Regularity of Edge Ideals Via Suspension

We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a key role in proof of the base case s =2.


Introduction
Let M be a finitely generated graded module over a polynomial ring R = K[x 1 , . . ., x n ], where K is a field.The Castelnuovo-Mumford regularity (or simply, regularity) reg(M) of M is defined as reg(M) = max{j − i | Tor R i (M, K) j = 0}.Regularity is an important invariant in commutative algebra and algebraic geometry that measures the complexity of ideals, modules, and sheaves.A question that has been studied by many is how the regularity behaves with respect to taking powers of homogenous ideals.It is known that in the long-run reg(I k ) is linear in k, that is, there exist integers a(I), b(I), c(I) such that reg(I k ) = a(I)k + b(I) for all k ≥ c(I) (see [7,18]).For various classes of ideals people have studied these integers and also have looked for various upper and lower bounds for reg(I k ).For monomial ideals these invariants as well as bounds reflect the underlying combinatorics (see e.g.[3,13,14,20,21,22] for various works under this theme).For monomial ideals I generated in same degree d, Kodiyalam [18] showed that a(I) = d.
One important class of monomial ideal is the class of edge ideals I(G) of finite simple graphs, namely the ideals generated by squarefree monomials of degree two.For edge ideals, c(I(G)) ≤ 2 for various cases: for example when the underlying graph is either cochordal or gap and cricket free or bipartite with reg(I(G)) ≤ 3 (see [1,2,3,12]).As of and it is conjectured (see e.g.[3,17]) that this inequality holds for any graph.For various classes of graphs (e.g cochordal) we have b(I(G)) = reg(I(G)) − 2 so this upper bound is tight if holds.Our Theorem 1.1(ii) below verifies inequality (1.1) for all G bipartite.Clearly this bound this sharp, for example if we take any complete bipartite graph with nonempty edge set then reg(I(G) s ) = 2s for all s by Theorem 2.6 below.
Our main theorem is the following: Theorem 1.1.(i) Let G be a finite simple graph.Then: (ii) Further, if G is also bipartite, then for all s ≥ 2 we have: Part (i) is proved topologically, via Hochster's formula and various uses of Mayer-Vietoris long exact sequence.Part (ii) for s > 2 is proved algebraically, via various uses of short exact sequences for related ideals.Our part two improves the main result of [16], which proves that if G is bipartite, then for all s ≥ 2 the reg(I(G) s ) ≤ 2s+Cochord(G)− 1, where Cochord(G) is the cochordal number of G (see [16] for definition).Their bound is known to be not sharp, whereas our bound is.
Outline: preliminaries are given in Section 2, Theorem 1.1 is proved in Section 3, and concluding remarks are given in Section 4.

Preliminaries
In this section, we set up the basic definitions and notation needed for the main results.Let M be a finitely generated graded R = K[x 1 , . . ., x n ]-module.Write the graded minimal free resolution of M in of the form: where p ≤ n, R(−j) indicates the ring R with the shifted grading such that, for all a ∈ Z, R(−j) a = R a−j .The non-negative integers β (i,j) (M) are called i th -graded Betti number of M in degree j.
The Castelnuovo-Mumford regularity (or regularity) of M is defined to be Let I be a nonzero proper homogeneous ideal of R. Then it follows from the definition that reg(R/I) = reg(I) − 1.
Let I be any ideal of R and a ∈ R any element, the the colon ideal (I : a) is defined by (I : Polarization is a process that creates a squarefree monomial out of a monomial, possibly in a larger ring.If f = x e 1 1 . . .x en n is a monomial in K[x 1 , . . ., x n ] then polarization of f is defined as f = x 11 . . .x 1e 1 x 21 . . .x 2e 2 . . .x n1 . . .x nen in the ring K[x 11 , . . .x 1e 1 , x 21 . . .x 2e 2 , . . ., x n1 . . ., x nen ].For convenience we identify the variable x i1 with x i , so the new polynomial ring extends the old one.For a monomial ideal I with minimal monomial generators {m 1 , . . ., m k }, we define the polarization of I as Ĩ := ( m1 , . . ., mk ) in a suitable ring, see e.g [14] or [19,Sec.1.6].In the special case where degree of a variable u = x i is two in some generator we call the unique new variable x i1 a whisker variable and denote it by u ′ for short.In this paper we repeatedly use one of the important properties of the polarization: One of the main technique that is used in this paper is that of short exact sequences.In particular we shall use the following well known result [2, Lem.2.11]: Theorem 2.2.(i) Let I be a homogeneous ideal in a polynomial ring R and m be an element of degree d in I. Then the following is a short exact sequence: In case I is square free and x a variable, then also reg(I, x) ≤ reg I.
Let G be a finite simple graph with V (G) = {x 1 . . ..x n } the varibles of R. The the edge ideal I(G) of G is defined as the ideal in R: For example, edge ideal of a 5-cycle is (x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 5 x 1 ).
The next couple of theorems allow for induction when increasing the power of an edge ideal.For bipartite graphs we further have:  Now we recall some basic definitions about graphs and simplicial complex that will be useful.
Let G be a finite simple graph with vertex set V (G) and edge set E(G).A subgraph H ⊆ G is called induced if {u, v} is an edge of H if and only if u and v are vertices of H and {u, v} is an edge of G.
Let G be a graph.We say 2 disjoint edges {f 1 , f 2 } form an 2K 2 in G if G does not have an edge with one endpoint in f 1 and the other in f 2 .A graph without 2K 2 is called 2K 2 -free or gap-free graph.The complement of a graph G, denoted by G c , is the graph on the same vertex set in which {u, v} is an edge of G c if and only if it is not an edge of G. Then G is gap-free if and only if G c contains no induced 4-cycle.
A graph G is chordal (also called triangulated) if every induced cycle in G has length 3, and is co-chordal if the complement graph G c is chordal.The following important theorem(s) characterizes the edge ideals with regularity 2, and the regularity of their powers.A simplicial complex ∆ on a vertex set {1, . . ., n} is a collection of subsets of {1, . . ., n} such that if τ ∈ ∆, σ ⊆ τ then we have σ ∈ ∆.The induced subcomplex ∆[A] of ∆ on vertex set A ⊂ {1, . . ., n} is the collection of faces τ ′ of ∆ such that τ ′ ⊂ A. Clearly the induced subcomplex is a simplicial complex itself.We denote by Vert(∆) the set of vertices of ∆.
The link of a face d in∆ is: The (open) star of a face σ in a simplicial complex ∆ is the set of all faces that contain σ, namely St Let G be a finite simple graph and ∆ = cl(G c ), where cl(G) denotes the corresponding clique complex of a graph G.
The following formulation of regularity follows from the so called Hochster's formula (See [20]for further details): Theorem 2.7 (Hochster's formula).For any finite simple graph G whose edge set is nonempty and ∆ = cl(G c ) we have:

Main Results
Let G be a finite simple graph, ab ∈ E(G) and G ′ = G ∪ {xy : x = y, ax, by ∈ E(G)}.Let ∆ = cl(G c ) and ∆ ′ = cl(G ′c ).We first prove reg(I(G) 2 : e) ≤ reg(I(G)) for every edge e of G.This will lead us to our main result via a series of short exact sequence arguments.For that we first prove: With this notation we observe the following: We get this equality simply from the definition of G ′ , where every neighbour of a is connected to every neighbour of b.
With these we have the following: Proof of Claim: Consider the following Mayer-Vietoris long exact sequence: If the first and the third terms of this sequence are zero then so will be the middle/second term by exactness.By taking suspension we note that Hl (Σ a,b (∆ ), which vanishes by the assumption on ∆.Thus we also have Hl−1 ∆ ′ [W C ] = 0. Hence the third term is zero.Also the first term is zero, by assumption, because Let D ′ be defined by W 2 ∩ D = D ′ ∪ {d} and denote X = ∆ ′ [W 2 ].We prove the claim using the Mayer-Vietoris long exact sequence corresponding to the union X = ast d X ∪ link d X St d X: Like before we prove that the end terms are zero.As ast d X = ∆ ′ [W 2 \{d}], by induction Hl ast d X is zero.Clearly Hl St d X is zero as the space is a cone.Thus the first term is zero.For the last term we note that by definition link We are now ready to prove Theorem 1.1.
Due to the asymptotic stability we have that for an homogeneous ideal I generated in degree d we have an integer c(I) such that reg(I s+1 ) − reg(I s ) = d for all s ≥ c(I).We have proved that for all bipartite graphs G we have reg(I(G) s ) − reg(I(G)) ≤ 2s − 2. However the behaviour of the sequence {reg(I s )} can be irregular for smaller s values even for edge ideals.In fact there are examples of bipartite graphs where reg(I(G) 2 ) = reg(I(G)) + 1 (for example one can check that this is the case for the bipartite edge ideal (x 1 y 1 , x 2 y 2 , x 3 y 3 , x 4 y 4 , x 1 y 2 , x 2 y 4 , x 3 y 1 , x 4 y 3 )).
Can c(I) be bounded by some simple invariants of I, for homogenous ideals?Conca [5] showed that for any given integer d > 1 there exists an ideal J generated by d + 5 monomials of degree d + 1 in 4 variables such that reg(J k ) = k(d + 1) for every k < d and reg(J d ) ≥ d(d + 1) + d − 1.In particular, c(I) cannot be bounded above in terms of the number of variables only, not even for monomial ideals in general.Further, a result of Raicu [25] gives binomial ideals I n on n 2 variables, generated in degree 2, with c(I n ) = n − 1.Thus, the following question arise: Question 4.2.For homogeneous ideals I on n variables, generated in degree d, is c(I) bounded above by a function of d and n?
It has been conjectured by Banerjee and Mukundan [4] that for all bipartite graphs G, we have c(I(G)) ≤ 2. It is known for cochordal, gap free plus cricket/diamond/4cycle free [2,15,10,11].Apart from edge ideals, it was shown by Conca and Herzog [6] that polymatroidal ideals have linear resolutions and powers of polymatroidal ideals are polymatroidal ideals.So for the class of all polymatroidal ideals c(I) = 1.
Finally we conclude by a discussion on a related conjecture by [23]: Theorem 2.3 was proved by Banerjee in his thesis to study this conjecture and related other problems, based on simple Theorem 2.2.We now explain why this inductive approach via colon ideals can not be used directly to settle Conjecture 4.3.Any 2-dimensional simplicial complex ∆ can be subdivided so that the resulted complex is flag-no-square, see [9] or [24, Lem.2.3], i.e. ∆ = cl(H) where H is a graph with no induced 4-cycles.In particular, we choose such H so that ∆ triangulates the dunce hat, a contractible 2-dimensional complex.Thus, all subcomplexes of D have vanishing homology in dimension ≥ 2. Further, the link of any vertex a ∈ ∆ is an induced subcomplex (as ∆ is a clique complex) with nonzero first homology.For an edge ab ∈ G := H c , the construction of ∆ ′ from Theorem 3.1 satisfies link a ∆ ′ = link a ∆ is an induced subcomplex of ∆ ′ .We conclude that reg(I(G)) = 3 and by Corollary 2.4(ii) for any edge ab ∈ G also reg((I(G) 2 : ab) = 3.Thus, if reg(I(G) 2 ) = 4 as Conjecture 4.3 suggests, then Theorem 2.2 can not be directly applied to prove it.
On the other hand, if reg(I(G) 2 ) > 4 then this will be a counter example.Unfortunately we could not verify the value of reg(I(G) 2 ) due to computational limitations.It will be great if this can be verified in future.
Most of this work was done when first author was visiting the Institute of Mathematics of Hebrew University and he would like to thank faculty and staff of Hebrew University for their hospitality.Also, first author was partially supported by DST INSPIRE (India) research grant (DST/INSPIRE/04/2017/000752) and he would like to acknowledge that.
Second author was partially supported by Israel Science Foundation grant ISF-1695/15, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288.
Further, (I(G) k+1 : e 1 • • • e k ) is an edge ideal of a bipartite graph on same bipartite partition of vertices as G.
by definition.Done.Claim Two: Hl ∆ ′ [W 2 ] = 0. Proof of Claim: We prove by induction on |D ∩ W 2 |.If the size of this intersection is zero then W 2 ⊆ C and the claim follows by assumption on ∆ as ∆ ′ [C] = ∆[C].