CW-complex Nagata Idealizations

We introduce a novel construction which allows us to identify the elements of the skeletons of a CW-complex $P(m)$ and the monomials in $m$ variables. From this, we infer that there is a bijection between finite CW-subcomplexes of $P(m)$, which are quotients of finite simplicial complexes, and some bigraded standard Artinian Gorenstein algebras, generalizing previous constructions in \cite{F:S}, \cite{CGIM} and \cite{G:Z}. We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree $(1,d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_1,d_2)$.


Introduction
Let X = V (f ) ⊂ P N K be a hypersurface, where the underlying field K has characteristic 0; the Hessian determinant of f (which we call the Hessian of f or the Hessian of X) is the determinant of the Hessian matrix of f .
Hypersurface with vanishing Hessian were studied for the first time in 1851 by O. Hesse; he wrote two papers ( [12,13]) according to which these hypersurfaces should be necessarily cones. In 1876 Gordan and Noether ( [9]) proved that Hesse's claim is true for N ≤ 3, and it is false for N ≥ 4. They and Franchetta classified all the counterexamples to Hesse's claim in P 4 (see [9,4,5,7]). In 1900, Perazzo classified cubic hypersurfaces with vanishing Hessian for N ≤ 6 ( [16]). This work was studied and generalized in [6], and the problem is still open in spaces of higher dimension.
Hessians of higher degree were introduced in [15] and used to control the so called Strong Lefschetz Properties (for short, SLP ). The Lefschetz properties have attracted a great attention in the last years. The basic papers of the algebraic theory of Lefschetz properties were the original ones of Stanley [17,18,19] and the book of Watanabe and others [10].
An algebraic tool that occurs frequently in these papers is the Nagata Idealization: it is a tool to convert any module M over a (commutative) ring (with unit) R to an ideal of another ring R ⋉ M . The starting point is the isomorphism between the idealization of an ideal I = (g 0 , . . . , g n ) of K[u 1 , . . . , u m ] and its level algebra see [10,Definition 2.72]. In this way, the new ring is a Standard Graded Introduction Artinian Gorenstein Algebra (SGAG algebra, for short). An explicit formula for the Macaulay generator f is: f = x 0 g 0 + · · · + x n g n ∈ K[x 0 , . . . , x n , u 1 , . . . , u m ] (1,d) .
A generalization of this construction is to consider polynomials of the form: f = x d 0 g 0 + · · · + x d n g n ∈ K[x 0 , . . . , x n , u 1 , . . . , u m ] (d,d+1) ; these are called Nagata polynomials of degree d. The Lefschetz properties for the relevant associated algebras A, the geometry of Nagata hypersurfaces of degree d, the interaction between the combinatorics of f and the structure of A were studied in [1], where the g i 's are square free monomials, using a simplicial complex associated to f .
In this paper we use the CW-complexes, to study Nagata polynomials of bidegree (d 1 , d 2 ). We study the Hilbert vector and we give a complete description of the ideal I for every case, also if the g i 's are not square free monomials.
The geometry of the Nagata hypersurface is very similar to the geometry of the hypersurfaces with vanishing Hessian.
More precisely, we introduce a new Construction 3.10 which allows us to identify each (monic) monomial of degree d in m variables with an element of the (d − 1)skeleton of a CW-complex that we call P (m). This CW-complex is constructed by generalizing the construction introduced in [3] which associates to a (monic) square-free monomial in m variables of degree d a unique (d − 1)-cell of the simplex of dimension m − 1, and vice versa. We consider an h-power u h i as a product of h linear forms:ũ 1 · · ·ũ h ; this corresponds to a (h − 1)-simplex, and we identify all the δ-faces, with δ < h − 1, of this simplex to just one δ-face, recursively, starting from δ = 0 to δ = h − 2: for δ = 0 we identify all the points to one, then if δ = 1 we obtain a bouquet of h-circles, and we identify all these circles, and so on. Generalizing this construction to a general monic monomial and attaching the corresponding CW-complexes along the common skeletons, we obtain P (m).
The paper is organized as follows: in Section 1 we recall some generalities about graded Artinian Gorenstein Algebras and Lefschetz Properties, with their connections with the vanishings of higher order Hessians. In Section 2 we recall what the Nagata idealization is, what we intend for a higher Nagata idealization and we show its connection with the Lefschetz Properties for bihomogeneous polynomials. Section 3 is the core of this article. After recalling generalities about bigraded algebras and the topological definitions that we need, we give the construction of the CW-complex P (m); then, we apply it to the Nagata polynomials ( Definition 2.5) in Theorems 3.16 and 3.18, which give Theorem 3.16 a precise description of the Artinian Gorenstein Algebra associated to a Nagata polynomial and Theorem 3.18 the generators of the annihilator of the polynomial. We show that from these theorems a generalization of the principal results of [1] follows: Corollaries 3.17 and 3. 19.
We think that the study of the Nagata hypersurfaces can be-among other things-a useful tool for the classification of the hypersurfaces with vanishing Hessian in P n .
• K is a field of characteristic 0. • R := K[x 0 , . . . , x n ] will always be the ring of polynomials in n + 1 variables x 0 , . . . , x n . • Q := K[X 0 , . . . , X n ] will be the the ring of differential operators of R, where • The subscript of a graded K-algebra will indicate the part of that degree; R d is the K-vector space of the homogeneous polynomials of degree d, and Q δ the K-vector space of the homogeneous differential operators of order δ.
We also recall the following definitions.   In particular, if I is generated by a homogeneous element f , we write Ann(I) = Ann(f ).
Let A = Q/ Ann(f ), where f is homogeneous. By construction A is a standard graded Artinian K-algebra; moreover A is Gorenstein.
Theorem 1.6 ( [14], §60ff, [15], Theorem 2.1). Let I be a homogeneous ideal of Q such that A = Q/I is a standard Artinian graded K-algebra. Then A is Gorenstein if and only if there exist d ≥ 1 and f ∈ R d such that A ∼ = Q/ Ann(f ). Remark 1.7. Using the notation as above, A is called the SGAG K-algebra associated to f . The socle degree d of A is the degree of f and the codimension is n + 1, since I 1 = 0. (1) The multiplication map ·L : A i → A i+1 is of maximal rank for all i, then A has the Weak Lefschetz Property (WLP, for short); (2) The multiplication map ·L k : A i → A i+k is of maximal rank for all i and k, then A has the Strong Lefschetz Property (SLP, for short); Definition 1.9. Let A be the SGAG K-algebra associated to an element f ∈ R d , and let B k = {α j ∈ A k | j ∈ {1, . . . , σ k }} be an ordered K-basis of A k . The k-th Hessian matrix of f with respect to B k is ). An element L = a 0 X 0 + · · · + a n X n ∈ A 1 is a strong Lefschetz element of A if and only if hess k f (a 0 , . . . , a n ) = 0 for all k ∈ 0, . . . , d 2 . In particular, if for some k one has hess k f = 0, then A does not have SLP. (r, m) + (s, n) = (r + s, m + n), (r, m) · (s, n) = (rs, sm + rn).

Bigraded Artinian Gorenstein Algebras. Let
since A is a Gorenstein ring, and the pair (d 1 , d 2 ) is said the socle bidegree of A. In this case we call A an SBAG algebra.
A homogeneous ideal I of S is a bihomogeneous ideal if: Let f ∈ S (d1,d2) , then I = Ann(f ) is a bihomogeneous ideal and using Theorem 1.6, Remark 2.3. Using the above notations, one has: Indeed, for all α ∈ T (i,j) with i > d 1 , j > d 2 , α(f ) = 0; as a consequence: provides the following short exact sequence: The following theorem, which links Nagata idealizations with bihomogeneous polynomials, holds.
From now on, we assume that n satisfies this condition. ♦ We will need the following propositions.
Let S and T be as in the previous subsection.
is called a simplicial Nagata polynomial of degree d 1 if the monomials g i are square free.
Remark 3.2. One needs n ≤ m d 2 otherwise the g i cannot be square free. 3.1.1. Abstract finite simplicial complexes.
The elements σ of ∆ are called faces or simplices; a face with q + 1 vertices is called q-face or face of dimension q and one writes dim σ = q; the maximal faces (with respect to the inclusion) are called facets; if all facets have the same dimension d ≥ 1 then one says that ∆ is of pure dimension d.
There is a natural bijection, introduced in [3], between the square free monomials, of degree d, in the variables u 1 , . . . , u m and the (2) Let f = n i=0 x d1 i g i ∈ S (d1,d2) be a simplicial Nagata polynomial; by hypothesis there is bijection between the monomials g i and the indeterminates x i . From this, we can associate to f a simplicial complex ∆ f with vertices For the topological background, we refer to [11]. We start by fixing some notations.
1 +· · ·+x 2 k = 1} and its closure, i.e. the closed (unitary) k-dimensional disk will be denoted by We recall the following Definition 3.6. A CW-complex is a topological space X constructed in the following way: (1) There exists a fixed and discrete set of points X 0 ⊂ X, whose elements are called 0-cells; α ; the elements of the kskeleton are the (closure of the) attached k cells; (3) X = k∈N ≥0 X k and a subset C of X is closed if and only if C ∩ X k is closed for any k (closed weak topology).
Definition 3.7. A subset Z of a CW-complex X is a CW-subcomplex if it is the union of cells of X, such that the closure of each cell is in Z. We will be interested mainly in finite CW-complexes. Example 3.9 (Geometric realization of an abstract simplicial complex). It is an obvious fact that to any simplicial complex ∆ one can associate a finite CW-complex ∆ via the geometric realization of ∆ as a simplicial complex (as a topological space) ∆. △ In what follows we will always identify abstract simplicial complexes with their corresponding simplicial complexes.
we have a bijection Alternatively, we can associate to u i1 · · · u i d the element of the (d − 1)-skeleton , so we have a bijection between the square-free monomials and the (d−1)-faces of the (topological) simplex ∆(m).
If j k ≤ 1 we do nothing, while if j k ≥ 2, we recursively identify, for ℓ varying from 0 to j k − 2, the ℓ-faces of the subsimplex 2 u 1 k ,...,u j k k ⊂ ∆(d): start with ℓ = 0, and we identify all the j k points to one point-call it u k . Then, for ℓ = 1, we obtain a bouquet of j k +1 2 circles, and we identify them in just one circle S 1 passing through u k , and so on, up to the facets of 2 u 1 k ,...,u j k k , i.e. its j k + 1 (j k − 1)-faces, which, by the construction, have all their boundary in common, and we identify all of them.
In this way, we obtain a finite CW-complex X = X g of dimension d − 1, with 0skeleton X 0 = {u i | j i = 0} ⊂ {u 1 , . . . , u m }, obtained from the (d − 1)-dimensional simplex ∆(d), with the above identification. Under this identification each closure of a (j k − 1)-cell u 1 k , . . . , u j k k becomes a point if j k = 1, a circle S 1 if j k = 2, a topological space with fundamental group Z 3 if j k = 2 (i.e. it is not a topological surface), etc. We will denote these spaces in what follows by ǫ j k −1 k , i.e. ǫ j k −1 k corresponds to u j k k , and vice versa: Proposition 3.11. Every power in u j1 1 · · · u jm m (up to a permutation of the variables) corresponds to a ǫ j k −1 k , and vice versa.
We can see X g as a (d − 1)-dimensional join between these spaces ǫ j k −1 k and the span of the 0-skeleton X 0 i.e. the simplex S X ⊂ ∆(m) associated to it; S X ∼ = ∆(ℓ), where ℓ = #X 0 ≤ m.
Remark 3.12. This last observation suggests we consider an alternative construction: recall that the cellular decomposition of the real projective space is obtained attaching a single cell at each passage; indeed, P n R is obtained from P n−1 R by attaching one n-cell with the quotient projection ϕ n−1 : S n−1 → P n−1 R as the attaching map.
Then, to each power u j k k we associate a real projective space of dimension j k − 1 P j k −1 k and immersions i k−1 : P j k −1 k ֒→ P j k k ; so P 0 k = u k ∈ P j k −1 k . Finally, to g = u j1 1 · · · u jm m we associate the join between the P j k −1 k and the S X defined above; if we call this join by X g , we can proceed in an equivalent way, by changing ǫ j k −1 k with P j k −1 k . ♦ It is clear how to glue two of these finite CW-complexes-say X = X u j 1 1 ···u jm m and Y = Y u k 1 1 ···u km m , of degree d = j 1 + · · ·+ j m and d ′ = k 1 + · · ·+ k m -along ∆(m): we simply attach X and Y via the inclusion maps S X ⊂ ∆(m) and S Y ⊂ ∆(m), where S X and S Y are the simplexes associated to, respectively, X and Y .
Finally, taking all these finite CW-complexes together, we obtain a CW-complex P in the following way:  In other words, if we define we have a bijection, using the above notation Proposition 3.14. X u j 1 if and only if u j1 1 · · · u jm m divides u k1 1 · · · u km m .
Let f = n i=0 x d1 i g i ∈ S (d1,d2) be a CW-Nagata polynomial; by hypothesis there is bijection between the monomials g i and the indeterminates x i . From this, we can associate to f a finite (d 2 − 1)-dimensional, CW-subcomplex of P (m), ∆ f where the (d 2 − 1)-skeleton is given by the X gi 's glued together with the above procedure. Each X gi can be identified with x d1 i as before. The previous construction generalizes the analogous one given in [1].

3.2.
The Hilbert Function of SBAG Algebras. The first main result of this paper is the following general theorem.
Remark 3.15. In order to state it, we observe that the canonical bases of given by monomials are dual bases each other, i.e.
Proof. We divide the proof into computing the dimension of A (i,j) and find a basis for it, as i varies: Then, by definition, if j ∈ {1, . . . , d 2 }, A (0,j) is generated by the (canonical images of the) monomials Ω s ∈ Q j = K[U 1 , . . . , U m ] j ∼ = Q (0,j) that do not annihilate f . This means that, if we write there exists an r s ∈ {0, . . . , n} such that g rs = u s1 1 · · · u sm m g ′ rs , where g ′ rs ∈ R d2−j is a (nonzero) monomial; this means that X u s 1 1 ···u sm m is an element of the (j − 1)-skeleton of the CW-complex ∆ f by Proposition 3.14.
We need to prove that these monomials are linearly independent over K: let {Ω 1 , . . . , Ω fj } be a system of monomials of Q (0,j) , where any Ω s = U s1 1 · · · U sm m with s 1 + · · · + s m = j, is associated to an element of the (j − 1)-skeleton of the CW-complex ∆ f ; take a linear combination of them and apply it to f : r Ω s (g r ) = n r=0 x d1 r fj s=1 c s Ω s (g r ).
By hypothesis, for any index s there exists an r s ∈ {0, . . . , n} such that Ω s (g rs ) = g ′ rs ∈ R d2−j \ {0}, then for any index s one has c s = 0, since the linear combinations in (2) are formed by linearly independent monomials (g r is fixed in each linear combination!). In other words, dim Therefore A (i,j) is generated by the only (canonical images of) the monomials Ω i,s1,...,sm s := X i s U s1 1 · · · U sm m ∈ Q (i,j) , with s 1 + · · · + s m = j, that do not annihilate f . In particular, a basis for A (i,0) is given by X i 0 , . . . , X i n and we can suppose from now on that j > 0. Since 1 · · · U sm m ) (g s ), in order to obtain that this is not zero, we must have that g s = u s1 1 · · · u sm m g ′ s , where g ′ s ∈ R d2−j is a nonzero monomial. This means X u s 1 1 ···u sm m ⊂ X gs by Proposition 3.14.
As above, we can prove that these monomials are linearly independent over K: let To find a basis for A (d1,j) , we consider the exact sequence (1) given by evaluation at f , which in this case reads then a basis for A (d1,j) is obtained in the following way: if {Ω 1 , . . . Ω f d 2 −j } is the basis for A (0,d2−j) ∼ = Q (0,d2−j) /I (0,d2−j) of the case i = 0, then a basis As a corollary of Theorem 3.16 we see that we can deduce the general case of the simplicial Nagata polynomial, which is a slight improvement of the first part of [1,Theorem 3.5]. x d1 r g r ∈ R (d1,d2) , with g r = x r1 · · · x r d 2 , be a simplicial Nagata polynomial of (positive) degree d 1 , where n ≤ m d 2 , let ∆ f be the simplicial complex associated to f and let A = Q/ Ann(f ). Then and moreover, ∀j ∈ {0, 1, . . . , where: • f j is the number of (j −1)-cells of the ∆ f (with the convention that f 0 = 1); • f j,r is the number of (j − 1)-subcells of ∆ gr , i.e. the (d 2 − 1)-cell of the ∆ f associated to g r (with the convention that f 0,r = 1, so that dim A (i,0) = n + 1).

Proof. Let
Again using the identification introduced in Remark 3.15, a basis for I β (i,j) is given by • The monomials X i r U s1 1 · · · U sm m such that s 1 +· · ·+s m = j, with r = s, where u s1 1 · · · u sm m divides g s (i.e. X u s 1 1 ···u sm m is an element of the (j − 1)-skeleton of X gs ), for i = 1, . . . , d 1 − 1, and • The monomials X i s U r1 1 · · · U rm m such that r 1 +· · ·+r m = j, where u r1 1 · · · u rm m does not divide g s (i.e. the element of the (j − 1)-skeleton of P (m), X u r 1 1 ···u rm m , is not contained in X gs ), for j ∈ {1, . . . , d 2 }.