HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR

HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR
ARTINIAN GORENSTEIN GRADED ALGEBRAS
FRANCESCO RUSSO
1. W EAK AND S TRONG L EFSCHETZ P ROPERTIES FOR STANDARD
ARTINIAN G ORENSTEIN GRADED ALGEBRAS
Let X ⊂ PN
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 dimC (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.
1.1. Theorem. (Hard Lefschetz Theorem) Let X ⊂ PN
C be a smooth irreducible complex projective variety of dimension n ≥ 1. Then ∀q = 1, . . . , n
•[H]q : H n−q (X) → H n+q (X)
(1.1)
is an isomorphism.
The following consequence of the Hard Lefschetz Theorem inspired the
algebraic notions we shall introduce in a moment.
1.2. 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]k
H i (X) −→ 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.
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FRANCESCO RUSSO
Analogously if i ≥ n − k, the composition
H 2n−i−2k (X)
•[H]i−n+k
−→
•[H]k
H i (X) −→ 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.
1.3. Definition. Let K be a field and let
A=
d
M
Ai
i=0
be an artinian associative and commutative graded K-algebra with A0 = K
and Ad 6= 0. Let
• : Ai × Ad−i → Ad
(α, β)
→ α•β
be the restriction of the multiplication in A.
We say that A satisfies the Poincarè Duality Property if:
(i) dimK (Ad ) = 1;
(ii) • : Ai ×Ad−i → Ad ' K is non-degenerate for every i = 0, . . . , [ d2 ].
The algebra A is said to be standard if
A'
K[x0 , . . . , xN ]
,
I
as graded algebras, with√I ⊂ K[x0 , . . . , xN ] a homogeneous ideal. Let us
remark that this implies I = (x0 , . . . , xN ) because (x0 , . . . , xN )l ⊆ I for
l ≥ d + 1.
To each artinian graded K algebra A = ⊕di=0 Ai as above, letting hi =
dimK Ai , we can associate its h-vector
h = (1, h1 , . . . , hd ).
For algebras satisfying the Poincarè Duality Property we have hd = 1
and hd−i = hi for every i = 1, . . . , [ d2 ].
The h-vector of A is said to be unimodal if there exists and integer t ≥ 1
such that
1 ≤ h1 ≤ . . . ≤ ht ≥ ht+1 ≥ . . . ≥ hd−1 ≥ 1.
HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS
3
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.
1.4. 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.
1.5. 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.
1.6. Example. Let
∂
∂
,...,
]
∂x0
∂xN
and let F (x) ∈ K[x0 , . . . , xN ]d be a homogeneous polynomial of degree
d ≥ 1. Then for every G ∈ Q we shall indicate by G(F ) ∈ K[x0 , . . . , xN ]
be polynomial obtained by applying the differential operator G to the the
polynomial F . Define
Q = K[
AnnQ (F ) = {G ∈ Q : G(F ) = 0} ⊂ Q.
Then AnnQ (F ) ⊂ Q is a homogenous ideal and
A=
Q
AnnQ (F )
is a standard artinian Gorenstein graded K-algebra with Ai = 0 for i > d
and Ad 6= 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].
1.7. Theorem. Let
A=
d
M
i=0
Ai '
K[x0 , . . . , xN ]
I
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FRANCESCO RUSSO
be an artinian standard graded K-algebra. Then A is Gorenstein if and
only if there exists F ∈ K[x0 , . . . , xN ]d such that A ' Q/ AnnQ (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].
1.8. Definition. Let K be a a field and let
A=
d
M
Ai
i=0
be an artinian associative and commutative graded K-algebra with Ad 6= 0.
The algebra A is said to have the Strong Lefschetz Property, briefly SLP ,
if there exits an element L ∈ A1 such that the multiplication map
•Lk : Ai → Ai+k
is of maximal rank, that is injective or surjective, ∀ 0 ≤ i ≤ d and
∀ 0 ≤ k ≤ d − k.
An element L ∈ A1 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 ∈ A1 such that the multiplication map
•L : Ai → Ai+1
is of maximal rank, that is injective or surjective, ∀ 0 ≤ i ≤ d − 1.
An element L ∈ A1 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 ∈ A1 such that the multiplication map
•Ld−2i : Ai → Ad−i
is an isomorphism ∀ i = 0, . . . , [ d2 ].
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 A1 , 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.
HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS
5
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 hi = hd−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.
1.10. Definition. Let
Q
AnnQ (F )
be a standard artinian Gorenstein graded K-algebra with Ai = 0 for i > d
and Ad 6= 0, F ∈ K[x0 , . . . , xN ]d .
Without loss of generality we can assume (AnnQ (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,
{x0 , . . . , xN } is a basis of A1 , that is h1 = N + 1 = hd−1 . Let us define
A=
hess(1) F = det(H(F )),
that is hess(1) F = h(F ).
(i)
(i)
Let 2 ≤ i ≤ [d/2] and let Bi = {α1 , . . . , αhi } be a basis of Ai . Let us
(i)
define the matrix HessBi F as the hi × hi matrix whose elements are given
by
∂
(i)
(i) ∂
( ) • αn(i) ( ))(F ) ∈ K[x0 , . . . , xN ]d−2i .
(HessBi F )m,n = (αm
∂x
∂x
Finally define
(i)
(i)
hessBi F = det(HessBi F ) ∈ K[x0 , . . . , xN ]hi (d−2i) .
The definition depends on the choice of the basis Bi ’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.
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FRANCESCO RUSSO
1.11. Lemma. Let G ∈ K[x0 , . . . , xN ]e and let L = a0 ∂x∂ 0 + . . . + aN ∂x∂N ∈
Q1 . Then
Le (G) = e! · G(a0 , . . . , aN ).
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.
1.12. Theorem. (Watanabe, [Wa], [MW]) Let notation be as above. An
element L = a0 x0 + . . . + aN xN ∈ A1 is a strong Lefschetz element of
A = Q/ AnnQ (F ) if and only if
(i) F (a0 , . . . , aN ) 6= 0 and
(ii) hess(i) F (a0 , . . . , aN ) 6= 0 for all i = 1, . . . , [d/2].
Proof. The identification Ad ' K is obtained by letting G ∈ Ad 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 : Ai → Ad−i is the same as giving a bilinear map φ : Ai × Ai → Ad ' K
via the identification between Ai and Ad−i given by multiplication in A.
Consider the multiplication linear map •Ld−2i : Ai → Ad−i with L as
in the statement. The associated bilinear map φ : Ai × Ai → Ad ' K is
defined by
φi (ξ, η) = [(Ld−2i • ξ) • η](F )
and it is symmetric by the commutativity of the product in A. Moreover,
•Ld−2i is an isomorphism if and only if φi is non-degenerate.
(i)
(i)
Choose a basis Bi = {α1 , . . . , αhi } of Ai . Then the symmetric matrix
H i associated to φi with respect to the basis Bi has elements
i
(i)
(i)
Hm,n
= (Ld−2i • αm
) • αm
](F ) =
(i)
(i)
= Ld−2i ([αm
)•αm
](F ))
Lemma 1.11
=
(i)
(i)
(d−2i)!·{([αm
)•αm
](F ))(a0 , . . . , aN )} =
(i)
= (d − 2)! · (HessBi F )m,n (a0 , . . . , aN )).
In conclusion •Ld−2i is an isomorphism for every i = 1, . . . , [d/2] if and
only if (i) and (ii) hold.
1.13. Corollary. (Watanabe, [Wa] and [MW]) Let notation be as above.
Then:
(1) A ' AnnQQ (F ) , F ∈ K[x0 , . . . , xN ]d such that (AnnQ (F ))1 = 0, satisfies the SLP if and only if hess(i) (F ) 6= 0 for every i = 1, . . . , [d/2].
HESSIANS AND THE STRONG LEFSCHETZ PROPERTY FOR ARTINIAN GORENSTEIN GRADED ALGEBRAS
7
(2) Let hypothesis and notation be as in (1). If d ≤ 4, then A satisfies
the SLP if and only if h(F ) 6= 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[x0 , . . . , xN ]d
with vanishing hessian and with (AnnQ (F ))1 = 0 produces an example of a graded artinian Gorenstein algebra A = ⊕di=o Ai not
satistying the SLP .
R EFERENCES
[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),
591–603.
[St]
R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner
property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184.
[HMMNWW] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe, The
Lefschetz properties, Lecture Notes in Mathematics 2080. 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, 171–178.
D IPARTIMENTO DI M ATEMATICA E I NFORMATICA , U NIVERSIT À DEGLI S TUDI DI
C ATANIA , V IALE A. D ORIA 6, 95125 C ATANIA , ITALY
E-mail address: [email protected]

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