Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles
Karim Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou
aa r X i v : . [ m a t h . C O ] J a n ANISOTROPY, BIASED PAIRINGS, AND THE LEFSCHETZ PROPERTY FORPSEUDOMANIFOLDS AND CYCLES
KARIM ALEXANDER ADIPRASITO, STAVROS ARGYRIOS PAPADAKIS, AND VASILIKI PETROTOUA
BSTRACT . We prove the hard Lefschetz property for pseudomanifolds and cycles inany characteristic with respect to an appropriate Artinian reduction. The proof is acombination of Adiprasito’s biased pairing theory and a generalization of a formula ofPapadakis-Petrotou to arbitary characteristic. In particular, we prove the Lefschetz theo-rem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecturedue to Stanley. We also provide a simplified presentation of the characteristic 2 case, andgeneralize it to pseudomanifolds and cycles.
1. I
NTRODUCTION
In [Adi18], Adiprasito introduced the Hall-Laman relations and biased pairing propertyto prove the hard Lefschetz theorem for homology spheres and homology manifoldswith respect to generic Artinian reductions and generic degree one elements acting asLefschetz operators. They replace, intuitively speaking, the Hodge-Riemann relationsin the Kähler setting, which we know cannot apply to general spheres outside the realmof polytope boundaries.The proof relies on the fact that the biased pairing properties, the property that thePoincaré pairing does not degenerate at certain ideals , specifically monomial ideals. Thisproperty allows for replacing the problem of proving the Lefschetz property with anequivalent one, which can be chosen much easier, as well as the idea of using the linearsystem as a variable. However, this replacement process is rather tedious geometrically,and is somewhat tricky to generalize to singular manifolds, and gets more tedious forobjects beyond mild singularities. So the proof is geometrically somewhat challenging.Recently, Papadakis and Petrotou [PP20] have built on the same two ideas, and studiedanisotropy of Stanley-Reisner rings, a stronger (in the sense of more restrictive) notion
Date : January 19, 2021.2010
Mathematics Subject Classification.
Primary 05E45, 13F55; Secondary 32S50, 14M25, 05E40, 52B70,57Q15.
Key words and phrases. hard Lefschetz theorem, pseudomanifolds, simplicial cycles, face rings. than the biased pairing property, demanding that the Poincaré pairing does not degen-erate at any ideal. Clearly, for this, the field has to be rather special, and cannot bealgebraically closed, for instance. Nevertheless implies a partial new proof of the Lef-schetz property for spheres, though only in characteristic 2, by observing that in certaintranscendental extensions, anisotropy is relatively easy to prove by linear algebra argu-ments using a formula they had found.Still, it is quite remarkable that both proofs of the generic Lefschetz property have theirkey ideas in common: ◦ Both proofs make use of the self-pairing in face rings. ◦ Both make use of infinitesimal deformations of the linear system.So it is natural to try to bring both together to compensate for each other’s weaknesses.Our first main result is the following. An orientable pseudomanifold is a pseudoman-ifold with a nontrivial fundamental class, with respect to a fixed characteristic. It isconnected if that fundamental class is furthermore unique. For simplicity of notation,all pseudomanifolds shall be orientable and connected for the entirety of the paper, thatis, we restrict to pseudomanifolds ∆ of dimension d − with H d − ( ∆, k ) ∼ = k . Theorem I.
Consider k any infinite field, Σ any ( d − -dimensional pseudomanifold over k ,and the associated graded commutative face ring k [ Σ ] .Then there exists an Artinian reduction A ( Σ ) , and an element ℓ in A ( Σ ) , such that for every k ≤ d / , we have the hard Lefschetz property: We have an isomorphism B k ( Σ ) · ℓ d − k −−−−→ B d − k ( Σ ) . Here, B denotes the Gorensteinification of A , that is, the quotient of A by the annihila-tor of the fundamental class. This generalizes the generic Lefschetz theorem for spheres(in which case B ∗ ( Σ ) = A ∗ ( Σ ) ) and manifolds in [Adi18] and the characteristic twocase for spheres in [PP20].The Lefschetz property has many applications, implying the Grünbaum-Kalai-Sarkariaconjecture, g-conjecture and many more, but we shall not discuss these here, and directthe interested reader to [Adi18] for a derivation of these implications. Towards theend of this paper, we will provide some new applications, to doubly Cohen-Macaulaycomplexes.Note that orientability is automatic over characteristic two. Alas, more miracles happenin that characteristic. We have the following stronger result. NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 3
Theorem II.
Consider k any field of characteristic two, Σ any ( d − -dimensional pseudo-manifold over k , and the associated graded commutative face ring k [ Σ ] . Then, for some fieldextension k ′ of k , we have an Artinian reduction A ( Σ ) that is anisotropic, i.e. for every element u ∈ A k ( Σ ) , k ≤ d , we have u = 0 . That this is stronger than the Lefschetz theorem is a consequence of the characterizationtheorem of biased pairing theory, or alternatively the Kronecker/perturbation lemma,both of which we shall recall. In general characteristic, we are unable to prove such astatement. Instead, we prove that every u pairs with another that is sufficiently similarto u , essentially related by a change in coefficients, so that they lie in the same monomialideal. We shall also generalize both results to simplicial cycles, and apply this to doublyCohen-Macaulay complexes in Section 6.2. B ASIC N OTIONS
Face rings are the main object of the paper. Our treatment is standard except for therelative case, in which we follow [AY20]. Fix a field k . Definition 2.1.
Let ∆ be a simplicial complex. Define the polynomial ring k [ x v | v ∈ ∆ (0) ] , with variables indexed by vertices of ∆ . The non-face ideal I ∆ of ∆ is the idealgenerated by all elements of the form x v · x v · . . . · x v j where { v , . . . , v j } is not a faceof ∆ . The face ring of ∆ is k [ ∆ ] := Σ [ x v | v ∈ ∆ (0) ] /I ∆ . If Ψ = ( ∆, Γ) is a relative complex, the relative face module of Ψ is defined by I Γ /I ∆ .This is an ideal of k [ ∆ ] .To further out notational abilities, let us recall the definition of a star in a (relative)simplicial complex (Ψ = ∆, Γ ) : The star of a simplex τ within Ψ is st τ Ψ = ( { σ ∈ ∆ : σ ∪ τ ∈ ∆ } , { σ ∈ Γ : σ ∪ τ ∈ Γ } ) . Similarly, the link is lk τ Ψ = ( { σ ∈ ∆ : σ ⊔ τ ∈ ∆ } , { σ ∈ Γ : σ ⊔ τ ∈ Γ } ) . Now, assume that k is infinite. Consider an Artinian reduction A ∗ ( ∆ ) of a face ring k [ ∆ ] with respect to a linear system of parameters Θ . It is instructive to think of A ∗ ( ∆ ) as ageometric realization of ∆ in k d , with the coefficients of x i in Θ giving the coordinatesof the vertex i , recorded in a matrix V . KARIM ADIPRASITO, STAVROS A. PAPADAKIS, AND VASILIKI PETROTOU
Now, consider the quotient H d − ( ∆ ; k ) → µ , the restriction to a single cohomologyclass in our ( d − -complex ∆ . In other words, we pick a simplicial cycle µ ∨ ֒ −→ H d − ( ∆ ; k ) and it’s dual map H d − ( ∆ ; k ) ։ µ . Via the canonical isomorphism H d − ( ∆ ; k ) ∼ = A d ( ∆ ) , see [Adi18, Section 3.9], [TW00], we can consider µ as a quotient of the latter.Then B kµ ( ∆ ) is the quotient of A k ( ∆ ) by the annihilator of the fundamental class, thatis, the annihilator in the pairing A k ( ∆ ) × A d − k ( ∆ ) −→ µ. For minimal homology cycles of dimension ( d − , we set µ to be the unique homologycycle. Notice that B ∗ µ ( ∆ ) is a Poincaré duality algebra, with a fundamental class in thedegree k of µ , and that the pairing of an element in B kµ ( ∆ ) with µ induces a map to k ,the degree map. The algebra A k ( ∆ ) has been Gorensteinified , for lack of a better word.It should be noted that the simplicial cycle µ ∨ plays a special role: In characteristic , itdefines the degree map deg : B dµ ( ∆ ) −→ k as A ∗ ( ∆ ) acts on A ∗ ( ∆ ) by differentials [Lee96], [Adi18, Section 3.1]. Hence, by pairingwith µ , seen as an element of A ∗ ( ∆ ) , we obtain the desired map deg . Convention.
Of course, without specifying the fundamental class, it is not possible toreally write down the map, but there is a canonical choice in the situation of pseudoman-ifolds, at least if we have the Artinian reduction Θ fixed. For this, recall that we writedown the coefficients of the linear system of parameters Θ in a matrix V ∈ k ∆ (0) × [ d ] ,where ∆ (0) denotes the vertices of ∆ .This canonical choice maps the monomial x τ of a cardinality d face τ , that is, a facet ofthe complex, to the inverse of the determinant of the ( d × d ) -minor of V correspondingto τ : deg( x τ ) = sgn µ ( τ ) | V | τ | where we fix an order on the vertices of ∆ and compute the sign with respect to thefundamental class. Two perspectives.
There are, of course, two perspectives that we shall make use of: Wecan consider A ∗ ( ∆ ) over k as functions in V , including the degree map in particular, NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 5 or we consider A ∗ ( ∆ ) over k ( V ) , the field of rational functions associated to the tran-scendental extension of k by the entries of V . It is useful to keep this dichotomy inmind. 3. B IASED PAIRINGS AND H ALL L AMAN RELATIONS
Let us recall the basics of biased pairing theory. More depth and breath is found in[Adi18, Section 5], but we repeat proofs where they are needed for our purposes.3.1.
Biased Poincaré pairings.
Recall: Let Σ be a pseudomanifold of dimension d − ,and B ∗ ( Σ ) its Gorensteinified Artinian reduction. Then we have a pairing B k ( Σ ) × B d − k ( Σ ) −→ B d ( Σ ) ∼ = R . We say that Σ satisfies biased Poincaré duality in degree k ≤ d with respect to someproper subcomplex ∆ , the pairing K k ( Σ, ∆ ) × K d − k ( Σ, ∆ ) −→ K d ( Σ, ∆ ) (1)is nondegenerate on the left. Here K k ( Σ, ∆ ) is nonface monomial of ∆ in B ∗ ( Σ ) . No-tice: Proposition 3.1.
For an ideal I in B ∗ ( Σ ) the following are equivalent:(1) The map I −→ B ∗ ( M ) . ann B ∗ ( M ) I is an injection in degree k .(2) For every x ∈ I k , there exists a y in I d − k such that x · y = 0 .(3) I satisfies biased Poincaré duality in degree k . We obtain immediately an instrumental way to prove biased Poincaré duality for mono-mial ideals.
Corollary 3.2. K ∗ ( Σ, ∆ ) satisfies biased Poincaré duality in degree k if and only if K k ( Σ, ∆ ) −→ B k ( Σ ) | ∆ is injective, where B ∗ ( Σ ) | ∆ is the Poincáre dual of K k ( Σ, ∆ ) , of is the quotient of B k ( Σ ) byelements that pair only trivially with elements of K k ( Σ, ∆ ) . Invariance under subdivisions.
An important tool is the invariance of biased pair-ing under subdivisions (and their inverses). We only need this for odd-dimensionalpseudomanifolds, and with respect to the middle pairing, i.e., a manifold of dimension
KARIM ADIPRASITO, STAVROS A. PAPADAKIS, AND VASILIKI PETROTOU k − , and regarding the pairing in degree k . Recall that a map of simplicial complexesis simplicial if the image of every simplex lies in a simplex of the image. A simplicialmap ϕ : Σ ′ → Σ of pseudomanifolds is a combinatorial subdivision (speak: Σ ′ is asubdivision of Σ ) if it maps the fundamental class to the fundamental class.For geometric simplicial complexes (that is, with respect to an Artinian reduction), werequire a geometric subdivision to map the linear span of a simplex to the linear spanof the simplex containing it, i.e. if σ ′ is an element of Σ ′ , and σ is the minimal face of σ containing ϕ ( σ ′ ) combinatorially, then in the geometric realization, we require that thelinear spans are mapped to corresponding linear spans, i.e. span ϕ ( σ ′ ) ⊆ span σ. In particular, we do not require the image of a face to lie within the combinatorial targetgeometrically, only that they span the same space.
Lemma 3.3.
A geometric subdivision ϕ : Σ ′ → Σ of (2 k − -dimensional pseudomanifoldsthat restricts to the identity on a common subcomplex ∆ preserves biased Poincaré duality, thatis, K ∗ ( Σ, ∆ ) satisfies biased Poincaré duality (in degree k ) if and only if K ∗ ( Σ ′ , ∆ ) does (indegree k ).Proof. The map induces a pullback ϕ ∗ : A ∗ ( Σ ) ֒ −→ A ∗ ( Σ ′ ) , that is compatible with the Poincaré pairing, so it induces a map ϕ ∗ : B ∗ ( Σ ) ֒ −→ B ∗ ( Σ ′ ) . Let us denote by G the orthogonal complement to the image in B k ( Σ ′ ) , the image ofthe Gysin, so that the decomposition B k ( Σ ) ⊕ G = B k ( Σ ′ ) is orthogonal. Notice then that ϕ ∗ is the identity on A ∗ ( ∆ ) , and its image in B ∗ ( Σ ) , so K k ( Σ, ∆ ) ⊕ G = K k ( Σ ′ , ∆ ) Hence, it induces an isomorphism on the orthogonal complements of K ∗ ( Σ, ∆ ) resp. K ∗ ( Σ ′ , ∆ ) . (cid:3) Hall-Laman relations, and the suspension trick.
Finally, biased Poincaré dualityallows us to formulate a Lefschetz property at ideals. We say that Σ a pseudomanifoldof dimension satisfies the Hall-Laman relations in degree k ≤ d and with respect to an NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 7 ideal I ∗ ⊂ B ∗ ( Σ ) if there exists an ℓ in B ( Σ ) , the pairing I k × I k −→ I d ∼ = R a b → deg( abℓ d − k ) (2)is nondegenerate. Note that the Hall-Laman relations coincide with the biased pairingproperty if k = d .If we wish to prove the Hall-Laman relations for a pair ( Σ, ∆ ) , where ∆ is a subcomplexof Σ a ( d − -pseudomanifold, specifically the Hall-Laman relations for K ∗ ( Σ, ∆ ) orits annihilator. Label the two vertices of the suspension n and s (for north and south).We then argue using the following observation. Let π denote the projection along n ,and let ϑ denote the height over that projection, and let A ∗ B denote the free join oftwo simplicial complexes A and B . We then have the following special case of thecharacterization theorem for biased pairings. Lemma 3.4 ([Adi18, Lemma 7.5]) . Considering susp Σ realized in R d +1 , and k < d , thefollowing two are equivalent:(1) The Hall-Laman relations for K k +1 (susp Σ, susp( ∆ ) ∪ s ∗ Σ ) with respect to x n .(2) The Hall-Laman relations for K k ( π Σ, π ∆ ) with respect to ϑ .Proof. Set ϑ = x n − x s in A ∗ (susp Σ ) . Consider then the diagram A k ( π Σ ) A d − k ( π Σ ) A k +1 (susp Σ, s ∗ Σ ) A d − k ( n ∗ Σ ) · ϑ d − k ∼ ∼· x d − k − n An isomorphism on the top is then equivalent to an isomorphism of the bottom map,and restricting to ideals and their Poincaré duals gives the desired. (cid:3)
Lefschetz elements via the perturbation lemma.
Let us note we can use anotherway to construct Lefschetz elements. For this, let us remember a Kronecker lemma of[Adi18], see also [Rin13]:
Lemma 3.5.
Consider two linear maps α , β : X −→ Y KARIM ADIPRASITO, STAVROS A. PAPADAKIS, AND VASILIKI PETROTOU of two vector spaces X and Y over R . Assume that β has image transversal to the image of α ,that is, β (ker α ) ∩ im α = 0 ⊂ Y . Then a generic linear combination α “+” β of α and β has kernel ker( α “+” β ) = ker α ∩ ker β . As observed in [Adi18, Section 6.6], this can be used to iteratively prove the existence ofLefschetz elements provided that B ( Σ ) satisfies the biased pairing property in the pull-back to any vertex link. Let us consider for simplicity the case of Σ a pseudomanifoldof dimension k − .The connection to the classical Hall matching theorem, which constructs stable match-ings in an discrete setting [Hal35]. This lemma is designed to do the same in the settingof linear maps. The idea is now to prove the following transversal prime property : for W a set of vertices in Σ if ker “ X v ∈ W ” x v = \ v ∈ W ker x v Note: proving the transversal prime property for all vertices together is equivalent tothe Lefschetz isomorphism X = B k ( Σ ) · ℓ −−→ Y = B k +1 ( Σ ) for ℓ the generic linear combination over all variables. This is because \ v vertex of Σ ker x v = 0 because of Poincaré duality.Note further, to see how the biased pairing property implies the transversal propertyby induction on the size of the set W , that when we try to apply the criterion by multi-plying with a new variable x v , adding a vertex v to the set W , then we are really pullingback to a principal ideal ideal h x v i in A ( Σ ) , and asked to prove that x v ker “ P v ∈ W ” and im “ P v ∈ W ” ∩ h x v i intersect only in .Note finally that both spaces are orthogonal complements. This is the case if and only ifthe Poincaré pairing is perfect when restricted to either (or equivalently both) of them.4. S OME USEFUL IDENTITIES ON RESIDUES AND DEGREES
We now prove and recall some useful identities on the degree.
NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 9
The square of a monomial.
Now that we know that we can restrict to minimalcycles of odd dimension k − d − , and biased pairings in them, we can go a littlefurther. The trick is to consider the degree of an element in B k ( ∆ ) of a pseudomanifoldof dimension d − as a function of V , which we think of as independent variables. Letus start with a formula due to Lee that describes the coefficients of the fundamentalclass: Lemma 4.1 ([Lee96, Theorem 11]) . We have deg( x τ )( V ) = X F facet containing τ deg( x F ) Y i ∈ τ [ F − i ] ! · Y i ∈ F \ τ [ F − i ] − where [ F − i ] is the volume element of F − i . To compute the volume element [ F − i ] , we can fix a general position vector v that isadded as an element to the matrix V in the i -th column, and compute the determinant.Now consider deg( x τ ) ( V ) , for τ a face of cardinality k . We now want to compute thepartial differential with respect to a ( k − -dimensional face σ of ∆ . For this, we picka basis of k ( V ) d by simply considering the vertices ¯ σ , . . . , ¯ σ k of σ and the faces for τ , . . . , τ k of τ . Denote the basis by B τσ , simply a labelling of V | τ ∪ σ . Let us denote, forfunctions in V , the partial differential ∂ τσ := deg( x σ x τ ) ∂ B τσ , where ∂ B τσ is the partial differential of V σ in the directions of V σ and V τ , the partialdifferential of V σ in directions V σ and V τ and so on. In other words, we are inter-ested in the behavior of a function f ( V ) when varying V σ in the directions of V σ and V τ , varying V σ in directions V σ and V τ etc. We have: Lemma 4.2.
Assume σ and τ are disjoint, but lie in a common face of ∆ . Then ∂ τσ deg( x τ ( V )) = ( − k (deg( x τ x σ )) Let us briefly note that this formula follows also quite simply from the global residueformula [Gri76, CCD97], appropriately generalized to the setting of pseudomanifoldsand face rings. This is not hard, but as Lee already provided a formula in the generalsetting, we shall work with him instead.
Proof.
This is rather simple: Using a volume preserving transformation, we may assumethat σ and τ span orthogonal complements, and that their vertices are orthogonal intheir spans. Using the linear relations, we can rewrite x τ using the linear combinations squarefreeterms generated by st τ ∆ only, as carried out in Lemma 4.1. The only one of those termsdepending on all the variables we differentiate after is deg( x τ x σ ) Y a i where a i is the coordinate of σ i in direction ¯ τ i . Differentiating gives the desired. (cid:3) We need another, different version, that is similarly simple to prove:
Lemma 4.3.
Assume σ and τ are any two faces, and consider a vertex v not in st τ ∪ σ ∆ . Then ∂ v deg( x τ ) ( V ) = 0 The general formula.
Set now e k := k ( V , V ′ ) , where k is any field, the fieldof rational functions with variables V , as well as a copy V ′ of the vertex coordinates.Consider now an element u of k ( V ) k [ ∆ ] that is the linear combination of squarefreemonomials. We say a face σ is compatible with u if ◦ st σ ∆ contains but one face in the support | u | of u , denoted by τ ( u, σ ) . The coefficientof u at τ ( u, σ ) is . ◦ Consider a face τ ′ of | u | that is not τ ( u, σ ) . Then the star of τ ′ intersects σ trivially, orin a facet σ τ ′ of τ , and the coefficient u τ ′ of u in τ ′ is independent of the variables of σ \ σ τ ′ .Associated to u = P λ τ x τ , we consider an element u ′ defined by replacing all oc-curences of a variable of V in λ τ with the corresponding variable in V ′ , obtaining λ ′ τ and u ′ .We now differentiate deg( u · u ′ ) , using the formula of the previous section. Lemma 4.4.
For u compatible and normalized with respect to σ , we have ∂ τ ( u,σ ) σ deg( u · u ′ ) = deg(( ∂ τ ( u,σ ) σ u ) · u ′ ) + ( − k (deg( x τ ( u,σ ) · x σ )) (3)Note that if we substitute V ′ → V on both sides of this equation, then we obtain Corollary 4.5. ∂ τ ( u,σ ) σ deg( u · u ′ ) u = u ′ − ( − k deg(( ∂ τ ( u,σ ) σ u ) · u ) = (deg( x τ ( u,σ ) · x σ )) (4) Key Observation:
Note that if u lies in some monomial ideal, then so does u ′ and ∂ τ ( u,σ ) σ u , as we only changed the coefficients of the monomials, and introduced no newmonomial. This is rather marvellous, and informs us how we want to prove the biasedpairing property. NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 11
5. P
ROVING THE L EFSCHETZ PROPERTY
To finish the proof of the Lefschetz theorem, over e k , we need to prove the biased pairingproperty in degree k for a pair ( Σ, ∆ ) , where Σ is an orientable pseudomanifold of odddimension d − k − , Σ ′ is a codimension one orientable pseudomanifold in it, and ∆ is a component of the complement.5.1. Characteristic two.
In characteristic two, one can prove a stronger statement thanjust biased pairing: We can prove that elements in u ∈ B k ( Σ ) over k ( V ) , for Σ apseudomanifold of dimension k − , have deg( u ) = 0 . It illustrates an importantprinciple: normalization.Consider u ∈ A k ( Σ ) . Consider the pairing with x σ for some cardinality k face σ . Normalization:
We may now assume that u is represented as P λ τ x τ so that only one τ of the sum lies in st σ Σ , and may further assume that this τ lies in lk σ Σ . This is because B k (st σ Σ ) ∼ = B k (lk σ Σ ) is of dimension one.Finally observe that ∂ τσ deg( u ) = ∂ τσ deg( X ( λ τ x τ ) )= X λ τ ∂ τσ deg( x τ )= X λ τ deg( x τ x σ )= X deg( X λ τ x τ x σ ) = deg( x σ u ) in characteristic . We conclude Proposition 5.1.
We have ∂ τσ deg( u ) = deg( x σ u ) This finishes the proof of the biased pairing property, as every u must pair with some x σ by Poincaré duality, so every element must pair with itself.5.2. General characteristic.
For the general case, let us consider the ideal K ∗ ( Σ, ∆ ) in e k [ Σ ] . The trick is to start with an element [ u ] in K k ( Σ, ∆ ) , and use the subdivisionlemma and normalizations to replace it by an equivalent, but compatible element.Note first that by the subdivision lemma, we may assume that K ∗ ( Σ, ∆ ) is generatedby a single indeterminate to start with, associated to a vertex v . Let us now consider a class [ u ] in K k ( Σ, ∆ ) , represented by an element u in the ideal h x v i ⊂ e k [ Σ ] as a sum u = X λ τ x τ , where the sum is over faces τ with common vertex v .We want to prove that u pairs nontrivially with some other element K k ( Σ, ∆ ) . If itdoes so with some element x σ , associated to a cardinality k face containing v , then weare done. Notice further that we may assume that the coefficients λ τ are from k ( V ) , asthe variables V ′ are transcendental over that field.On the other hand, it has to pair with some face of Σ by Poincaré duality. It has to beone of the faces σ of Σ ′ , as all other faces annihilate K ∗ ( Σ, ∆ ) . Normalization:
Now, we use the following observation:
Observation 5.2.
The quotient K k (st σ Σ, st σ ∆ ) of K k ( Σ, ∆ ) is one-dimensional. In particular, we can assume that the restriction of u to st σ Σ is supported in a singleface of cardinality k . We shall use observations like this repeatedly.Note that we can also normalize u , so that the coefficient of u in that face is . Subdivision:
Now, recall that we can subdivide and unsubdivide to make the com-plex ( Σ, ∆ ) , and with it u , more manageable. First consider Σ ′ − σ , and consider it’sdouble, glued along the faces induced by the vertices of ∂ st σ Σ ′ . Cone over the result-ing complex, call the result T , glue it to ( Σ \ Σ ′ − σ ) as the boundary component ofits compactification. Up to further subdivision, we may assume that the natural map T → Σ ′ − σ is simplicial, and that [ v ∈ σ st v ( ∂T ) \ ( Σ ′ − σ ) naturally deformation retracts to ∂ ( Σ ′ − σ ) = ∂ st σ Σ ′ We obtain a new pseudomanifold e Σ , in which ∆ is naturally embedded, and a thick-ened complex e ∆ = T ∪ ∆ . The natural map e Σ −→ Σ. Consider the natural pullback map A ( Σ ) ֒ −→ A ( e Σ ) , which takes u to an element e u supported in ( e Σ, e ∆ ) . NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 13
Unsubdivision:
Consider now the boundary of e ∆ in e Σ , denoted by e Σ ′ . Then [ v ∈ σ st v e Σ ′ deformation retracts naturally along the radial projection to st σ Σ ′ . Consequently, themapping cylinder defines an unsubdivision of e Σ ′ , and with it e Σ , so that the inducedmap ∂ [ v ∈ σ st v e Σ ′ ! −→ ∂ st σ Σ ′ is a combinatorial isomorphism. Consider the resulting complex b Σ , and the subcomplex e ∆ corresponding to e ∆ .We have a natural injection A ( b Σ ) ֒ −→ A ( e Σ ) , whose image can be assumed to contain e u . Indeed, otherwise, it has a natural compo-nent in the orthogonal complement, which pairs with itself, and we are done. Denotethe preimage by b u . Final steps:
Now, notice that for every facet σ ′ of σ , K k (st σ ′ ∪ v b Σ, st σ ∪ v b Σ ) is one dimensional, so we can assume that b u is supported in a single face τ ′ σ of therespective links (though we can no longer normalize, as we would have to do so simul-taneously).Now, we can finally assume that b u pairs with trivially with x σ ′ ∪ v , where σ ′ is again anyfacet of σ , as we would be done otherwise. Hence, the coefficient in τ ′ σ , if we specializethe coordinates for it to coincide with τ ( u, σ ) , is − . Hence, we can assume that b u iscompatible with σ .Consequently, by Corollary 4.5 we have that one of the two terms, deg( b u · b u ′ ) or deg(( ∂ τ ( b u,σ ) σ b u ) · b u ) is nontrivial. Which proves biased pairing for b u and therefore also u . (cid:3)
6. C
YCLES AND DOUBLY C OHEN -M ACAULAY COMPLEXES
Cycles and more cycles.
The main theorems extend to simplicial cycles. For this,consider a simplicial complex Σ of dimension d − , and a simplicial homology ( d − -cycle µ ∨ over some field k on it. Then we induce associated to it a Gorenstein ring B ∗ µ ( Σ ) . Note that the ring is readily described, and in fact defined, by the degree map: For a cardinality d face τ , we have deg( x τ ) = µ ∨ τ | V | τ | , where µ ∨ τ is the oriented coefficient of µ ∨ in τ . Theorem III. B ∗ µ ( Σ ) has the Lefschetz property over e k , and over any infinite field extensionof k with respect to a sufficiently general position of parameters. Similarly, we have:
Theorem IV. B ∗ µ ( Σ ) is totally anisotropic over k ( V ) in characteristic two. The proof is the same, save for a weighting coming from the coefficients of µ on eachfacet (replacing the standard sgn( τ ) -weighting in the case of pseudomanifolds).Consider now a collection M ∨ of cycles in H d − ( Σ ) , that is, a map H d − ( Σ ) −→ M. Then we have the quotient B ∗ M ( Σ ) of A ∗ ( Σ ) induced as before as the quotient by theannihilator under the pairing A k ( Σ ) × A d − k ( Σ ) −→ M. Corollary 6.1. B ∗ M ( Σ ) has the top-heavy Lefschetz property over e k , and any infinite fieldextension of k with respect to a sufficiently general position of parameters, that is, there is an ℓ ∈ B M ( Σ ) so that B kM ( Σ ) · ℓ d − k −−−−→ B d − kM ( Σ ) is injective.Proof. Consider a stratification M −→ µ i where the µ i are one-dimensional subspaces of M and the stratification is injective.Then we have an injection B kM ( Σ ) −→ M B kmu i ( Σ ) , and hence B kM ( Σ ) B d − kM ( Σ ) L B kµ i ( Σ ) L B d − kµ i ( Σ ) · ℓ d − k · ℓ d − k which implies injectivity on the top if it is present on the bottom. (cid:3) NISOTROPY, BIASED PAIRINGS AND LEFSCHETZ FOR CYCLES 15
Doubly Cohen-Macaulay complexes.
A simplicial complex is called s -Cohen-Macaulay if it is Cohen-Macaulay and after the removal of s vertices, the complex is still Cohen-Macaulay of the same dimension.For instance, a triangulated homology sphere is -Cohen-Macaulay, also called doublyCohen-Macaulay [Sta96, Chapter III.3]. Stanley showed that doubly Cohen-Macaulaycomplexes are level, that is, for such a complex ∆ of dimension d − , the socle is con-centrated in degree d . In other words, if M = A d ( ∆ ) , we have B kM ( Σ ) = A k ( Σ ) . From the last result, we conclude
Corollary 6.2.
Consider a doubly Cohen-Macaulay complex Σ of dimension d − . Then A ∗ M ( Σ ) has the top-heavy Lefschetz property over e k , and any infinite field extension of k withrespect to a sufficiently general position of parameters, that is, there is an ℓ ∈ A M ( Σ ) so that A kM ( Σ ) · ℓ d − k −−−−→ A d − kM ( Σ ) is injective. In particular, the g -vector of a doubly Cohen-Macaulay complex is an M -vector, since A kM ( Σ ) · −→ A k +1 M ( Σ ) is injective for k ≤ d . 7. O PEN PROBLEMS
Concerning problems surrounding the g-theorem, there is a stronger conjecture avail-able: It has been conjectured that stronger, for s -Cohen-Macaulay complexes, we havethis injection for k ≤ d + s − , but the approach offers no clue how to do it. In particular,the combinatorial conjecture is: Conjecture 7.1.
Consider an s -Cohen-Macaulay complex of dimension d − . Then the vector ( h , h − h , . . . , h ⌈ d + s ⌉ − h ⌈ d + s ⌉− ) is an M -vector. It is equally an open problem to extend the anisotropy to general characteristic. Weconjecture this is true, but have no good idea for an approach.
Conjecture 7.2.
Consider k any field, Σ any ( d − -dimensional pseudomanifold over k , andthe associated graded commutative face ring k [ Σ ] . Then, for some field extension k ′ of k , we have an Artinian reduction A ( Σ ) that is totally anisotropic, i.e. for every element u ∈ A k ( Σ ) , k ≤ d , we have u = 0 . Of course, special cases over characteristic , such as the anisotropicy of integral homol-ogy spheres over k = Q can be established using standard mixed characteristic tricksand reduction to characteristic two, but more seems difficult. A generalization of theanisotropy to characteristic p that should be more immediate is the following: Conjecture 7.3.
Consider k any field of characteristic p , Σ any ( d − -dimensional pseudo-manifold over k , and the associated graded commutative face ring k [ Σ ] . Then, for some fieldextension k ′ of k , we have an Artinian reduction A ( Σ ) that is totally p -anisotropic, i.e. forevery element u ∈ A k ( Σ ) , k ≤ dp , we have u p = 0 . Finally, both [Adi18] and the present work only prove the existence of Lefschetz ele-ments on a generic linear system of parameters, and it would be interesting to havespecific ones. One candidate is, in our opinion, the moment curve ( t, t , . . . , t d ) . Open Problem 7.4.
Do distinct points on the moment curve provide a linear system with theLefschetz property (on any (PL)-sphere, pseudomanifold or cycle)?
Acknowledgements.
K. A. was supported by the European Research Council underthe European Union’s Seventh Framework Programme ERC Grant agreement ERC StG716424 - CASe and the Israel Science Foundation under ISF Grant 1050/16. S. A. P. thanksChristos Athanasiadis for suggesting the problem, and David Eisenbud for useful con-versations. We benefited from experiments with the computer algebra program Macaulay2[GS]. This work is part of the Univ. of Ioannina Ph.D. thesis of V. P., financially sup-ported by the Special Account for Research Funding (E.L.K.E.) of the University of Ioan-nina under the program with code number 82561 and title
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