring R is called Gorenstein if the localization at each prime ideal is a local Gorenstein ring. The following is a nice characterization of artinian graded Gorenstein algebras Proposition. ([GHMS], [MW, Proposition 2.1]) Let A be a graded artinian K-algebra. Then A satisfies the Poincarè Duality Property if and only if it is Gorenstein Example. Let Q = K[,..., ] x 0 x N and let F (x) K[x 0,..., x N ] d be a homogeneous polynomial of degree d 1. Then for every G Q we shall indicate by G(F ) K[x 0,..., x N ] be polynomial obtained by applying the differential operator G to the the polynomial F. Define Ann Q (F ) = {G Q : G(F ) = 0} Q. Then Ann Q (F ) Q is a homogenous ideal and Q Ann Q (F ) is a standard artinian Gorenstein graded K-algebra with A i = 0 for i > d and A d 0. The Theory of Inverse Systems developed by Macaulay yields the following nice characterization of standard artinian Gorenstein graded K-algebras. This result is surely well known to the experts in the field. A short proof of a little bit more general result can be found in [MW, Theorem 2.1] Theorem. Let d i=0 A i K[x 0,..., x N ] I

4 4 FRANCESCO RUSSO be an artinian standard graded K-algebra. Then A is Gorenstein if and only if there exists F K[x 0,..., x N ] d such that A Q/ Ann Q (F ). We now come to the definition of Lefschetz Properties, originally developed by R. Stanley in [St], see also [HMMNWW] for an expanded treatment. In the sequel we shall follow strictly the presentation in [Wa] and in [MW] Definition. Let K be a a field and let d i=0 be an artinian associative and commutative graded K-algebra with A d 0. The algebra A is said to have the Strong Lefschetz Property, briefly SLP, if there exits an element L A 1 such that the multiplication map A i L k : A i A i+k is of maximal rank, that is injective or surjective, 0 i d and 0 k d k. An element L A 1 satisfying the previous property will be called a strong Lefschetz element of A. The algebra A is said to have the Weak Lefschetz Property, briefly W LP, if there exists an element L A 1 such that the multiplication map L : A i A i+1 is of maximal rank, that is injective or surjective, 0 i d 1. An element L A 1 satisfying the last property will be called a Lefschetz element of A. A is said to have the Strong Lefschetz Property in the narrow sense if there exists an element L A 1 such that the multiplication map is an isomorphism i = 0,..., [ d 2 ]. L d 2i : A i A d i 1.9. Remark. Since we shall always deal with infinite fields (more precisely algebraically closed fields of characteristic zero), if there exists a Lefschetz element or a strong Lefschetz element, then the general element of A 1, in the sense of the Zariski topology, shares the same property. If A satisfies the W LP, then the h-vector of A is unimodal. The contrary is not true as shown by simple examples, see [MW]. Moreover, there are artinian Gorestein algebras whose h-vector is not unimodal.

5 HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS5 If a graded artinian K-algebra A satisfies the SLP in the narrow sense, then the h-vector of A is unimodal and symmetric, that is h i = h d 1 (see the proof of Corollary 1.2). On the contrary for a graded artinian K-algebra having a symmetric Hilbert function, the notion of SLP and SLP in the narrow sense coincide. This is the case of Gorenstein artinian graded K-algebras on which we shall mainly focus. In a moment we shall see examples of artinian Gorenstein graded algebras having unimodal h-vector but not satisfying the SLP (and neither the W LP ), whose construction depend on the existence of homogeneous polynomial with vanishing hessian determinant depending on all the variables modulo linear changes of coordinates Definition. Let Q Ann Q (F ) be a standard artinian Gorenstein graded K-algebra with A i = 0 for i > d and A d 0, F K[x 0,..., x N ] d. Without loss of generality we can assume (Ann Q (F )) 1 = 0, that is that there does not exists a linear change of coordinates such that F does not depend on all the variables; equivalently the partial derivates of F are linearly independent. Under this hypothesis, which we shall assume from now on, {x 0,..., x N } is a basis of A 1, that is h 1 = N + 1 = h d 1. Let us define hess (1) F = det(h(f )), that is hess (1) F = h(f ). Let 2 i [d/2] and let = {α (i) 1,..., α (i) h i } be a basis of A i. Let us define the matrix Hess (i) F as the h i h i matrix whose elements are given by (Hess (i) F ) m,n = (α m (i) ( x ) α(i) n ( x ))(F ) K[x 0,..., x N ] d 2i. Finally define hess (i) F = det(hess (i) F ) K[x 0,..., x N ] hi (d 2i). The definition depends on the choice of the basis s. Choosing different basis the value of the (i)-hessian of F is altered by the multiplication of a non-zero element of K. Since we are mainly interested in the vanishing of these polynomial we could omit the reference to the basis and simply write hess (i) F. We shall need the following elementary result, known as differential Euler Identity, whose proof is left to the reader.

6 6 FRANCESCO RUSSO Lemma. Let G K[x 0,..., x N ] e and let L = a 0 x a N Q 1. Then L e (G) = e! G(a 0,..., a N ). x N The connection between the huge amount of algebraic definitions introduced so far and the contents of this chapter is finally made clear by the next result Theorem. (Watanabe, [Wa], [MW]) Let notation be as above. An element L = a 0 x a N x N A 1 is a strong Lefschetz element of Q/ Ann Q (F ) if and only if (i) F (a 0,..., a N ) 0 and (ii) hess (i) F (a 0,..., a N ) 0 for all i = 1,..., [d/2]. Proof. The identification A d K is obtained by letting G A d act on F as differential operator. Since deg(g) = d = deg(f ) we get G(F ) K. The Poincarè Duality Property holds for A so that to define a linear map f : A i A d i is the same as giving a bilinear map φ : A i A i A d K via the identification between A i and A d i given by multiplication in A. Consider the multiplication linear map L d 2i : A i A d i with L as in the statement. The associated bilinear map φ : A i A i A d K is defined by φ i (ξ, η) = [(L d 2i ξ) η](f ) and it is symmetric by the commutativity of the product in A. Moreover, L d 2i is an isomorphism if and only if φ i is non-degenerate. Choose a basis = {α (i) 1,..., α (i) h i } of A i. Then the symmetric matrix H i associated to φ i with respect to the basis has elements H i m,n = (L d 2i α (i) m ) α (i) m ](F ) = = L d 2i ([α m (i) ) α m (i) Lemma 1.11 ](F )) = (d 2i)! {([α m (i) ) α m (i) ](F ))(a 0,..., a N )} = = (d 2)! (Hess (i) F ) m,n (a 0,..., a N )). In conclusion L d 2i is an isomorphism for every i = 1,..., [d/2] if and only if (i) and (ii) hold Corollary. (Watanabe, [Wa] and [MW]) Let notation be as above. Then: Q (1) A, F K[x Ann Q (F ) 0,..., x N ] d such that (Ann Q (F )) 1 = 0, satisfies the SLP if and only if hess (i) (F ) 0 for every i = 1,..., [d/2].

7 HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS7 (2) Let hypothesis and notation be as in (1). If d 4, then A satisfies the SLP if and only if h(f ) 0. In particular for N 3, every such A satisfies the SLP. (3) For every N 4 and for d = 3, 4 a polynomial F K[x 0,..., x N ] d with vanishing hessian and with (Ann Q (F )) 1 = 0 produces an example of a graded artinian Gorenstein algebra d i=oa i not satistying the SLP. REFERENCES [GHMS] A. V. Geramita, T. Harima, J. C. Migliore, Y. S. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007) 139 pp. [MW] T. Maeno, J. Watanabe, Lefschetz elements of artinian Gorenstein algebras and hessians of homogeneous polynomials, Illinois J. Math. 53 (2009), [St] R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, [HMMNWW] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe, The Lefschetz properties, Lecture Notes in Mathematics Springer, Heidelberg, 2013, xx+250 pp. [Wa] J. Watanabe, A remark on the Hessian of homogeneous polynomials, in The Curves Seminar at Queens, Volume XIII, Queens Papers in Pure and Appl. Math., Vol. 119, 2000, DIPARTIMENTO DI MATEMATICA E INFORMATICA, UNIVERSITÀ DEGLI STUDI DI CATANIA, VIALE A. DORIA 6, CATANIA, ITALY address: frusso@dmi.unict.it