aa r X i v : . [ m a t h . A C ] F e b LINEAR STRANDS SUPPORTED ON CELL COMPLEXES
KELLER VANDEBOGERT
Abstract.
In this paper, we study ideals I whose linear strand can be sup-ported on a polyhedral cell complex. We provide a sufficient condition for thelinear strand of an arbitrary subideal of I to remain supported on an easilydescribed subcomplex. In particular, we prove that a certain class of rainbowmonomial ideals always have linear strand supported on a cell complex, includ-ing any initial ideal of the ideal of maximal minors of a generic matrix. Thisfollows from a general statement on the cellularity of complexes whose associ-ated poset forms a meet semilattice. We also provide a sufficient condition forthese ideals to have linear resolution, which is also an equivalence under mildassumptions. We then employ a result of Almousa, Fløystad, and Lohne toapply these results to polarizations of Artinian monomial ideals. We concludewith further questions relating to cellularity of certain classes of squarefreemonomial ideals and the relationship between initial ideals of maximal minorsand algebra structures on certain resolutions. Introduction
A common theme in the study of monomial ideals is the association of some kindof combinatorial object P that can recover the ideal of interest I . It is then oftenthe case that properties desirable from a combinatorial perspective for P translateinto properties that are desirable from an algebraic perspective for I . This kindof correspondence has been used to great effect in, for instance, Stanley-Reisnertheory, where any squarefree monomial ideal has a bijective correspondence witha simplicial complex. This correspondence yields nontrivial consequences, such asHochster’s formula or Reisner’s criterion for Cohen-Macaulayness.From a homological perspective, the object that best captures the behavior ofa monomial ideal I is a free resolution. Thus, a similar line of thinking suggeststhat the association of a free resolution F • of I to some combinatorial object P willyield a dictionary for translating properties of P into properties of F • . This wasfirst explored by Bayer, Peeva, and Sturmfels [3], where free resolutions that couldbe supported on simplicial complexes were studied. This means that the differen-tial of F • and the natural differential on the simplicial complex induce each other,and there is a bijective correspondence between basis elements in a given homo-logical degree and faces of a given dimension. Examples of well-known complexessupported on simplicial complexes include the Taylor resolution and the Scarf com-plex. However, simplicial complexes turn out to be too restrictive in general tosupport more than the most well-behaved complexes.This leads to the idea of considering cellular resolutions , that is, resolutionssupported on so-called cell complexes, introduced by Bayer and Sturmfels in [4].Cellular resolutions have turned out to occupy a “goldilocks” zone of generality; Date : February 16, 2021. they are general enough to support minimal free resolutions of many large classes ofmonomial ideals, including the well-known Eliahou-Kervaire resolution (see [21]),but not so general as to be combinatorially incomprehensible. There are manyclasses of ideals whose minimal free resolutions can be supported on cellular com-plexes; see, for instance [8], [2], [16], [11], [12], [13], or [9].It turns out that not all monomial ideals have cellular resolutions; indeed, thereare monomial ideals whose minimal free resolution cannot even be supported ona CW-complex (this is due to Velasco in [32]). Reiner and Welker (see [28]) havealso constructed a relatively simple monomial ideal with linear resolution, but suchthat no change of basis has differentials with coefficients equal to 0 or ±
1. Minimalfree resolutions of arbitrary monomial ideals are not completely devoid of structure,however; a result of Clark and Tchernev (see [7]) shows that every monomial idealhas minimal free resolution that can be supported on a poset.In [23], a specific cell complex called the complex of boxes was used to sup-port what is now called the box polarization of squarefree strongly stable ideals.In general, polarizations are a method of associating to any monomial ideal I asquarefree monomial e I in a larger polynomial ring that is homologically indistin-guishable. There are many different ways to polarize a monomial ideal; indeed,Almousa, Fløystad, and Lohne in [1] describe all such polarizations for powers ofthe graded maximal ideal in terms of spanning trees of certain maximal “downtriangles” associated to the graph of linear syzygies. Moreover, they also showthat the study of polarizations of an Artinian monomial ideals is equivalent viaAlexander duality to the study of ideals generated by rainbow monomials with lin-ear resolution. Rainbow monomial ideals also appear as defining ideals for sets ofpoints in multiprojective spaces, where each color class corresponds to a distinctcopy of P n ; see the work by Favacchio, Guardo, and Migliore [14] or [15] for rainbowmonomials from this perspective.In this paper, we use these connections as motivation for studying a proper sub-class of all equigenerated rainbow monomial ideals, which we call rainbow determi-nantal facet ideals (or rainbow DFIs). These ideals are inspired by determinantalfacet ideals (DFIs), a generalization of binomial edge ideals (see [10], [24], [18],and [20]); the main difference is that rainbow DFIs depend on both a term order < and a parametrizing simplicial complex ∆. We prove that the linear strand of any rainbow DFI is supported on a cell complex; in particular, any rainbow DFIwith linear resolution (hence Alexander dual to a polarization) has a cellular reso-lution. Moreover, we formulate a sufficient condition for any rainbow DFI to havelinear resolution; if the Alexander dual ∆ ∨ of the associated simplicial complex hasnonmaximal overlap, this becomes an equivalence.The techniques involved in proving these statements are of independent interestand involve the formulation of more general statements on the structure of theminimal free resolution of any initial ideal of the ideal of maximal minors, theconstruction of linear strands of arbitrary modules, and cellularity of complexeswhose associated poset forms a meet-semilattice. Moreover, we examine conditionsguaranteeing that a sequence that is regular on R/I is also regular on
R/J for anyfixed J ⊂ I ; this is used for computing the graded Betti numbers of rainbow DFIs(and is later used to give a characterization of Cohen-Macaulayness).The paper is organized as follows. In Section 2, we first introduce some nec-essary background and notation relating to (rainbow) DFIs, the Eagon-Northcott INEAR STRANDS SUPPORTED ON CELL COMPLEXES 3 complex, and arbitrary initial ideals of ideals of maximal minors. We perform amore detailed study of so-called sparse
Eagon-Northcott complexes, including anexplicit description of the differentials and the multidegrees appearing in each ho-mological degree.In Section 3, we consider the construction of linear strands in general. We firstprove, using the machinery of iterated trimming complexes (introduced in [31]),that the linear strand of any subideal with G ( J ) ⊆ G ( I ) is obtained by taking anappropriate subcomplex of the linear strand of I . We then apply this result to thecase that the linear strands in question are supported on cellular complexes; weintroduce the notion of support chains centered at a vertex v of a cellular complexto construct a morphism of complexes. Taking the kernel of this morphism ofcomplexes will yield the linear strand upon deleting of the generator correspondingto the vertex v . In the case that the cellular complex is linearly connected , theresulting linear strand is obtained by simply restricting to the subcomplex inducedby deleting v .In Section 4, we study conditions guaranteeing that certain posets associated toa complex F • give rise to a cell complex supporting F • . Assuming that the frame associated to F • is sufficiently well-behaved, cellularity can be checked by showingthat the associated poset forms a meet semilattice . We use this result to prove thatany initial ideal of the ideal of maximal minors has cellular resolution. Even moregenerally, we combine this with the results of Section 3 to prove that the linearstrand of every rainbow DFI is supported on a cell complex.In Section 5, we study conditions on the simplicial complex ∆ associated to arainbow DFI implying linearity of the minimal free resolution. This condition isformulated in terms of the existence of an ordering on the elements of ∆ ∨ such thatthey form a so-called free sequence ; moreover, if elements of ∆ ∨ have nonmaximaloverlap, then every ordering yields a free-sequence and this condition is also anequivalence. In this latter situation, we also compute the Betti numbers explicitly;we already know by the result of Section 4 that these ideals have cellular resolution.Finally, in Section 6, we apply our results to polarizations of Artinian monomialideals. Taking Alexander duals, all of the conditions guaranteeing linearity of Sec-tion 5 directly translate to the property of being the polarization of an Artinianmonomial ideal. Moreover, assuming that all elements of ∆ ∨ have nonmaximaloverlap, we show that rainbow DFIs are Alexander dual to a polarization of apower of the maximal ideal if and only if | ∆ ∨ | = 0. We conclude with a variety ofquestions: we first ask about whether or not the admission of an order for whichthe elements of ∆ ∨ form a free sequence is a necessary condition for the associatedrainbow DFI to have linear resolution. We also ask about the existence of termorders for which a pre-selected set of generators are free vertices of the cellular com-plex supporting the resolution of the initial ideal of maximal minors. Lastly, wepose a curious potential connection between the selection of a multigraded algebrastructure on the minimal resolution of the ideal of all squarefree monomials, andan arbitrary initial ideal of an ideal of maximal minors.2. Background and Sparse Eagon-Northcott Complexes
In this section, we introduce some notation related to (rainbow) determinantalfacet ideals that will be in play for the rest of the paper. We then delve into so-called sparse
Eagon-Northcott complexes, which arise by homogenizing the classical
KELLER VANDEBOGERT
Eagon-Northcott complex with respect to a variable t and term order < , thensetting t = 0. It turns out that one can say precisely what the differentials andmultidegrees must look like for any given term order. These complexes will be usedin later sections for proving cellularity and establishing criteria for having linearresolution.Throughout this paper, all complexes will be assumed to be nontrivial only innonnegative homological degrees. Notation 2.1.
Let R = k [ x ij | i n, j m ] be a polynomial ring overan arbitrary field k . Let M be an n × m matrix of variables in R where n m .For indices a = { a , . . . , a r } and b = { b , . . . , b r } such that 1 a < . . . < a r n and 1 b < · · · < b r m , set[ a | b ] = [ a , . . . , a r | b , . . . , b r ] = det x a ,b · · · x a ,b r ... . . . ... x a r ,b · · · x a r ,b r where [ a | b ] = 0 if r > n . When r = n , use the simplified notation [ a ] = [1 , . . . , n | a ].The ideal generated by the r -minors of M is denoted I r ( M ). Definition 2.2.
For a simplicial complex ∆ and an integer i , the i -th skeleton ∆ ( i ) of ∆ is the subcomplex of ∆ whose faces are those faces of ∆ with dimension atmost i . Let S denote the set of simplices Γ with vertices in [ m ] with dim(Γ) > n − ( n − ⊂ ∆.Let Γ , . . . , Γ c be maximal elements in S with respect to inclusion, and let ∆ i :=Γ ( n − i . Each Γ i is called a maximal clique , and any induced subcomplex of Γ i is a clique . The simplicial complex ∆ clique whose facets are the maximal cliques of ∆ iscalled the clique complex associated to ∆. The decomposition ∆ = ∆ ∪ · · · ∪ ∆ c is called the maximal clique decomposition of ∆. Definition 2.3.
Let ∆ be a pure ( n − m ]. Let R = k [ x ij | i n, j m ] be a polynomial ring over anarbitrary field k . Let M be an n × m matrix of variables in R . The determinantalfacet ideal (or DFI ) J ∆ ⊆ R associated to ∆ is the ideal generated by determinantsof the form [ a ] where a supports an ( n − a ]correspond to the vertices of some facet of ∆.Let < be an arbitrary term order. The rainbow DFI rain < ( J ∆ ) associated to∆ is the monomial ideal generated by all monomials of the form in < ([ a ]), where a supports and ( n − M from which all minors are taken.That is, J ∆ = J ∆ + · · · + J ∆ r . Notation 2.4.
Let ∆ be a pure ( n − m ]. The notation ∆ ∨ will denote the Alexander dual of ∆; that is, ∆ ∨ is the unique ( n − ∪ ∆ ∨ equal to all n -subsets of[ m ].Recall that the set of maximal minors forms a universal Gr¨obner basis for I n ( M )by a result of Sturmfels and Zelevinsky (see [29]). This immediately yields thefollowing: INEAR STRANDS SUPPORTED ON CELL COMPLEXES 5
Observation 2.5.
Let < be an arbitrary term order. For any ( n − < ( J ∆ ) + rain < ( J ∆ ∨ ) = in < I n ( M ) . Next we define the linear strand , which is one of the central objects of study inthis paper. These will be dealt with more intensively in Section 3.
Definition 2.6.
Let F • be a minimal graded R -free complex with F having initialdegree d . Then the linear strand of F • , denoted F lin • , is the complex obtained byrestricting d Fi to ( F i ) d + i for each i > Remark . Observe that the minimality assumption in Definition 2.6 ensuresthat the linear strand is well defined. Choosing bases, the linear strand can beobtained by restricting to the columns where only linear entries occur in the matrixrepresentation of each differential.The following result, due to Boocher, shows that with respect to any term order < , the ideal in < I n ( M ) specializes to the ideal of all squarefree monomials of degree n in m variables. Theorem 2.8 ([5, Proof of Theorem 3.1]) . For any term order < , the sequence ofvariable differences { x − x , . . . , x − x n } ∪ · · · ∪ { x m − x m , . . . , x m − x nm } forms a regular sequence on R/ in < I n ( M ) . In particular, β ij ( R/I n ( M )) = β ij ( R/ in < I n ( M )) for all i, j. The following definition is the definition of the well-known Eagon-Northcot com-plex. Recall that the Eagon-Northcott complex is a minimal free resolution of thequotient ring defined by the ideal of maximal minors I n ( M ); it will turn out to bethe basis from which all resolutions of R/ in < I n ( M ) can be built. Definition 2.9 (Eagon-Northcott complex) . Let φ : F → G be a homomorphismof free modules of ranks n and m , respectively, with n > m . Let c φ be the image of φ under the isomorphism Hom R ( F, G ) ∼ = −→ F ∗ ⊗ G . The Eagon-Northcott complex is the complex0 → D m − n ( G ∗ ) ⊗ m ^ F → D m − n − ( G ∗ ) ⊗ m − ^ F → · · · → G ∗ ⊗ n +1 ^ F → n ^ F → n ^ G with differentials in homological degree > c φ ∈ F ∗ ⊗ G , and the map V g F → V g G is V g φ . Notation 2.10.
Let E • denote the Eagon-Northcott complex of Definition 2.9. If F has basis f , . . . , f m and G has basis g , . . . , g n , then define g ∗ ( α ) ⊗ f I := g ∗ ( α )1 · · · g ∗ ( α n ) n ⊗ f i ∧ · · · ∧ f i n + ℓ , where α = ( α , . . . , α n ) and I = ( i < · · · < i n + ℓ ). Observe that E • inherits a Z n × Z m -grading by settingmdeg( g ∗ ( α ) ⊗ f I ) = (1 + α ǫ + · · · + α n ǫ n , ǫ i + · · · ǫ i n + ℓ ) , where ǫ k denotes the appropriately sized vector with 1 in the i th spot and 0 else-where, and 1 denotes a length n vector of 1s. KELLER VANDEBOGERT
Corollary 2.11.
Let F • denote a multigraded resolution of in < I n ( M ) . Then forevery multidegree α , β α ( R/ in < I n ( M )) . Proof.
By Theorem 2.8, a minimal free resolution of R/ in < I n ( M ) may be obtainedby setting some of the entries in the matrix representation of the differentials ofDefinition 2.9 equal to 0. With respect to the Z n × Z m -grading of the Eagon-Northcott complex E • , one has β α ( R/ in < I n ( M )) . Since any Z nm -graded minimal free resolution is also Z n × Z m -graded, the resultfollows. (cid:3) Definition 2.12.
Let R = k [ x ij | i n, j m ] be a polynomial ring overan arbitrary field k , endowed with some term order < . The sparse Eagon-Northcott E < • complex is the complex obtained by:(1) homogenizing E • with a new variable t , with respect to the term order < ,then(2) setting t = 0 in the resolution.As observed by Boocher, the minimal free resolution of R/ in < I n ( M ) may beobtained as the sparse Eagon-Northcott complex E < • . Definition 2.13.
Let F be a free R -module with some fixed basis B ⊂ F . The support of an element f = P b ∈ B c b b ∈ F with respect to the basis B is defined tobe: Supp B ( f ) := { b ∈ B | c b = 0 } , where c b ∈ R . When the basis B is understood, the notation Supp( f ) will be usedinstead.When dealing with the sparse Eagon-Northcott complex E < • , the support of anyelement will be assumed to be with respect to the basis elements of Notation 2.10. Observation 2.14.
Let < be any term order. Then the associated sparse Eagon-Northcott complex can be given a Z nm -multigrading. Proof.
Assign the multidegrees inductively. In homological degree 1, f I has mul-tidegree in < ([ I ]); for basis elements in higher homological degree, the multidegreeis assigned as the lcm of the multidegrees appearing in the support of the imageunder the differential. (cid:3) Notation 2.15.
Let < be a term order. Every basis element of the sparse Eagon-Northcott complex E < • can be assigned a unique multidegree g ∗ ( α ) ⊗ f I ←→ n Y i =1 (cid:16) x ib | α i − | + i · · · x ib | α i | + i (cid:17) , where I = { b , . . . , b n + ℓ } . For ease of notation, the above multidegree will bewritten n Y i =1 (cid:16) x ib | α i − | + i · · · x ib | α i | + i (cid:17) = x b · · · x n b n , where b i = ( b | α i − | + i , . . . , b | α i | + i ). INEAR STRANDS SUPPORTED ON CELL COMPLEXES 7
Theorem 2.16.
Let < be any term order. Then the complex E < • has differentialsof the form g ∗ ( α ) ⊗ f I X i ∈ Supp( α ) j ∈ b i sgn( j ∈ I ) x ij g ∗ ( α − ǫ i ) ⊗ f I \ j , where mdeg( g ∗ ( α ) ⊗ f I ) = x b · · · x n b n , and f I in < ([ I ]) . Proof.
This follows by induction on the homological degree, where the base casefor ℓ = 2 is clear. Let ℓ >
2; since ( E < • , d • ) is a complex, by definition d ℓ − ◦ d ℓ = 0.Assume for sake of contradiction that Supp( d ℓ ( g ∗ ( α ) ⊗ f I ) does not contain g ∗ ( α − ǫ i ) ⊗ f I \ j for some i ∈ Supp( α ), j ∈ b i . Let r be any integer and s ∈ b r such that g ∗ ( α − ǫ r ) ⊗ f I \ s ∈ Supp( d ℓ ( g ∗ ( α ) ⊗ f I )); by the inductive hypothesis, g ∗ ( α − ǫ r − ǫ i ) ⊗ f I \ s,j ∈ Supp( g ∗ ( α − ǫ r ) ⊗ f I \ s ). However, if g ∗ ( α − ǫ i ) ⊗ f I \ j / ∈ Supp( d ℓ ( g ∗ ( α ) ⊗ f I )),then the term x ij x rs g ∗ ( α − ǫ r − ǫ i ) ⊗ f I \ s,j does not cancel with any other term in theimage d ℓ − ◦ d ℓ ( g ∗ ( α ) ⊗ f I ), contradicting the fact that E < • is a complex. (cid:3) Theorem 2.16 immediately yields an explicit description of the multidegrees ap-pearing in each homological degree. This result will be essential later for showingthat the associated poset forms a meet semilattice.
Corollary 2.17.
The multidegrees appearing in homological degree ℓ > of thecomplex E < • may be constructed by choosing all degree n + ℓ − monomials x b · · · x n b n with the property that x b i x b i · · · x nb in ∈ G ( in < I n ( M )) for all b i j ∈ b j , j = 1 , . . . , n .Proof. This follows by induction on the homological degree combined with thedifferential of Theorem 2.16. (cid:3) On the Construction of Linear Strands
In this section, we consider the construction of linear strands. We first formulatea more general statement on constructing the linear strand of certain classes ofideals with the machinery of iterated trimming complexes . With this more generalresult, we consider ideals whose linear strand is supported on a cell complex. Afterdeveloping a fair amount of machinery, this section culminates in Theorems 3.19and 3.20. These results show that ideals whose linear strand is supported on a linearly connected cell complex have the property that every subideal is supportedon a cell complex that is simple to describe.We begin this section with a general result on linear strands originally due toHerzog, Kiani, and Madani; intuitively, this says that a linear complex arises as alinear strand if and only if the homology is concentrated in sufficienly large degrees.
Theorem 3.1 ([19], Theorem 1.1) . Let R be a standard graded polynomial ringover a field k . Let G • be a finite linear complex of free R -modules with initialdegree n . Then the following are equivalent:(1) The complex G • is the linear strand of a finitely generated R -module withinitial degree n .(2) The homology H i ( G • ) i + n + j = 0 for all i > and j = 0 , . KELLER VANDEBOGERT
We will temporarily adopt the following general setup for the next two results.In this setup, we are choosing specific generators of a given ideal I that we wishto trim off, then “rescale” by some new ideal. Theorem 3.3 shows how a (gener-ally nonminimal) resolution can be computed for this new ideal based on the freeresolutions of all other ideals involved. Setup 3.2.
Let R be a standard graded polynomial ring over a field k . Let I ⊆ R be a homogeneous ideal and ( F • , d • ) denote a homogeneous free resolution of R/I .Write F = F ′ ⊕ (cid:16) L ti =1 Re i (cid:17) , where, for each i = 1 , . . . , t , e i generates a freedirect summand of F . Using the isomorphism Hom R ( F , F ) = Hom R ( F , F ′ ) ⊕ (cid:16) t M i =1 Hom R ( F , Re i ) (cid:17) write d = d ′ + d + · · · + d t , where d ′ ∈ Hom R ( F , F ′ ) and d i ∈ Hom R ( F , Re i ) .For each i = 1 , . . . , t , let a i denote any homogeneous ideal with d i ( F ) ⊆ a i e i , and ( G i • , m i • ) be a homogeneous free resolution of R/ a i .Use the notation K ′ := im( d | F ′ : F ′ → R ) , K i := im( d | Re i : Re i → R ) , andlet J := K ′ + a · K + · · · + a t · K t . Theorem 3.3.
Adopt notation and hypotheses as in Setup 3.2. Then there is amorphism of complexes (3.1) · · · d k +1 / / F kQ k − (cid:15) (cid:15) d k / / · · · d / / F d ′ / / Q (cid:15) (cid:15) F ′ d (cid:15) (cid:15) · · · L m ik / / L ti =1 G ik − L m ik − / / · · · L m i / / L ti =1 G i − P ti =1 m i ( − ) · d ( e i ) / / R. Moreover, the mapping cone of 3.1 is a free resolution of
R/J . Definition 3.4.
The iterated trimming complex associated to the data of Setup3.2 is the complex of Theorem 3.3.Restricting the diagram 3.1 to the linear strands of all complexes involved, oneimmediately obtains the following:
Corollary 3.5.
Adopt notation and hypotheses as in Setup 3.2. Assume that thecomplexes F • , G j • ( j = 1 , . . . , t ) are minimal and that I has initial degree ℓ . For i > , let C i := Ker (cid:16) ( Q i − ) i + ℓ : ( F i ) i + ℓ → t M j =1 ( G ji ) i + ℓ (cid:17) . Then the complex · · · d k +1 −−−→ C k d k −→ · · · d −→ C d ′ −→ F ′ → is the linear strand of J . INEAR STRANDS SUPPORTED ON CELL COMPLEXES 9
Observe that if one chooses a i := ( K ′ : K i ) in Setup 3.2, the the mapping coneof 3.1 will be a (necessarily nonminimal) resolution of R/K ′ . This fact is what willallow us to construct linear strands for subideals of a given ideal K .For the remainder of this section, we will transition into dealing with cell com-plexes supporting linear complexes. We will have to build up a decent amount ofmachinery before constructing a morphism of complexes whose kernel can recoverthe linear strand of certain subideals. We begin with the definition of a polyhedralcell complex; for more precise details, see [6, Chapter 6.2] or [22, Chapter 4]. Definition 3.6. A polyhedral cell complex P is a finite collection of convex poly-topes (called cells or faces of P ) in some Euclidean space, satisfying the followingtwo properties: • if H is a polytope in P , then every face of H also lies in P , and • if H i , H j are both in P , then H i ∩ H j is a face of both H i and H j .Denote by V ( P ) the set of vertices (or 0-dimensional cells) of P . If X ⊆ V ( P ),the induced subcomplex of P on X is the subcomplex { F ∈ P | V ( F ) ⊆ X } . The f -vector of a d -dimensional polyhedral cell complex P is the vector ( f , f , . . . , f d ),where f i is the number of i -dimensional cells of P .Let P denote a cell complex supporting a linear complex. Let v be a vertex of P and let N v denote the neighbors of v ; that is, all vertices connected to v in thegraph of linear syzygies ( v is not considered a neighbor of itself). The incidencefunction associated to any given cell complex will be denoted by ǫ .Lemmas 3.7 and 3.8 will be essential for establishing the existence so-calledsupport chains centered at a vertex (see Definition 3.11). Lemma 3.7.
Let P be a cell complex supporting a linear complex. Then the cor-respondence P V ( P ) is injective.Proof. Assume P and Q are faces of P with the property that V ( P ) = V ( Q ).Observe that since the multidegree of the basis elements e P and e Q are completelydetermined by their vertex sets, P and Q must have the same dimension. Since P is a cell complex, P ∩ Q is a face of strictly smaller dimension such that e P ∩ Q hasthe same multidegree. Since P is linear, no such basis element can exist. (cid:3) Lemma 3.8.
Let P be a cell complex supporting a linear complex. Then for anyvertex p ∈ V ( P ) , there exists a unique codimension face Q ⊂ P with p / ∈ V ( Q ) .Proof. Observe first that for any codimension 1 face Q ⊂ P , | V ( Q ) | = | V ( P ) | − V ( Q ), then the induced cell complex differential would have degree at least 2.Thus, if any Q as in the statement were to exist, it must be unique by Lemma 3.8.It remains to show that such a Q exists. One may assume without loss of gen-erality that there exists some codimension 2 face R with p / ∈ V ( R ), since otherwiseevery face of P must contain p . Since P is a cell complex, R may be written as theintersection of two codimension 1 faces of P . This implies that one of these faces cannot contain p , whence the result. (cid:3) Notation 3.9.
Let R be a standard graded polynomial ring over a field k endowedwith some term order < . Let v ∈ P be a vertex of some cellular complex supportinga linear complex. For every v ′ ∈ N v , define x v ′ := m v,v ′ m v . There is an induced order on elements of N v defined by v < v ⇐⇒ x v < x v , v , v ∈ N v . Lemma 3.10.
Let P be a cell complex supporting a linear complex and let v be anyvertex of P . Let P be an n -dimensional face of P with V ( P ) ∩ N v = { v < · · · < v n } .Then there exists a unique chain P = P n ⊃ P n − ⊃ · · · ⊃ P ⊃ P = v such that:(1) P i is a codimension face of P i +1 for each i = 0 , . . . , n − , and(2) P i ∩ N v = { v n − i < · · · < v n } .Proof. This is just an iteration of Lemmas 3.7 and 3.8. (cid:3)
Definition 3.11.
Let P be an n -dimensional face of P with V ( P ) ∩ N v = { v < · · · < v n } . Then the support chain centered at v associated to P is the unique chainof subfaces of Lemma 3.10.Given a face P as above, define c ( P ) := ǫ ( P n , P n − ) · ǫ ( P n − , P n − ) · · · ǫ ( P , P ) , where P = P n ⊃ P n − ⊃ · · · ⊃ P ⊃ P = v is the associated support chain centered at v . Lemma 3.12.
Let P be an n -dimensional face of P with V ( P ) ∩ N v = { v < · · · Let P = P n ⊃ P n − ⊃ · · · ⊃ P ⊃ P = v and Q = Q n − ⊃ Q n − ⊃ · · · ⊃ Q ⊃ Q = v be the support chains centered at v associated to P and Q , respectively. Observethat P n − i +1 = Q n − i +1 ; moreover, since P n − i +1 is a codimension 2 face of Q n − i +3 , Q n − i +2 and P n − i +2 must be the unique codimension 1 faces of Q n − i +3 containing P n − i +1 . By definition of the incidence function, ǫ ( Q n − i +2 , Q n − i +1 ) ǫ ( P n − i +1 , Q n − i +2 ) = − ǫ ( P n − i +2 , Q n − i +1 ) ǫ ( P n − i +1 , P n − i +2 ) . Repeating this argument i more times, the result follows. (cid:3) With the existence of support chains established, there are a few more stepsrequired for defining the relevant morphism of complexes. The most important ofthese steps is Lemma 3.15, which will imply that the morphism of Lemma 3.16 iswell-defined. INEAR STRANDS SUPPORTED ON CELL COMPLEXES 11 Observation 3.13. Let P be a cell complex supporting the linear strand of someideal ( I, g ), where g ∈ R . Then the linear strand of ( I : g ) is obtained as the linearstrand of the ideal ( x v ′ | v ′ ∈ N v ). Setup 3.14. Let R be a standard graded polynomial ring over a field endowed withsome term order < . Let P be a cellular complex supporting a linear complex F • and suppose that v ∈ P is some vertex. Let K • denote the Koszul complex resolving R/ ( x v ′ | v ′ ∈ N v ) . Lemma 3.15. Let P be a cellular complex supporting a linear complex. Let P bean n -dimensional face of P supported on v . Then, | V ( P ) ∩ N v | = n . In the following proof, we will say that P is maximally supported on v if | V ( P ) ∩ N v | = n , as in the statement of the Lemma. Proof. The proof follows by induction on the dimension of any given face of P . If P is a 1-dimensional face containing v , then by definition P contains a neighbor of v .Let P be an i -dimensional face and assume by induction that all faces of dimension i − v are maximally supported on v . By Lemma 3.8, there exists acodimension 1 face containing v that is also not maximally supported on v , whichis a contradiction. (cid:3) Lemma 3.16 defines the morphism of complexes whose kernel can be used tocompute the linear strand of certain subideals, as guaranteed by Corollary 3.5. Lemma 3.16. Adopt notation and hypotheses as in Setup 3.14. Define Q n − : F n → K n − e P ( c ( P ) e V ( P ) ∩ N v if v ∈ V ( P )0 otherwise . Then for all i > , the following diagram commutes: (3.1) F i +1 Q i (cid:15) (cid:15) / / F iQ i − (cid:15) (cid:15) K i / / K i − Proof. Observe first that each Q i is well defined by Lemma 3.15. Going clockwisearound the diagram 3.1, one obtains: e P X Q ⊂ P ǫ ( Q, P ) m P /m Q e Q X | V ( Q ) ∩ N v | = n − ǫ ( Q, P ) c ( Q ) m P /m Q e V ( Q ) ∩ N v . Going counterclockwise around 3.1, e P c ( P ) e V ( P ) ∩ N v n X i =1 ( − i +1 c ( P ) x v ji e V ( P ) ∩ N v \ v ji . The result then follows from Lemma 3.12. (cid:3) It turns out that the kernels of the maps Q i as above may not yield a complexsupported on a cellular complex in general. The following Definition will end upbeing a property sufficient for guaranteeing that the resulting kernel can be easilydescribed as an induced subcomplex. Definition 3.17. A cellular complex P is linearly connected with respect to v if forall v i , v j ∈ N v , ( v i , v j ) is an edge of P whenever m v i and m v j have a linear syzygy. Lemma 3.18. Let P be a cellular complex supporting the linear strand of somemodule. If P is linearly connected with respect to v , then x v i = x v j for all distinct v i , v j ∈ N v .Proof. Let v , v ∈ N v be such that x v = x v and suppose for sake of contradictionthat { v , v } is a face of P . One may assume that the differentials in the presentingmatrix have the following form: e v,v m v,v m v e v − m v,v m v e v ,e v,v m v,v m v e v − m v,v m v e v ,e v ,v m v ,v m v e v − m v ,v m v e v . Since x v = x v , one has m v,v = m v,v = m v ,v . It is then clear that e v,v − e v,v + e v ,v is a cycle that cannot possibly be a boundary. By Theorem 3.1, nosuch element can exist. (cid:3) Theorem 3.19. Let P be a cellular complex supporting the linear strand of somemonomial ideal I . Assume that P is linearly connected with respect to v . Then thecellular complex induced by removing all faces supported on v is the linear strandof the monomial ideal with minimal generating set G ( I ) \ m v .Proof. Let J denote the ideal with minimal generating set G ( I ) \ m v . By Corollary3.5, the linear strand of J is obtained by restricting to the kernel of each Q i mapas in Lemma 3.16. Moreover, for every V ⊂ N v , there is at most one face withsupport containing V ; if there were more than one such face, then the intersectionof these faces would also be supported on V , a contradiction to Lemma 3.15. Thus,by Lemma 3.18, the only possible basis elements contained in the kernel are thosebasis elements corresponding to faces that do not contain v . (cid:3) Theorem 3.20. Let P be a cellular complex supporting the linear strand of somemonomial ideal I , and assume that every -dimensional face of P has a uniquemultidegree. Then the linear strand of any monomial ideal J with G ( J ) ⊆ G ( I ) issupported on the cellular complex induced by restricting to the vertices correspondingto generators of J .In particular, if J has linear minimal free resolution, then J has a cellular min-imal free resolution.Proof. The assumption that every 1-dimensional face has a unique multidegreeimplies that for any vertex v ∈ P , every pair of distinct vertices v , v ∈ N v hasthe property that x v = x v . The proof then proceeds identically as in the proof ofTheorem 3.19. (cid:3) INEAR STRANDS SUPPORTED ON CELL COMPLEXES 13 Meet Semilattices and Cellularity In this section, we consider the question of when a particular complex may besupported on a cellular complex. We first recall the construction of so-called frames of monomial ideals, as described by Peeva and Velasco in [27]. We formulate a moregeneral result on cellularity, Lemma 4.7, which says that a complex is supportedon a cellular complex so long as the associated poset forms a meet semilattice andthe frame has coefficients 0 or ± 1. It turns out that the sparse Eagon-Northcottcomplex E < • will satisfy these hypotheses, whence Corollary 4.9 immediately estab-lishes that E < • is cellular for any term order < . Combining this with the resultsof Section 3, we obtain the previously mentioned fact that every rainbow DFI haslinear strand supported on a cellular complex (see Corollary 4.11). Definition 4.1. An r -frame U • is a finite complex of finite k -vector spaces withdifferential ∂ and a fixed basis satisfying(1) U = k ,(2) U = k r ,(3) ∂ ( w j ) = 1 for each basis vector w j ∈ U .Recall that for any choice of r monomials m , . . . , m r , an r -frame U • may be ho-mogenized to a (not necessarily acyclic) complex F • with H ( F • ) = R/ ( m , . . . , m r ).This is accomplished inductively by assigning a multidegree to a basis element e P by taking the lcm of the multidegrees of the basis elements appearing in the sup-port of ∂ ( e P ). Once the multigrading has been assigned, the differential may becomputed as follows: d ( e P ) = X e Q ∈ Supp( ∂ ( e P )) a P,Q mdeg( e P )mdeg( e Q ) e Q , where the coefficients a P,Q are determined by the associated r -frame. Completedetails are found in Construction 55 . Theorem 4.2 ([27, Theorem 4.14]) . Let R be a polynomial ring over a field and I ⊆ R a monomial ideal. Then the I -homogenization of any frame of the multigradedminimal free resolution F • of R/I is F • . Corollary 4.3. Let R be a polynomial ring over a field and I ⊆ R a monomialideal. Then the I -homogenization of any frame of the multigraded linear strand F lin • of R/I is F lin • .Proof. Viewing F lin • as a subcomplex of the minimal free resolution, this followsimmediately from Theorem 4.2. (cid:3) The following Definition is a method of associating any resolution (with a fixedbasis in each homological degree) to a poset whose elements correspond to the basiselements. There is a natural way to try to associate any such poset to a face latticeof some geometric object, but in general this will not yield a cell complex. Definition 4.4. Let F • be a complex with a fixed basis B i in each homologicaldegree i . The associated poset po( F • ) is defined as follows: given basis elements e ∈ F i , f ∈ F i − , f e ⇐⇒ f ∈ Supp B i − ( d F ( f )) . Extending transitively yields a partial order on basis elements appearing in allhomological degrees. Example 4.5. Let ψ : F → R be an R -module homomorphism, where F is a free R -module. The induced Koszul complex is the complex K • with K i = V i F anddifferential i ^ F comultiplication −−−−−−−−−−→ F ⊗ i − ^ F ψ ⊗ −−−→ R ⊗ i − ^ F = i − ^ F. Let F have basis f , . . . , f n . If one chooses the standard bases for each exteriorpower K i , the poset po( K • ) is precisely the power set 2 [ n ] partially ordered byinclusion. Definition 4.6. Let S be a poset with respect to some binary relation . Then S is a meet-semilattice if for all x, y ∈ S , there exists a greatest lower bound w ∈ S for x and y ; that is, for any other w ′ ∈ S , w ′ w whenever w ′ x, y .As previously mentioned, choosing a face lattice defined by po( F • ) does notalways yield a cell complex. However, the following Lemma shows that if the posetis a meet semilattice and the frame has coefficients 0 or ± 1, then it does. Lemma 4.7. Let R be a polynomial ring over a field. Let F • be the minimal freeresolution (or linear strand) of some monomial ideal J . Assume:(1) all coefficients of the associated frame are contained in {− , , } , and(2) po( F • ) is meet semilattice.Then F • is supported on a cellular complex.Proof. By Theorem 4.2 (or Corollary 4.3), F • is obtained by homogenizing itsassociated frame. Recall that in order to define a cell complex, it suffices to describeits face lattice. Thus, the associated cellular complex is defined as follows: the 0-cells are labelled by the minimal monomial generators of J . To specify an i -face of P , it suffices to specify all of its codimension 1 faces. Then, P has codimension 1 faces Q , . . . , Q ℓ ⇐⇒ Supp( d F ( e P )) = { e Q , . . . , e Q ℓ } . By definition, any face of some face P in P is also a face of P . To prove cellularity,it remains to prove that the intersection of any two faces is still a face P . However,the claim that P ∩ Q = ∅ is equivalent to the claim that the chains below the basiselements e P and e Q intersect nontrivially. Since po( F • ) is assumed to be a meetsemilattice, there exists a unique face of P ∩ Q containing all other faces of P ∩ Q .Finally, since the differentials of the homogenized frame agree with the inducedcell differentials and the coefficients are 0 or ± F • is supported on P . (cid:3) Lemma 4.8. Adopt notation and hypotheses as in Theorem 2.16. Then po( E < • ) isa meet semilattice.Proof. The sparse Eagon-Northcott complex has the property that if e P and e Q aretwo basis elements with mdeg( P ) | mdeg( Q ), then e P e Q . Assume that the chainbelow any two basis elements e P and e Q intersects nontrivially. By Corollary 2.17,there exists a basis element e H of multidegree gcd(mdeg( e P ) , mdeg( e Q )). Moreover, INEAR STRANDS SUPPORTED ON CELL COMPLEXES 15 x x x x x x x x x x x x x x x x x x x x x x x x Figure 1. Two examples of cellular resolutions arising from initialideals of maximal minors of a 2 × e H ′ e P , e Q must have multidegree dividing e H , whence e H ′ e H . (cid:3) Combining the previous two results immediately yields the following: Corollary 4.9. Let R = k [ x ij | i n, j m ] be a polynomial ring overan arbitrary field k and M be an n × m matrix of variables in R where n m . Let < denote any term order. Then R/ in < I n ( M ) has minimal free resolution supportedon a cellular complex.Proof. Observe that the frame associated to the sparse Eagon-Northcott complexhas coefficients 0 or ± 1. Combining Lemmas 4.7 and 4.8, the result follows imme-diately. (cid:3) Remark . For the remainder of the paper, no distinction will be made betweenthe poset po( E < • ) and the cell complex that it induces.Using the results of Section 3, Corollary 4.9 immediately implies the muchstronger result that every rainbow DFI has linear strand supported on a cellularcomplex. Corollary 4.11. Adopt notation and hypotheses as in Corollary 4.9. Let ∆ be an ( n − -dimensional pure simplicial complex. Then the linear strand of rain < ( J ∆ ) is supported on a cellular complex.In particular, if rain < ( J ∆ ) has linear resolution, then rain < ( J ∆ ) has a cellularresolution.Proof. This is immediate by Corollary 4.9 combined with Theorem 3.20 and Corol-lary 2.11. (cid:3) Preservation of Regularity for Subideals and Linearity Corollary 4.11 establishes that any rainbow DFI with linear resolution musthave a cellular resolution. This immediately begs the question of which rainbowDFIs have linear resolution. In this section, we consider conditions that guaranteelinearity. We first study rainbow DFIs for which the sequence of variable differences { x − x , . . . , x − x n } ∪ · · · ∪ { x m − x m , . . . , x m − x nm } forms a regular sequence on the associated quotient ring. This can be used to showthat certain ideals for which the squarefree complement (see Definition 5.1) has nolinear relations must have cellular resolution.Next, using a result of Nagel and Reiner [23], we see that rainbow DFIs withthe property that the Alexander dual ∆ ∨ admits an ordering for which it is a freesequence (see Definition 5.8) must have linear resolution. Moreover, if the elementsof the Alexander dual have nonmaximal overlap, this condition is an equivalence;the Betti table for these ideals can also be computed explicitly.We begin with some definitions and notation related to monomial ideals andtheir complementary ideals. Definition 5.1. Let R = k [ x , . . . , x n ] be a standard graded polynomial ring overa field k . Let K denote an equigenerated monomial ideal with generators in degree d . Define G ( K ) := unique minimal generating set of K consisting of monic monomials.Given a monomial ideal K , define the (squarefree) complementary ideal K to bethe ideal with minimal generating set: G ( K ) = ( { degree d squarefree monomials }\ G ( K ) if K squarefree , { degree d monomials }\ G ( K ) otherwise. Lemma 5.2 ([30]) . Let K ′ be a squarefree equigenerated momomial ideal of degree n with K ′ = ( x I , . . . , x I r ) such that | I i ∩ I j | < n − . Then R/K ′ has a linearminimal free resolution. It is worth noting that [30] not only shows that ideals as in the above statementhave linear resolution, but also produces the minimal free resolution explicitly. Aspreviously mentioned, we wish to study when the sequence of variable differences { x − x , . . . , x − x n } ∪ · · · ∪ { x m − x m , . . . , x m − x nm } forms a regular sequence on the quotient defined by a rain DFI. This problem canbe easily abstracted to the following question, which we call the preservation ofregularity :If J ⊆ I and a is regular on R/I , then when is a regular on R/J ?The initial answer to this question, although simple, turns out to be surprisinglyuseful. Lemma 5.3. Let R be a commutative ring and let a be regular on R/ ( J, g ) . Then a is regular on R/J if and only if a is regular on ( J : g ) .Proof. Without loss of generality assume that a = a has length 1. The proof followsimmediately from the string of inclusions:Ass( J : g ) ⊆ Ass( J ) ⊆ Ass( J : g ) ∪ Ass( J, g ) . (cid:3) If we assume furthermore that the ideal I is a monomial ideal with a convenientsplitting, then Lemma 5.3 can be upgraded considerably. Corollary 5.4. Let R be a polynomial ring over a field. Let I be a linearly pre-sented, equigenerated monomial ideal of degree d . Assume that I = J + K , where J and K are monomial ideals with β ,d +1 ( K ) = 0 . Let a be regular on R/I . INEAR STRANDS SUPPORTED ON CELL COMPLEXES 17 Then a is regular on R/J if and only if a is regular on R/ ( J : g ) for all g ∈ G ( K ) .Proof. Write G ( K ) = { g , . . . , g t } . Since I is linearly presented and K has no linearrelations on any of its generators (this is the assumption β ,d +1 ( K ) = 0), one finds( J + ( g , . . . , g i ) : g i +1 ) = ( J : g i +1 ) for all i = 0 , . . . , t − . The result then follows by iterating Lemma 5.3. (cid:3) The following Proposition is a standard exercise which we include here for con-venience. Proposition 5.5. Let M be a finitely generated R -module. Assume a is an M -regular sequence and let F • be any free resolution of M over R . Then F • ⊗ R/ a isa free resolution of M/ a M over R/ a .Proof. By induction it is of no loss of generality to assume a = a has length 1.There is a short exact sequence of complexes:0 → F • a −→ F • → F • /aF • → . The long exact sequence of homology implies H i ( F • /aF • ) = 0 for i > 2, and H ( F • /aF • ) = 0 by the assumption that a is regular on M . (cid:3) Setup 5.6. Let R = k [ x ij | i n, j m ] be a polynomial ring overan arbitrary field k and M be an n × m matrix of variables in R where n m .Let ∆ be an ( n − -dimensional pure simplicial complex and assume that < is anarbitrary term order on R . Assume that ∆ ∨ satisfies the following:(*) for all σ, τ ∈ ∆ ∨ , | σ ∩ τ | < n − . Corollary 5.7. Adopt notation and hypotheses as in Setup 5.6. Then the sequenceof variable differences { x − x , . . . , x − x n } ∪ · · · ∪ { x m − x m , . . . , x m − x nm } forms a regular sequence on R/ rain < ( J ∆ ) if and only if grade(rain < ( J ∆ ) : g ) = m − n for all g ∈ rain < ( J ∆ ∨ ) .Proof. Observe that grade(rain < ( J ∆ ) : g ) > m − n for all g ∈ rain < ( J ∆ ∨ ) byTheorem 2.8. The grade after specializing must be m − n , so the variable differencesare regular if and only if grade(rain < ( J ∆ ) : g ) = m − n for all g ∈ rain < ( J ∆ ∨ ). Theresult then follows by Corollary 5.4. (cid:3) The next definition introduces the notion of a free sequence of vertices in a cellcomplex. This definition will end up being the “correct” condition for characterizingrainbow DFIs with linear resolution. Definition 5.8. Let P be a cellular complex. A vertex is free if it is contained ina unique facet of P . A free sequence is a sequence of vertices { v , . . . , v k } ⊆ P suchthat v i is a free vertex of P\{ v , . . . , v i − } for each 1 = 1 , . . . , k . Lemma 5.9 ([23]) . Let C be a polytopal complex and v a vertex in C that lies ina unique facet P , with dim P > . Then the subcomplex induced by deleting thevertex v is homotopy equivalent to C . Given an ( n − E < • ). We will tacitly employ thisassociation for the remainder of the paper. Corollary 5.10. Adopt notation and hypotheses as in Corollary 4.9. Let ∆ be an ( n − -dimensional pure simplicial complex. If the facets of ∆ ∨ admit an orderingfor which they are a free sequence in po( E < • ) , then rain < ( J ∆ ) has linear minimalfree resolution supported on a cellular complex.Proof. The fact that the minimal free resolution is cellular is Corollary 4.11. Thelinearity follows from Lemma 5.9; since in < I n ( M ) has linear resolution, remov-ing a free vertex at each step leaves the homology unchanged. This implies thatrain < ( J ∆ ) has linear resolution. (cid:3) Theorem 5.11. Adopt notation and hypotheses as in Setup 5.6. Then rain < ( J ∆ ) has a linear minimal free resolution if and only if grade(rain < ( J ∆ ) : g ) = m − n forall g ∈ G (rain < ( J ∆ ∨ )) (that is, each element of ∆ ∨ is a free vertex of po( E < • ) ).Proof. = ⇒ : Argue by contraposition. Recall that a free resolution of R/ rain < ( J ∆ )may be obtained by an iterated trimming complex taking the form of a mappingcone of a morphism of complexes: · · · d k +1 / / F kQ k − (cid:15) (cid:15) d k / / · · · d / / F d ′ / / Q (cid:15) (cid:15) F ′ d (cid:15) (cid:15) · · · L m ik / / L ti =1 K ik − L m ik − / / · · · L m i / / L ti =1 K i − P ti =1 m i ( − ) · d ( e i ) / / R. In the above, the complex on the top row is built from the minimal free resolutionof R/ in < I n ( M ), which has length m − n + 1. The bottom row is a direct sum of theminimal free resolutions of (rain < ( J ∆ ) : g ) for each g ∈ rain < ( J ∆ ∨ ); for completedetails on this construction see [31]. The minimal free resolution will be linear ifand only if the comparison maps Q i are surjective in each homological degree. Ifgrade(rain < ( J ∆ ) : g ) > m − n for some g ∈ rain < ( J ∆ ∨ ), however, then Q m − n +1 will not be surjective because F m − n +2 = 0. ⇐ = : The assumption grade(rain < ( J ∆ ) : g ) = m − n for all g ∈ rain < ( J ∆ ∨ )implies that each g ∈ ∆ ∨ is a free vertex in po( E < • ). The conclusion follows fromCorollary 5.10. (cid:3) Corollary 5.12. Adopt notation and hypotheses as in Setup 5.6. Then R/ rain < ( J ∆ ) has Betti table · · · ℓ · · · m − n + 10 1 · · · · · · · · · ... · · · · · · · · · · n − · (cid:0) mn (cid:1) − r · · · (cid:0) n + ℓ − ℓ − (cid:1)(cid:0) mn + ℓ − (cid:1) − r (cid:0) m − nℓ − (cid:1) · · · (cid:0) m − m − n (cid:1) − r. Proof. This follows from Corollary 5.7 and Proposition 5.5 combined with the Bettitables given in [30, Corollary 5.12]. (cid:3) Applications to Polarizations In this section, we consider applications of the machinery and results producedin the previous sections to polarizations of Artinian monomial ideals. We recallthe result of Almousa, Fløystad, and Lohne that shows Alexander duality pro-vides an equivalence between rainbow monomial ideals and polarizations of Ar-tinian monomial ideals. This immediately implies Proposition 6.4 and Corollary INEAR STRANDS SUPPORTED ON CELL COMPLEXES 19 J ∆ with ∆ ∨ consist-ing only of free vertices is a polarization of a power of the maximal ideal if and onlyif rain < J ∆ = in < I n ( M ). We conclude with a variety of questions related to theresults of this paper.We begin with the definition of an Alexander dual of a monomial ideal. Definition 6.1. Let I be a monomial ideal. The Alexander dual I ∨ is the idealwith generators consisting of all monomials with nontrivial common divisor withevery generator of I . Example 6.2. Let I = ( x x , x x , x x , x x , x x , x x ). Then one computes: I ∨ = ( x x x , x x x , x x x , x x x ) . Next, we give the definition of a polarization. As previously mentioned, theintuition behind polarization is to replace an ideal with a squarefree ideal thatis homologically indistinguishable. The existence of polarizations translates theproblem of minimal free resolutions of arbitrary monomial ideals into the problemof minimal free resolutions of squarefree monomial ideals. Definition 6.3. Let I be an Artinian monomial ideal in the polynomial ring S = k [ x , . . . , x n ], such that for every index i , x m i i is a minimal generator. Let ˇ X i = { x i , x i , . . . , x im i } be a set of variables, and let ˜ S = k [ ˇ X , . . . ˇ X n ] be the polynomialring in the union of these variables. An ideal ˜ I ⊂ ˜ S is a polarization of I if σ = { x − x , x − x , . . . , x − x m } ∪ { x − x , . . . , x − x m } ∪ . . . ∪{ x n − x n , . . . , x n − x n,m n } is a regular sequence in ˜ S/ ˜ I and ˜ I ⊗ ˜ S/σ ∼ = I . Proposition 6.4 ([1]) . Let J be an ideal generated by rainbow monomials withlinear resolution (with all variables in the ambient ring occurring in some generatorof J ). Then J ∨ is a polarization of an Artinian monomial ideal. Corollary 6.5. Let ∆ be an ( n − -dimensional simplicial complex such that ∆ ∨ admits an ordering for which it is a free sequence of po( E < • ) . Then (rain < ( J ∆ )) ∨ is the polarization of an Artinian monomial ideal.Proof. Combine Proposition 6.4 with Corollary 5.10. (cid:3) Corollary 6.6. Adopt notation and hypotheses as in Setup 5.6. Then:(1) (rain < ( J ∆ )) ∨ is the polarization of an Artinian monomial ideal if and onlyif grade (cid:0) rain < ( J ∆ ) : g (cid:1) = m − n for all g ∈ G (rain < ( J ∆ ∨ )) , and(2) in the situation of (1) , (rain < ( J ∆ )) ∨ is a polarization of a power of thegraded maximal ideal if and only if | ∆ ∨ | = 0 (that is, rain < ( J ∆ ) = in < I n ( M ) ).Proof. The statement of (1) is immediate from Theorem 5.11. To prove (2), itsuffices to show that if | ∆ ∨ | > 0, then the monomial ideal K with G ( K ) = { x a | a ∈ ∆ ∨ } is not Cohen-Macaulay. That is, K ∨ does not have linear resolution. Let x a = x a · · · x a n ∈ G ( K ) and write [ m ] \ a = { b < · · · < b m − n } . It is clear that x b · · · x b m − n ∈ K ∨ . By Corollary 5.12, R/K ∨ has regularity m − n + 1, whence R/K ∨ does not have linear resolution. (cid:3) The following illustrates many of the results in this paper in a simple example. Example 6.7. Let < denote the standard diagonal term order, with n = 3 and m = 5. Assume that ∆ ∨ = { (1 , , , (3 , , } . The associated cellular resolution isprecisely the complex of boxes introduced by Nagel and Reiner. One can verify:rain < ( J ∆ ) : x x x = ( x , x ) , rain < ( J ∆ ) : x x x = ( x , x );this shows that (1 , , 3) and (3 , , 5) are free vertices in the complex of boxes. Asguaranteed by Corollary 5.12, R/ rain < ( J ∆ ) has Betti table0 1 2 3total: 1 8 11 40: 1 . . .1: . . . .2: . 8 11 4and one can verify using Macaulay2 [17] that (rain < ( J ∆ )) ∨ is a polarization of( x , x x , x , x x x , x x , x ) . Question . Is Corollary 5.10 an equivalence? That is, if a rainbow DFI rain < ( J ∆ )has linear resolution, then do the elements of ∆ ∨ admit an ordering for which theyare a free sequence in po( E < • )?Question 6.8, if answered in the affirmative, yields an interesting connectionbetween polarizations and free sequences on the cell complex po( E < • ). Moreover,using the combinatorics of the initial ideals provided by Sturmfels and Zelevinsky[29], it may be possible to explicitly enumerate all free sequences on po( E < • ), thusconstructing all possible polarizations with Alexander dual contained in in < I n ( M )for a given term order. Question . Let ∆ be an ( n − | σ ∩ τ | < n − σ, τ ∈ ∆ ∨ . Does there exist a term order < on R suchthat every facet of ∆ ∨ corresponds to a free vertex of po( E < • )?A positive answer to Question 6.9 would immediately yield that a large class ofequigenerated monomial ideals (considered in [30]) has cellular resolution, since onecan employ Corollary 5.7 to specialize. More precisely, to the authors knowledge,it is not known if every equigenerated squarefree monomial ideal whose squarefreecomplement has no linear relations must have cellular resolution. Question . Let < be any term order. Then does the sparse Eagon-Northcottcomplex admit the structure of an associative DG-algebra? If so, then how doesthis product compare to the product on the squarefree Eliahou-Kervaire resolutionconstructed by Peeva [25] upon specialization?The first part of Question 6.10 is likely not as difficult to answer as the secondpart. It would be interesting to see how multigraded DG-products on E < • specializeto give DG-products on the resolution of all squarefree monomial ideals of a givendegree. It would be even more interesting if these products were all distinct. If so,this would also yield a fascinating correspondence between choices of initial idealsof in < ( M ) and choices of algebra structures on the minimal free resolution of allsquarefree monomials of degree n in m variables. Up to taking linear combinations,this may provide a method of parametrizing all such multigraded products. INEAR STRANDS SUPPORTED ON CELL COMPLEXES 21 Acknowledgements Thanks to Ayah Almousa for very helpful discussions related to this work.Thanks to Paolo Mantero for helpful comments and suggestions on an earlier draftof this paper. References 1. Ayah Almousa, Gunnar Fløystad, and Henning Lohne, Polarizations of powers of gradedmaximal ideals , arXiv preprint arXiv:1912.03898 (2019).2. Ekkehard Batzies and Volkmar Welker, Discrete morse theory for cellular resolutions , Journalfur die Reine und Angewandte Mathematik (2002), 147–168.3. Dave Bayer, Irena Peeva, and Bernd Sturmfels, Monomial resolutions , Mathematical ResearchLetters (1998), no. 1, 31–46.4. Dave Bayer and Bernd Sturmfels, Cellular resolutions of monomial modules , Journal f¨ur diereine und angewandte Mathematik (1998), no. 502, 123–140.5. Adam Boocher, Free resolutions and sparse determinantal ideals , Math. Res. Lett (2012),no. 04, 805–821.6. Winfried Bruns and J¨urgen Herzog, Cohen-macaulay rings , no. 39, Cambridge universitypress, 1998.7. Timothy Clark and Alexandre Tchernev, Minimal free resolutions of monomial ideals and oftoric rings are supported on posets , Transactions of the American Mathematical Society (2019), no. 6, 3995–4027.8. Alastair Craw and Alexander Quintero V´elez, Cellular resolutions of noncommutative toricalgebras from superpotentials , Advances in Mathematics (2012), no. 3, 1516–1554.9. Mike Develin and Josephine Yu, Tropical polytopes and cellular resolutions , ExperimentalMathematics (2007), no. 3, 277–291.10. Persi Diaconis, David Eisenbud, and Bernd Sturmfels, Lattice walks and primary decomposi-tion , Mathematical Essays in Honor of Gian-Carlo Rota, Springer, 1998, pp. 173–193.11. Anton Dochtermann and Alexander Engstr¨om, Cellular resolutions of cointerval ideals , Math-ematische Zeitschrift (2012), no. 1-2, 145–163.12. Anton Dochtermann, Michael Joswig, and Raman Sanyal, Tropical types and associated cel-lular resolutions , Journal of Algebra (2012), no. 1, 304–324.13. Anton Dochtermann and Fatemeh Mohammadi, Cellular resolutions from mapping cones ,Journal of Combinatorial Theory, Series A (2014), 180–206.14. Giuseppe Favacchio, Elena Guardo, and Juan Migliore, On the arithmetically cohen-macaulayproperty for sets of points in multiprojective spaces , Proceedings of the American Mathemat-ical Society (2018), no. 7, 2811–2825.15. Giuseppe Favacchio and Juan Migliore, Multiprojective spaces and the arithmetically cohen–macaulay property , Mathematical Proceedings of the Cambridge Philosophical Society, vol.166, Cambridge University Press, 2019, pp. 583–597.16. Gunnar Fløystad, Cellular resolutions of cohen-macaulay monomial ideals , Journal of Com-mutative Algebra (2009), no. 1, 57–89.17. Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research inalgebraic geometry , Available at .18. J¨urgen Herzog, Takayuki Hibi, Freyja Hreinsd´ottir, Thomas Kahle, and Johannes Rauh, Bino-mial edge ideals and conditional independence statements , Advances in Applied Mathematics (2010), no. 3, 317–333.19. J¨urgen Herzog, Dariush Kiani, and Sara Saeedi Madani, The linear strand of determinantalfacet ideals , The Michigan Mathematical Journal (2017), no. 1, 107–123.20. Sara Saeedi Madani, Binomial edge ideals: A survey , The 24th National School on Algebra,Springer, 2016, pp. 83–94.21. Jeffrey Mermin, The eliahou-kervaire resolution is cellular , Journal of Commutative Algebra (2010), no. 1, 55–78.22. Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra , vol. 227, SpringerScience & Business Media, 2004.23. Uwe Nagel and Victor Reiner, Betti numbers of monomial ideals and shifted skew shapes , theelectronic journal of combinatorics (2009), R3–R3. 24. Masahiro Ohtani, Graphs and ideals generated by some 2-minors , Communications in Algebra (2011), no. 3, 905–917.25. Irena Peeva, , Journal of Algebra (1996), no. 3, 945–984.26. , Graded syzygies , vol. 14, Springer Science & Business Media, 2010.27. Irena Peeva and Mauricio Velasco, Frames and degenerations of monomial resolutions , Trans-actions of the American Mathematical Society (2011), 2029–2046.28. Victor Reiner and Volkmar Welker, Linear syzygies of stanley-reisner ideals , MathematicaScandinavica (2001), 117–132.29. Bernd Sturmfels and Andrei Zelevinsky, Maximal minors and their leading terms , Advancesin Mathematics (1993), no. 1, 65–112.30. Keller VandeBogert, Minimal free resolutions of certain equigenerated monomial ideals , arXivpreprint arXiv:2007.02373 (2020).31. , Trimming complexes and applications to resolutions of determinantal facet ideals ,Communications in Algebra (2020), 1–20.32. Mauricio Velasco, Minimal free resolutions that are not supported by a cw-complex , Journalof Algebra319