CW-complex Nagata Idealizations
aa r X i v : . [ m a t h . A C ] J un CW-COMPLEX NAGATA IDEALIZATIONS
ARMANDO CAPASSO, PIETRO DE POI, AND GIOVANNA ILARDI
Abstract.
We introduce a construction which allows us to identify the el-ements 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, andcertain bigraded standard Artinian Gorenstein algebras, generalizing previousconstructions of Faridi and ourselves.We apply this to a generalization of Nagata idealization for level algebras.These algebras are standard graded Artinian algebras whose Macaulay dualgenerator is given explicitly as a bigraded polynomial of bidegree (1 , d ). Weconsider the algebra associated to polynomials of the same bidegree ( d , d ). Introduction
Let X = V ( f ) ⊂ P N K be a hypersurface, where the underlying field K has char-acteristic 0; the Hessian determinant of f (which we call the Hessian of f or theHessian of X ) is the determinant of the Hessian matrix of f .Hypersurface with vanishing Hessian were studied for the first time in 1851 byO. Hesse; he wrote two papers ([12, 13]) according to which these hypersurfacesshould be necessarily cones. In 1876 Gordan and Noether ([9]) proved that Hesse’sclaim is true for N ≤
3, and it is false for N ≥
4. They and Franchetta classifiedall the counterexamples to Hesse’s claim in P (see [9, 4, 5, 7]). In 1900, Perazzoclassified cubic hypersurfaces with vanishing Hessian for N ≤ Strong Lefschetz Properties (for short,
SLP ). The Lefschetz properties haveattracted a great attention in the last years. The basic papers of the algebraictheory of Lefschetz properties were the original ones of Stanley [17, 18, 19] and thebook of Watanabe and others [10].An algebraic tool that occurs frequently in these papers is the
Nagata Idealiza-tion : 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 be-tween the idealization of an ideal I = ( g , . . . , g n ) of K [ u , . . . , u m ] and its levelalgebra see [10, Definition 2.72]. In this way, the new ring is a Standard Graded
Date : June 29, 2020.2010
Mathematics Subject Classification.
Primary 13A30, 05E40; Secondary 57Q05, 13D40,13A02, 13E10.
Key words and phrases.
Lefschetz properties, Artinian Gorenstein Algebra, Nagata idealiza-tion, CW-complex.P.D.P. & G.I. are members of INdAM - GNSAGA and P.D.P is supported by PRIN2017“Advances in Moduli Theory and Birational Classification”.
Artinian Gorenstein Algebra ( SGAG algebra , for short). An explicit formula forthe Macaulay generator f is: f = x g + · · · + x n g n ∈ K [ x , . . . , x n , u , . . . , u m ] (1 ,d ) . A generalization of this construction is to consider polynomials of the form: f = x d g + · · · + x dn g n ∈ K [ x , . . . , x n , u , . . . , u m ] ( d,d +1) ;these are called Nagata polynomials of degree d . The Lefschetz properties for therelevant associated algebras A , the geometry of Nagata hypersurfaces of degree d , the interaction between the combinatorics of f and the structure of A werestudied in [1], where the g i ’s are square free monomials, using a simplicial complexassociated to f .In this paper we use the CW-complexes , to study Nagata polynomials of bidegree( d , d ). 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 thehypersurfaces with vanishing Hessian.More precisely, we introduce a new Construction 3.10 which allows us to identifyeach (monic) monomial of degree d in m variables with an element of the ( d − P ( m ). This CW-complex is constructedby generalizing the construction introduced in [3] which associates to a (monic)square-free monomial in m variables of degree d a unique ( d − m −
1, and vice versa. We consider an h -power u hi as a product of h linear forms: ˜ u · · · ˜ u h ; this corresponds to a ( h − δ -faces, with δ < h −
1, of this simplex to just one δ -face, recursively, startingfrom 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 soon. Generalizing this construction to a general monic monomial and attaching thecorresponding CW-complexes along the common skeletons, we obtain P ( m ).The paper is organized as follows: in Section 1 we recall some generalities aboutgraded Artinian Gorenstein Algebras and Lefschetz Properties, with their connec-tions with the vanishings of higher order Hessians. In Section 2 we recall whatthe Nagata idealization is, what we intend for a higher Nagata idealization and weshow its connection with the Lefschetz Properties for bihomogeneous polynomials.Section 3 is the core of this article. After recalling generalities about bigradedalgebras and the topological definitions that we need, we give the construction ofthe CW-complex P ( m ); then, we apply it to the Nagata polynomials ( Definition2.5) in Theorems 3.16 and 3.18, which give Theorem 3.16 a precise description ofthe Artinian Gorenstein Algebra associated to a Nagata polynomial and Theorem3.18 the generators of the annihilator of the polynomial. We show that from thesetheorems a generalization of the principal results of [1] follows: Corollaries 3.17 and3.19.We think that the study of the Nagata hypersurfaces can be—among otherthings—a useful tool for the classification of the hypersurfaces with vanishing Hes-sian in P n . Acknowledgments.
We thank the anonymous referee for the careful reading and thevaluable suggestions.
Notations.
In this the paper we fix the following notations and assumptions: raded Artinian Gorenstein Algebras and Lefschetz Properties 3 • K is a field of characteristic 0. • R := K [ x , . . . , x n ] will always be the ring of polynomials in n + 1 variables x , . . . , x n . • Q := K [ X , . . . , X n ] will be the the ring of differential operators of R , where X i = ∂∂x i . • 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 .1. Graded Artinian Gorenstein Algebras and Lefschetz Properties
Graded Artinian Gorenstein Algebras are Poincar´e Algebras.Definition 1.1.
Let I be a homogeneous ideal of R such that A = R/I = d M i =0 A i isa graded Artinian K -algebra, where A d = 0. The integer d is the socle degree of A .The algebra A is said standard if it is generated in degree 1. Setting h i = dim K A i ,the vector Hilb( A ) = (1 , h , . . . , h d ) is called Hilbert vector of A . Since I = 0, then h = n + 1 is called codimension of A .We also recall the following definitions. Definition 1.2.
A graded Artinian K -algebra A = L di =0 A i is a Poincar´e algebra if · : A i × A d − i → A d is a perfect pairing for i ∈ { , . . . , d } . Definition 1.3.
A graded Artinian K -algebra A is Gorenstein if (and only if)dim K A d = 1 and it is a Poincar´e algebra. Remark . The Hilbert vector of a Poincar´e algebra A is symmetric with respectto h (cid:4) d (cid:5) , that is Hilb( A ) = (1 , h , h , . . . , h , h , ♦ Graded Artinian Gorenstein Quotient Algebras of Q . For any d ≥ δ ≥ K -bilinear map B : R d × Q δ → R d − δ defined by differenti-ation B ( f, α ) = α ( f ) Definition 1.5.
Let I = h f , . . . , f ℓ i —where f , . . . , f ℓ are forms in R —be a finitedimensional K -vector subspace of R . The annihilator of I in Q is the followinghomogeneous ideal Ann( I ) := { α ∈ Q | ∀ f ∈ I, α ( f ) = 0 } . 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 standardgraded Artinian K -algebra; moreover A is Gorenstein. Theorem 1.6 ([14], § . Let I be a homogeneous ideal of Q such that A = Q/I is a standard Artinian graded K -algebra. Then A is Gorensteinif and only if there exist d ≥ and f ∈ R d such that A ∼ = Q/ Ann( f ) . Graded Artinian Gorenstein Algebras and Lefschetz Properties
Remark . Using the notation as above, A is called the SGAG K -algebra associ-ated to f . The socle degree d of A is the degree of f and the codimension is n + 1,since I = 0. ♦ Lefschetz Properties and the Hessian Criterion.
Let A = d M i =0 A i be agraded Artinian K -algebra. Definition 1.8.
If there exists an L ∈ A such that:(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 ∈ { , . . . , σ k }} be an ordered K -basis of A k . The k -thHessian matrix of f with respect to B k isHess kf = ( α i α j ( f )) σ k i,j =1 . The k -th Hessian of f with respect to B k ishess kf = det (cid:16) Hess kf (cid:17) . Theorem 1.10 ([20] Theorem 4) . An element L = a X + · · · + a n X n ∈ A is a strong Lefschetz element of A if and only if hess kf ( a , . . . , a n ) = 0 for all k ∈ (cid:26) , . . . , (cid:22) d (cid:23)(cid:27) . In particular, if for some k one has hess kf = 0 , then A does nothave SLP. Higher Order Nagata Idealization
Nagata Idealizations.Definition 2.1.
Let A be a ring and let M be an A -module. The Nagata idealiza-tion A ⋉ M of M is the ring with underlying set A × M and operations defined asfollow: ( r, m ) + ( s, n ) = ( r + s, m + n ) , ( r, m ) · ( s, n ) = ( rs, sm + rn ) . Bigraded Artinian Gorenstein Algebras.
Let A = d M i =0 A i be a SGAG K -algebra, it is bigraded if: A d = A ( d ,d ) ∼ = K , A i = i M h =0 A ( i,h − i ) for i ∈ { , . . . , d − } , since A is a Gorenstein ring, and the pair ( d , d ) is said the socle bidegree of A . Inthis case we call A an SBAG algebra . Remark . By Definition 1.3, A i ∼ = A ∨ d − i = Hom K ( A d − i , K ) and since the dualitycommutes with direct sums, one has A ( i,j ) ∼ = A ∨ ( d − i,d − j ) for any pair ( i, j ). ♦ igher Order Nagata Idealization 5 We fix notation as in Theorem 2.4: • S := R ⊗ K K [ u , . . . , u m ] = K [ x , . . . , x n , u , . . . , u m ] is the bigraded ring ofpolynomials in m + n + 1 variables x , . . . , x n , u , . . . , u m ; • We have chosen the natural bigrading of S : x i has bidegree (1 ,
0) and u j has bidegree (0 , • Define S ( d ,d ) to be the K -vector space of bihomogeneous polynomials f of bidegree ( d , d ); that is, f can be written as p X i =0 a i b i , where a i ∈ R d = K [ x , . . . , x n ] d and b i ∈ K [ u , . . . , u m ] d . • T := Q ⊗ K K [ U , . . . , U m ] = K [ X , . . . , X n , U , . . . , U m ] is the (bigraded)ring of differential operators of S , where X i = ∂∂x i and U j = ∂∂u j ; X i hasbidegree (1 ,
0) and U j has bidegree (0 , I of S is a bihomogeneous ideal if: I = ∞ M i,j =0 I ( i,j ) , where ∀ i, j ∈ N ≥ , I ( i,j ) = I ∩ S ( i,j ) . Let f ∈ S ( d ,d ) , then I = Ann( f ) is a bihomogeneous ideal and using Theorem 1.6, A = T / (Ann( f )) is a SBAG K -algebra of socle bidegree ( d , d ) (and codimension m + n + 1). Remark . Using the above notations, one has: ∀ i > d , j > d , I ( i,j ) = T ( i,j ) . Indeed, for all α ∈ T ( i,j ) with i > d , j > d , α ( f ) = 0; as a consequence: ∀ k ∈ { , . . . , d + d } , A k = M ≤ i ≤ d ≤ j ≤ d i + j = k A ( i,j ) . Moreover, the evaluation map α ∈ T ( i,j ) α ( f ) ∈ A ( d − i,d − j ) provides the fol-lowing short exact sequence:(1) 0 / / I ( i,j ) / / T ( i,j ) / / A ( d − i,d − j ) / / . ♦ The following theorem, which links Nagata idealizations with bihomogeneouspolynomials, holds.
Theorem 2.4 ([10], Theorem 2.77) . Let S ′ := K [ u , . . . , u m ] and S := R ⊗ K S ′ berings of polynomials, let T ′ = K [ U , . . . , U m ] and T := Q ⊗ K T ′ be the associatedring of differential operators, where X i = ∂∂x i and U j = ∂∂u j . Let g , . . . , g n behomogeneous elements of S ′ of degree d , let I be the T ′ -submodule of S ′ generatedby { ∂ ( g i ) ∈ R | ∂ ∈ T, i ∈ { , . . . , n }} and let A ′ := T ′ / Ann( I ) . Define f = x g + · · · + x n g n ∈ R , it is a bihomogeneous polynomial of bidegree (1 , d ) , and let A := T /
Ann( f ) . Considering I as an A ′ -module, A ′ ⋉ I ∼ = A . Higher Order Nagata Idealization
Lefschetz Properties for Higher Nagata Idealizations.Definition 2.5.
A bihomogeneous polynomial f = n X i =0 x d i g i ∈ S ( d ,d ) is called a CW-Nagata polynomial of degree d ≥ g i ∈ K [ u , . . . , u m ], i =0 , . . . , n , are linearly independent monomials of degree d ≥ Remark . One needs n ≤ (cid:18) m + d − d (cid:19) otherwise the g i cannot be linearlyindependent.From now on, we assume that n satisfies this condition. ♦ We will need the following propositions.
Proposition 2.7 ([5] Proposition 2.5) . Let n + 1 ≥ m ≥ , d > d ≥ and s > (cid:18) m + d − d (cid:19) ; for any j ∈ { , . . . , s } , let f j ∈ S ( d , , g j ∈ S (0 ,d ) . Then theform f = f g + · · · + f s g s of degree d + d satisfies hess d f = 0; that is, A = T /
Ann( f ) does not have the SLP condition. Proposition 2.8 ([1] Proposition 2.7) . Let n + 1 ≥ m ≥ , d ≥ d . Then L = n X i =0 X i is a Weak Lefschetz Element ; that is, A = T /
Ann( f ) has the WLPcondition. CW-complex Nagata Idealization of Bidegree ( d , d )Let S and T be as in the previous subsection. Definition 3.1.
A bihomogeneous CW-Nagata polynomial f = n X i =0 x d i g i ∈ S ( d ,d ) is called a simplicial Nagata polynomial of degree d if the monomials g i are squarefree. Remark . One needs n ≤ (cid:18) md (cid:19) otherwise the g i cannot be square free. ♦ CW-complexes and bihomogeneous polynomials.
Abstract finite simplicial complexes.
Definition 3.3.
Let V = { u , . . . , u m } be a finite set. An abstract simplicialcomplex ∆ with vertex set V is a subset of 2 V such that(1) ∀ u ∈ V ⇒ { u } ∈ ∆,(2) ∀ σ ∈ ∆ , τ $ σ, τ = ∅ ⇒ τ ∈ ∆. W-complex Nagata Idealization of bidegree ( d , d ) 7 The elements σ of ∆ are called faces or simplices ; a face with q + 1 verticesis called q -face or face of dimension q and one writes dim σ = q ; the maximalfaces (with respect to the inclusion) are called facets ; if all facets have the samedimension d ≥ of pure dimension d . The set ∆ k of facesof dimension at most k is called k -skeleton of ∆. 2 V is called simplex (of dimension m − Remark . (1) (cfr. [1, Remark 3.4]) There is a natural bijection, introduced in [3], betweenthe square free monomials, of degree d , in the variables u , . . . , u m and the( d − V , with vertex set V = { u , . . . , u m } . In fact, asquare free monomial g = u i · · · u i d corresponds to the subset { u i , . . . , u i d } of 2 V . Vice versa, to any subset F of V with d elements one associates thefree square monomial m F = Y u i ∈ F u i of degree d .(2) Let f = n X i =0 x d i g i ∈ S ( d ,d ) be a simplicial Nagata polynomial; by hypoth-esis 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 u , . . . , u m where the facet which corresponds to g i is identified with x d i . ♦ CW-complexes.
For the topological background, we refer to [11]. We startby fixing some notations.
Definition 3.5.
Let k ∈ N ≥ . A topological space e k homeomorphic to the open(unitary) ball { ( x , . . . , x k ) ∈ R k | x + · · · + x k < } of dimension k (with thenatural topology induced by R k +1 ) is called a k -cell . Its boundary, i.e. the ( k − -dimensional sphere will be denoted by S k − = { ( x , . . . , x k ) ∈ R k | x + · · · + x k = 1 } and its closure, i.e. the closed (unitary) k -dimensional disk will be denoted by D k := { ( x , . . . , x k ) ∈ R k | x + · · · + x k ≤ } .We recall the following Definition 3.6. A CW-complex is a topological space X constructed in the fol-lowing way:(1) There exists a fixed and discrete set of points X ⊂ X , whose elements arecalled 0 -cells ;(2) Inductively, the k -skeleton X k of X is constructed from X k − by attaching k -cells e kα (with index set A k ) via continuous maps ϕ kα : S k − α → X k − (the attaching maps ). This means that X k is a quotient of Y k = X k − [ α ∈ A k D kα under the identification x ∼ ϕ α ( x ) for x ∈ ∂ D kα ; the elements of the k -skeleton are the (closure of the) attached k cells;(3) X = [ k ∈ N ≥ X k and a subset C of X is closed if and only if C ∩ X k is closedfor any k ( closed weak topology ). Definition 3.7.
A subset Z of a CW-complex X is a CW-subcomplex if it is theunion of cells of X , such that the closure of each cell is in Z . CW-complex Nagata Idealization of Bidegree ( d , d ) Definition 3.8.
A CW-complex is finite if it consists of a finite number of cells.We will be interested mainly in finite CW-complexes.
Example 3.9 (Geometric realization of an abstract simplicial complex) . It is anobvious fact that to any simplicial complex ∆ one can associate a finite CW-complex e ∆ via the geometric realization of ∆ as a simplicial complex (as a topological space) e ∆. △ In what follows we will always identify abstract simplicial complexes with theircorresponding simplicial complexes.
Construction 3.10.
In Remark 3.4, we saw that to any degree d square-free mono-mial u i · · · u i d ∈ K [ u , . . . , u m ] d one can associate the ( d − { u i , . . . , u i d } ofthe abstract ( m − m ) := 2 { u ,...,u m } , and vice versa: ifwe call ρ d := { f ∈ K [ u , . . . , u m ] d | f = 0 is a square-free monic monomial } D ( m ) d := ∆( m ) d \ ∆( m ) d − , we have a bijection σ d : ρ d → D ( m ) d u i · · · u i d
7→ { u i , . . . , u i d } . Alternatively, we can associate to u i · · · u i d the element of the ( d − { u i , . . . , u i d } ∈ ^ ∆( m ) d − , so we have a bijection σ d : ρ d → ^ ∆( m ) d − u i · · · u i d
7→ { u i , . . . , u i d } between the square-free monomials and the ( d − ^ ∆( m ).Using CW-complexes, we will extend this construction to the non-square-freemonic monomials . We proceed as follows. Let g := u j · · · u j m m be a genericdegree d := j + · · · + j m monomial. Consider the following finite set: W := n u , . . . , u j , . . . , u m , . . . , u j m m o , and if ∆( d ) := 2 W is the abstract associated (fi-nite) simplex, we consider the corresponding (topological) simplex (which is a CW-complex) ] ∆( d ).If j k ≤ j k ≥
2, we recursively identify, for ℓ varyingfrom 0 to j k −
2, the ℓ -faces of the subsimplex ^ n u k ,...,u jkk o ⊂ ] ∆( d ): start with ℓ = 0, and we identify all the j k points to one point—call it u k . Then, for ℓ = 1, weobtain a bouquet of (cid:0) j k +12 (cid:1) circles, and we identify them in just one circle S passingthrough u k , and so on, up to the facets of ^ n u k ,...,u jkk o , i.e. its j k + 1 ( j k − j , . . . , j m ; in this way, we obtain a finiteCW-complex X = X g of dimension d −
1, with 0-skeleton X = { u i | j i = 0 } ⊂ W-complex Nagata Idealization of bidegree ( d , d ) 9 { u , . . . , u m } , obtained from the ( d − ] ∆( d ), with the aboveidentification.In this way, we obtain a finite CW-complex X = X g of dimension d −
1, with 0-skeleton X = { u i | j i = 0 } ⊂ { u , . . . , u m } , obtained from the ( d − ] ∆( d ), with the above identification. Under this identification each closureof a ( j k − n u k , . . . , u j k k o becomes a point if j k = 1, a circle S if j k = 2, atopological space with fundamental group Z if j k = 2 (i.e. it is not a topologicalsurface), etc. We will denote these spaces in what follows by ǫ j k − k , i.e. ǫ j k − k corresponds to u j k k , and vice versa: Proposition 3.11.
Every power in u j · · · u j m m (up to a permutation of the vari-ables) corresponds to a ǫ j k − k , and vice versa. We can see X g as a ( d − join between these spaces ǫ j k − k and thespan of the 0-skeleton X i.e. the simplex S X ⊂ ^ ∆( m ) associated to it; S X ∼ = ] ∆( ℓ ),where ℓ = X ≤ m . Remark . This last observation suggests we consider an alternative construc-tion: recall that the cellular decomposition of the real projective space is obtainedattaching a single cell at each passage; indeed, P n R is obtained from P n − R by attach-ing one n -cell with the quotient projection ϕ n − : S n − → P n − R as the attachingmap.Then, to each power u j k k we associate a real projective space of dimension j k − P j k − k and immersions i k − : P j k − k ֒ → P j k k ; so P k = u k ∈ P j k − k .Finally, to g = u j · · · u j m m we associate the join between the P j k − k and the S X defined above; if we call this join by X g , we can proceed in an equivalent way, bychanging ǫ j k − k with P j k − k . ♦ It is clear how to glue two of these finite CW-complexes—say X = X u j ··· u jmm and Y = Y u k ··· u kmm , of degree d = j + · · · + j m and d ′ = k + · · · + 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: C := G u j ··· u jmm ∈ K [ u ,...,u m ] X u j ··· u jmm P ( m ) := C/ ∼ where ∼ is the equivalence relation induced by the above gluing. Proposition 3.13.
There is bijection between the monomials of degree d in K [ u , . . . , u m ] and the elements of the ( d − -skeleton of P ( m ) . In other words, if we define ρ ′ d := { f ∈ K [ u , . . . , u m ] d | f = 0 is a monic monomial } d , d ) we have a bijection, using the above notation σ ′ d : ρ ′ d → P ( m ) d − u j · · · u j m m X u j ··· u jmm . Proposition 3.14. X u j ··· u jmm ⊂ X u k ··· u kmm if and only if u j · · · u j m m divides u k · · · u k m m . Let f = n X i =0 x d i g i ∈ S ( d ,d ) be a CW-Nagata polynomial; by hypothesis there isbijection between the monomials g i and the indeterminates x i . From this, we canassociate to f a finite ( d − P ( m ), ∆ f wherethe ( d − X g i ’s glued together with the above procedure.Each X g i can be identified with x d 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 thispaper is the following general theorem.
Remark . In order to state it, we observe that the canonical bases of S ( d ,d ) = K [ x , . . . , x n ] d ⊗ K [ u , . . . , u m ] d and T ( d ,d ) = K [ X , . . . , X n ] d ⊗ K [ U , . . . , U m ] d given by monomials are dual bases each other, i.e. X k · · · X k n n U ℓ · · · U ℓ m m ( x i · · · x i n n u j · · · u j m m ) = δ i ,...,i n ,j ,...,j m k ,...,k n ,ℓ ,...,ℓ m where i + · · · + i n = k + · · · + k n = d , j + · · · + j m = ℓ + · · · + ℓ m = d and δ i ,...,i n ,j ,...,j m k ,...,k n ,ℓ ,...,ℓ m is the Kronecker delta.This simple observation allows us to identify—given a CW-Nagata polynomial f = n X r =0 x d r g r ∈ S ( d ,d ) — the dual differential operator G r of the monomial g r —i.e. the monomial G r ∈ K [ U , . . . , U m ] d such that G r ( g r ) = 1 and G r ( g ) = 0 forany other monomial g ∈ K [ u , . . . , u m ] d —with the same element of the ( d − f associated to g r . In other words, we associate to g r = u j · · · u j m m and to G r = U j · · · U j m m the CW-subcomplex of ∆ f ⊂ P ( m ), X u j ··· u jmm . Theorem 3.16.
Let f = n X r =0 x d r g r ∈ R ( d ,d ) , with g r = u j · · · u j m m , be a CW-Nagata polynomial of (positive) degree d , where n ≤ (cid:18) md (cid:19) , let ∆ f be the CW-complex associated to f and let A = Q/ Ann( f ) . Then A = d = d + d M h =0 A h where A h = A ( h, ⊕ · · · ⊕ A ( p,q ) ⊕ · · · ⊕ A (0 ,h ) , p ≤ d , q ≤ d , A d = A ( d ,d )W-complex Nagata Idealization of bidegree ( d , d ) 11 and moreover, ∀ j ∈ { , , . . . , d } , dim A ( i,j ) = a i,j = f j i = 0 n X r =0 f j,r i ∈ { , . . . , d − } ,f d − j i = d where: • f j is the number of the elements of the ( j − -skeleton of the CW-complex ∆ f (with the convention that f = 1 ); • f j,r is the number of the elements of the ( j − -skeleton of the CW complex X G r (with the convention that f ,r = 1 , so that dim A ( i, = n + 1 ).More precisely, a basis for A ( i,j ) , ∀ j ∈ { , , . . . , d } , is given by(1) If i = 0 , { Ω , . . . , Ω f j } , where any Ω s := U s · · · U s m m , with s + · · · + s m = j ,is associated to the element X u s ··· u smm of the ( j − -skeleton of ∆ f ;(2) If i = 1 , . . . , d − , (cid:8) Ω i,s ,...,s m s (cid:9) s ∈{ ,...,n } s k ≤ ,r k ,k =1 ,...,m P k s k = j where Ω i,s ,...,s m s := X is · U s · · · U s m m is associated to the element X u s ··· u smm of the ( j − -skeletonof X g s ;(3) If i = d , n X d Ω ( f ) , . . . X d n Ω f d − j ( f ) o , where (cid:8) Ω , . . . Ω f d − j (cid:9) is the ba-sis for A (0 ,d − j ) of case (1) .In the cases (1) and (2) the basis are given by monomials, in the case (3) , ingeneral, not.Proof. We divide the proof into computing the dimension of A ( i,j ) and find a basisfor it, as i varies: i = 0: A (0 , ∼ = K .Then, by definition, if j ∈ { , . . . , d } , A (0 ,j ) is generated by the (canon-ical images of the) monomials Ω s ∈ Q j = K [ U , . . . , U m ] j ∼ = Q (0 ,j ) that donot annihilate f . This means that, if we writeΩ s = U s · · · U s m m s + · · · + s m = j, there exists an r s ∈ { , . . . , n } such that g r s = u s · · · u s m m g ′ r s , where g ′ r s ∈ R d − j is a (nonzero) monomial; this means that X u s ··· u smm is an element ofthe ( j − f by Proposition 3.14.We need to prove that these monomials are linearly independent over K : let { Ω , . . . , Ω f j } be a system of monomials of Q (0 ,j ) , where any Ω s = U s · · · U s m m with s + · · · + s m = j , is associated to an element of the( j − f ; take a linear combination of themand apply it to f :0 = f j X s =1 c s Ω s ( f ) = f j X s =1 c s n X r =0 x d r Ω s ( g r ) = n X r =0 x d r f j X s =1 c s Ω s ( g r ) . By the linear independence of the x d r ’s f j X s =1 c s Ω s ( g r ) = 0 , ∀ r ∈ { , . . . , n } . (2) d , d ) By hypothesis, for any index s there exists an r s ∈ { , . . . , n } such thatΩ s ( g r s ) = g ′ r s ∈ R d − j \ { } , then for any index s one has c s = 0, since thelinear combinations in (2) are formed by linearly independent monomials( g r is fixed in each linear combination!). In other words, dim A (0 ,j ) = f j .0 < i < d : Observe that X a X b ( f ) = 0 if a = b . Therefore A ( i,j ) is generated by theonly (canonical images of) the monomials Ω i,s ,...,s m s := X is U s · · · U s m m ∈ Q ( i,j ) , with s + · · · + s m = j , that do not annihilate f . In particular, abasis for A ( i, is given by X i , . . . , X in and we can suppose from now onthat j >
0. SinceΩ i,s ,...,s m s ( f ) = x d − is ( U s · · · U s m m ) ( g s ) , in order to obtain that this is not zero, we must have that g s = u s · · · u s m m g ′ s ,where g ′ s ∈ R d − j is a nonzero monomial. This means X u s ··· u smm ⊂ X g s byProposition 3.14.As above, we can prove that these monomials are linearly independentover K : let (cid:8) Ω i,s ,...,s m s (cid:9) s ∈{ ,...,n } s k ≤ ,r k ,k =1 ,...,m P k s k = j be a system of monomials of Q ( i,j ) , where any Ω i,s ,...,s m s = X is · U s · · · U s m m is associated to the element X u s ··· u smm of the ( j −
1) skeleton of X g s ⊂ ∆ f ,i.e. X u s ··· u smm ⊂ X g s ⊂ ∆ f by Proposition 3.14.Take a linear combination of them and apply it to f :(3)0 = X s ∈{ ,...,n } s k ≤ ,r k ,k =1 ,...,m P k s k = j c i,s ,...,s m s Ω i,s ,...,s m s ( f ) = n X s =0 x d − i X s k ≤ ,r k ,k =1 ,...,m P k s k = j c i,s ,...,s m s g i,s ,...,s m s where g i,s ,...,s m s ∈ R d − j is the nonzero monomial such that g s = u s · · · u s m m g i,s ,...,s m s .From (3) we deduce, as in the preceding case, that X s k ≤ ,r k ,k =1 ,...,m P k s k = j c i,s ,...,s m s g i,s ,...,s m s = 0 s = 0 , . . . , n ;(4) as before, given one choice of s , . . . , s m there exists an s ∈ { , . . . , n } such g i,s ,...,s m s ( f ) is a nonzero monomial, and the (nonzero) g i,s ,...,s m s ’s in (4)are linearly independent since are obtained by a fixed g s . i = d : By duality, see Remark 2.2, A ( d ,j ) ∼ = A ∨ (0 ,d − j ) so dim A ( d ,j ) = f d − j .To find a basis for A ( d ,j ) , we consider the exact sequence (1) given byevaluation at f , which in this case reads(5) 0 → I (0 ,d − j ) → Q (0 ,d − j ) → A ( d ,j ) → , then a basis for A ( d ,j ) is obtained in the following way: if { Ω , . . . Ω f d − j } is the basis for A (0 ,d − j ) ∼ = Q (0 ,d − j ) /I (0 ,d − j ) of the case i = 0, then a basisfor A ( d ,j ) is n X d Ω , . . . , X d n Ω f d − j ( f ) o . (cid:3) W-complex Nagata Idealization of bidegree ( d , d ) 13 As a corollary of Theorem 3.16 we see that we can deduce the general case ofthe simplicial Nagata polynomial, which is a slight improvement of the first part of[1, Theorem 3.5].
Corollary 3.17.
Let f = n X r =0 x d r g r ∈ R ( d ,d ) , with g r = x r · · · x r d , be a simplicialNagata polynomial of (positive) degree d , where n ≤ (cid:18) md (cid:19) , let ∆ f be the simplicialcomplex associated to f and let A = Q/ Ann( f ) . Then A = d = d + d M h =0 A h where A h = A ( h, ⊕ · · · ⊕ A ( p,q ) ⊕ · · · ⊕ A (0 ,h ) , p ≤ d , q ≤ d , A d = A ( d ,d ) and moreover, ∀ j ∈ { , , . . . , d } , dim A ( i,j ) = a i,j = f j i = 0 n X r =0 f j,r i ∈ { , . . . , d − } ,f d − j i = d where: • f j is the number of ( j − -cells of the ∆ f (with the convention that f = 1 ); • f j,r is the number of ( j − -subcells of ∆ g r , i.e. the ( d − -cell of the ∆ f associated to g r (with the convention that f ,r = 1 , so that dim A ( i, = n + 1 ).More precisely, a basis for A ( i,j ) , ∀ j ∈ { , , . . . , d } , is given by(1) If i = 0 , { Ω , . . . , Ω f j } , where any Ω s := U s · · · U s j is associated to the ( j − -subcell { u s , . . . , u s j } of ∆ f ;(2) If i = 1 , . . . , d − , n Ω i,s ,...,s j s o s ∈{ ,...,n } s ,...,s j ∈ { r ,...,r d } where Ω i,s ,...,s j s := X is U s · · · U s j is associated to the ( j − -subcell { u s , . . . , u s j } of ∆ g s ( ⊂ ∆ f ) ;(3) If i = d , n X d Ω ( f ) , . . . X d n Ω f d − j ( f ) o , where { Ω , . . . Ω f d − j } is the ba-sis for A (0 ,d − j ) of case (1) .In the cases (1) and (2) the bases are given by monomials, in the case (3) , ingeneral, not. Theorem 3.18.
Let f = n X r =0 x d r g r ∈ S ( d ,d ) , with g r = x r · · · x r m m such that r + · · · + r m = d , be a CW-Nagata polynomial whose associated CW-complex is ∆ f , as in the preceding theorem.Then I := Ann( f ) is generated by:(1) X i X j and X d +1 k , for i, j, k ∈ { , . . . , n } , i < j ;(2) h U , . . . , U m i d +1 , i.e. all the (monic) monomials of degree d + 1 ; d , d ) (3) The monomials U s · · · U s m m such that s + · · · + s m = j , where X u s ··· u smm is a (minimal) element of the ( j − -skeleton of P ( m ) not contained in ∆ f (for j ∈ { , . . . , d } );(4) The monomials X r U i , where u i does not divide g r (i.e. { u i } is not anelement of the -skeleton of X g r );(5) The monomials X s U r · · · U r m m such that r + · · · + r m = j , where u r · · · u r m m is minimal among those that do not divide g s (i.e. the (minimal) elementof the ( j − -skeleton of P ( m ) , X u r ··· u rmm , is not contained in X g s ), for j ∈ { , . . . , d } ;(6) The binomials X d r U ρ · · · U ρ m m − X d s U σ · · · U σ m m with ρ + · · · + ρ m = σ + · · · + σ m = j such that g r,s = GCD( g r , g s ) and g r = u ρ · · · u ρ m m g r,s , g s = u σ · · · u σ m m g r,s (i.e. X g r,s is the element of the ( d − j − -skeleton of ∆ f which represents the intersection of X g r and X g s : X g r,s = X g r ∩ X g s ).Proof. Let A := T /I , where T = K [ X , . . . , X n , U , . . . , U m ].By Theorem 3.16, (1) a basis for A (0 ,j ) , ∀ j ∈ { , . . . , d } , is { Ω , . . . , Ω f j } , whereΩ s := U s · · · U s m m , with s + · · · + s m = j , is associated to the element X u s ··· u smm of the ( j − f . Therefore, using the identification introduced inRemark 3.15,a basis for I (0 ,j ) is given by the monomials U s · · · U s m m such that s + · · · + s m = j , where X u s ··· u smm is an element of the ( j − P ( m )not contained in ∆ f (for j ∈ { , . . . , d } );Observe that X i X j ( f ) = 0 if i = j and X d +1 k ( f ) = 0 = U i · · · U i m m ( f ) with P mj =1 i j = d + 1, for degree reasons. Set β := ( X X , . . . , X n − X n , X d +10 , . . . , X d +1 n , h U , . . . , U m i d +1 );this is a homogeneous ideal such that β ⊂ I and A ∼ = Tβ / Iβ .By Theorem 3.16, (2), if i = 1 , . . . , d −
1, a basis for A ( i,j ) ∀ j ∈ { , . . . , d } , isgiven by (cid:8) Ω i,s ,...,s m s (cid:9) s ∈{ ,...,n } s k ≤ ,r k ,k =1 ,...,m P k s k = j where Ω i,s ,...,s m s := X is · U s · · · U s m m is associated to the element X u s ··· u smm of the( j − X g s .Again using the identification introduced in Remark 3.15, a basis for (cid:18) Iβ (cid:19) ( i,j ) is given by • The monomials X ir U s · · · U s m m such that s + · · · + s m = j , with r = s , where u s · · · u s m m divides g s (i.e. X u s ··· u smm is an element of the ( j − X g s ), for i = 1 , . . . , d −
1, and • The monomials X is U r · · · U r m m such that r + · · · + r m = j , where u r · · · u r m m does not divide g s (i.e. the element of the ( j − P ( m ), X u r ··· u rmm , is not contained in X g s ),for j ∈ { , . . . , d } .It remains to find the generators of I of bidegree ( d , j ), with j ∈ { , . . . , d } .This is more complicated since the generators of A ( d ,j ) are not monomials. Let γ be the homogeneous ideal generated by the monomials of the cases (1), (2), (3),(3.2) and (3.2), i.e. the generators that we have found so far. We have β ⊂ γ ⊂ I W-complex Nagata Idealization of bidegree ( d , d ) 15 and the exact sequence (1) given by evaluation at f becomes0 → (cid:18) Iγ (cid:19) ( d ,j ) → (cid:18) Tγ (cid:19) ( d ,j ) → A (0 ,d − j ) → , since we identify A ∼ = Tγ / Iγ . Then, if ρ + · · · + ρ m = σ + · · · + σ m = j , X d r U ρ · · · U ρ m m − X d s U σ · · · U σ m m ∈ (cid:16) Tγ (cid:17) ( d ,j ) is in (cid:18) Iγ (cid:19) ( d ,j ) if and only if X d r U ρ · · · U ρ m m = X d s U σ · · · U σ m m ∈ A (0 ,d − j ) , which means U ρ · · · U ρ m m ( g r ) = U σ · · · U σ m m ( g s ). Since A (0 ,d − j ) is generated by the monomials Ω s := U s · · · U s m m , with s + · · · + s m = d − j , associated to the elements of the ( d − j − f , we obtain case(6). (cid:3) As we have done for Theorem 3.16, we give, as a corollary of Theorem 3.18 thecase of the simplicial Nagata polynomial, giving an improvement of the second partof [1, Theorem 3.5]; we also correct that statement, since the authors forgot thegenerators X i X j , i = j . Corollary 3.19.
Let f = n X r =0 x d r g r ∈ R ( d ,d ) , with g r = x r · · · x r d , be a simplicialNagata polynomial whose associated simplicial complex is ∆ f , as in the precedingtheorem.Then I := Ann( f ) is generated by:(1) X i X j and X d +1 k , for i, j, k ∈ { , . . . , n } , i < j ;(2) U , . . . , U m ;(3) The monomials U s · · · U s j , where { u s , . . . , u s j } is a (minimal) ( j − -cellof { u ,...,u m } not contained in ∆ f (for j ∈ { , . . . , d } );(4) The monomials X r U i , where u i does not divide g s (i.e. { u i } / ∈ ∆ g r );(5) The binomials X d r U ρ · · · U ρ j − X d s U σ · · · U σ j such that g r,s GCD( g r , g s ) , g r = u ρ · · · u ρ j g r,s , g s = u σ · · · u σ j g r,s (i.e. g r,s represents the ( d − j − -face given by the intersection ∆ g r ∩ ∆ g s of the facets of g r and g s : ∆ g r,s =∆ g r ∩ ∆ g s ).Proof. We note only that we have to add the squares of case (2) although they donot correspond to cells, since the polynomials g i are square-free. The rest followsfrom Theorem 3.18. We observe that these squares are in case (2) of Theorem3.18. (cid:3) Example 3.20.
Let f = x d u u u + x d u u u + x d u u u + x d u u u + x d u u u + x d u u u + x d u u u + x d u u u be a bihomogeneous bidegree ( d,
3) polynomial with d ≥
1; it is a simplicial Nagatapolynomial, whose associated simplicial complex is in the following figure: d , d ) u u u u u u x d x d x d x d x d x d x d x d We have: A = A ⊕ A ⊕ . . . ⊕ A d +3 . We want firstly to compute the Hilbert vector by applying Corollary 3.17; first ofall, a , = 8 a , = 6 , and therefore h = h d +3 = 1 h = h d +2 = a , + a , = 8 + 6 = 14 . Then, we analyze the possible cases depending on the degree d : • If d = 1, then a , = 8 · a , = 12 h = a , + a , = 36and the Hilbert vector is (1 , , , , • If d = 2, then, recalling bigraded Poincar´e duality, a , = a , = 8 a , = a , = 12and therefore h = a , + a , + a , = 8 + 8 · ,h = 0 + a , + a , + a , = 8 + 8 · , , , , , • If d = 3, then, again by bigraded Poincar´e duality, a , = a , = 8 , a , = a , = 8 · , a , = a , = 12 , a , = a , = 24 , a , = a , = 8 , therefore h = a , + a , + a , = 44 ,h = a , + a , + a , + a , = 64 h = 0 + a , + a , + a , = 44 h = h in accordance with Poincar´e duality and the Hilbert vector is(1 , , , , , , W-complex Nagata Idealization of bidegree ( d , d ) 17 • In general, let d ≥
4; by hypothesis h d +1 = h = a , + a , + a , = 44 , and h k = a k, + a k − , + a k − , + a k − , ∀ k ∈ { , . . . , d } , where a k, = 8 a k − , = 8 · a k − , = 8 · a k, = 8 . Again using the Poincar´e duality we have: h d +3 − k = h k = 64 ∀ k ∈ (cid:26) , . . . , (cid:22) d + 32 (cid:23)(cid:27) and the Hilbert vector is (1 , , , , . . . , , , , f ), by applying Corollary 3.19.Behaviour depends on d : • If d = 1, by Corollary 3.19 Ann( f ) is (minimally) generated by:(1) h X , . . . , X i = X , X X , . . . ;(2) U , . . . , U ;(3) U U , U U , U U ;(4) X U , X U , X U , X U , X U , X U , X U , X U , X U , X U , X U , X U ,X U , X U , X U , X U , X U , X U , X U , X U , X U , X U , X U , X U ;(5) X U − X U , X U − X U , X U − X U , X U − X U , X U − X U ,X U − X U , X U − X U , X U − X U , X U − X U , X U − X U ,X U − X U , X U − X U . • If d ≥
2, by Corollary 3.19 Ann( f ) is (minimally) generated by(1) h X , . . . , X i d +1 and X h X k where h, k ∈ { , . . . , } , h < k ;(2) U , . . . , U ;(3) U U , U U , U U ;(4) X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U ,X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U , X d U ;(5) X d U − X d U , X d U − X d U , X d U − X d U , X d U − X d U , X d U − X d U ,X d U − X d U , X d U − X d U , X d U − X d U , X d U − X d U , X d U − X d U ,X d U − X d U , X d U − X d U . Example 3.21.
Let f = x d u u + x d u + x d u u be a bihomogeneous bidegree ( d,
2) polynomial, with d ≥
1; it is a CW-Nagatapolynomial whose CW-complex is the following: x d x d u u u x d
28 CW-complex Nagata Idealization of Bidegree ( d , d ) We have: A = A ⊕ A ⊕ . . . ⊕ A d +2 and we want to find its Hilbert vector; first of all, a , = 3 a , = 3and therefore h = h d +2 = 1 h = h d +1 = a , + a , = 6 . Therefore, if d = 1, then Hilbert vector is (1 , , , d = 2, we have a , = 2 + 1 + 2 = 5 , so h = dim A = a , + a , + a , = 3 + 5 + 3 = 11and the Hilbert vector is (1 , , , , d = 3 then, by bigraded Poincar´e duality a , = a , = 3 a , = 3so h = a , + a , + a , = 11 h = a , + a , + a , + a , = 3 + 5 + 3 = 11and the Hilbert vector is (1 , , , , , d ≥
4; by hypothesis h d = h = a , + a , + a , = 11 , and h k = dim A ( k, + dim A ( k − , + dim A ( k − , ∀ k ∈ { , . . . , d } , so, since a k, = 3 a k − , = 5 a k − , = 3using Poincar´e duality we have: h d +2 − k = h k = a k, + a k − , + a k − , = 11 ∀ k ∈ (cid:26) , . . . , (cid:22) d + 22 (cid:23)(cid:27) , and the Hilbert vector is (1 , , , . . . , , , d = 1, by Theorem 3.18 Ann( f ) is (minimally) generated by: • h X , X , X i , U , U , U U , U ; • X U , X U , X U , X U , X U ; • X U − X U , X U − X U .Let d ≥
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Scuola Politecnica e delle Scienze di Base, Universit`a degli Studi di Napoli “Fed-erico II”, corso Protopisani Nicolangelo 70, Napoli (Italy), C.A.P. 80146; [email protected]
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit`a degli Studidi Udine, via delle Scienze 206, Udine (Italy) C.A.P. 33100; [email protected]