Classifying Nearly Complete Intersection Ideals Generated in Degree Two
CCLASSIFYING NEARLY COMPLETE INTERSECTION IDEALS GENERATEDIN DEGREE TWO
CHARLIE MILLER AND BRANDEN STONE
Abstract.
Nearly complete intersection ideals were introduced in [BS18] and defines a specialclass of monomial ideals in a polynomial ring. These ideals were used to give a lower bound ofthe total sum of betti numbers that appear a minimal free resolution of a monomial ideal. In thisnote we give a graph theoretic classification of nearly complete intersection ideals generated indegree two. In doing so, we define a novel graph operation (the inversion) that is motivated bythe definition of this new class of ideals. Introduction
Let I be a homogeneous ideal in a polynomial ring R over a field k . We denote the rank of the i -th free module in a minimal free resolution of R/I as β i ( R/I ). The long-standing conjecture ofBuchsbaum-Eisenbud [BE77] and Horrocks [Har79] states that if I has height c , then β i ( R/I ) (cid:62) (cid:18) ci (cid:19) . While the case when c (cid:62) (cid:88) β i ( R/I ) (cid:62) c , known as the Total Rank Conjecture , has been completely solved for arbitrary ideals (with char k (cid:54) = 2) by M. Walker [Wal17]. At the same time, a special case of this conjecture was independentlyshown for monomial ideals by A. Boocher and J. Seiner [BS18]. In particular, they show that if I isnot a complete intersection, then (cid:88) β i ( R/I ) (cid:62) c + 2 c − . In order to achieve this lower bound, the authors reduce to a special class of ideals they define asnearly complete intersections (NCI) (see Definition 2.2). Our main theorem, Theorem 3.7, gives acomplete characterization of NCI ideals generated in degree 2 by examining the associated graph G .For example, a squarefree monomial ideal I generated in degree 2 is not a nearly complete intersectionif P is an induced subgraph of G . Section 2 gives the necessary background information. The mainclassification theorem is proved in Section 3 as well as a new graph operation, the inversion of avertex (Definition 3.1), motivated by the definition of this new class of ideals. Acknowledgments.
This project is the result of undergraduate summer research supported byHamilton College. We are thankful to the faculty in the Math department for their encouragementthroughout the process. In particular, we are grateful to Courtney Gibbons for her continual guid-ance and feedback. The first author would also like to thank the organizers and speakers at theThematic Program in Commutative Algebra and its Interaction with Algebraic Geometry held atNotre Dame in the summer of 2019. This workshop introduced the foundational material relevantto this work.
The first author was supported by McGowan Family Fund, a summer research award for students at HamiltonCollege, Clinton, NY. a r X i v : . [ m a t h . A C ] J a n C. MILLER AND B. STONE Preliminaries
Unless otherwise noted, we let R = k [ x , x , . . . , x n ] be a standard graded polynomial ring over afield k in n variables. Given a monomial ideal I ⊆ R , the support of I (denoted Supp( I )) will referto the set of variables appearing in at least one minimal monomial generator. The following factabout the support is helpful throughout the note and we state it without proof. Lemma 2.1.
Let R = k [ x , x , . . . , x n ] and I = ( m , m , . . . , m t ) ⊆ R be a monomial ideal. If m ∈ R is a monomial such that Supp ( m ) ∩ (cid:34) t (cid:91) i =1 Supp ( m i ) (cid:35) = ∅ , then m ∈ R/I is a non-zero divisor.
Using notation defined in [BS18], I ( x = 1) is the ideal defined by setting x = 1 for some variable x in the support of I . As such, I ⊆ I ( x = 1). E.g., if I = ( ab, bc, ac ) ⊆ k [ a, b, c ], then I ( a = 1) = ( b, c )and I ⊆ I ( a = 1). The following was defined in [BS18] and is the main object of study in this note. Definition 2.2 ([BS18]) . A squarefree monomial ideal I ⊆ R is a nearly complete intersection if(1) it is generated in degree at least two,(2) is not a complete intersection, and(3) for each variable x in the support of I , I ( x = 1) is a complete intersection.For example, let I = ( ab, ac, bc ) ⊆ k [ a, b, c ]. We see that I is generated in degree 2 and is not acomplete intersection. Further, for each element of Supp( I ), I ( a = 1) = ( b, c ), I ( b = 1) = ( a, c ), and I ( c = 1) = ( a, b ) are complete intersections. Thus I is a nearly complete intersection.The main result, Theorem 3.7, completely classifies the NCIs generated in degree two via theirassociated graphs. Throughout, a finite graph G is a pair G = ( V ( G ) , E ( G )) where V ( G ) = { x , x , . . . , x n } is the set of vertices of G , and E ( G ) is a collection edges of G consisting of twoelement subsets of V ( G ). We will further assume all graphs are simple, i.e. not allowing loops andmultiple edges between vertices.There exists a one-to-one correspondence between finite simple graphs and monomial ideals gen-erated degree two. In particular, given a graph G , the edge ideal I ( G ) is typically defined by I ( G ) = ( x i x j | { x i , x j } ∈ E ( G )) ⊆ k [ x , x , . . . , x n ] , where V ( G ) = { x , x , . . . , x n } . For bookkeeping reasons, we slightly modify the standard definitionof edge ideal to allow for singletons, while at the same time preserving the one-to-one correspondence.In particular, in this note the edge ideal of a graph G is defined as I ( G ) = ( x i x j , x k | { x i , x j } ∈ E ( G ) , x k ∈ V ( G ) is a singleton) ⊆ k [ x , x , . . . , x n ] . For example, the graph G below corresponds to the edge ideal I ( G ) = ( ab, ac, bc, d ) ⊆ k [ a, b, c, d ]. a b c dG :Abusing notation we will often refer to an element u k ∈ I ( G ) both as u k = x i k x j k ∈ I ( G ) and u k = { x i k , x j k } ∈ E ( G ). Using this correspondence, we say a graph G is a nearly complete intersection(NCI) if the edge ideal I ( G ) is a nearly complete intersection. As such classifying the NCI graphswill in turn classify the NCI ideals generated in degree two.We end this section with a standard fact about graphs associated to complete intersections. Lemma 2.3.
Let G be a simple graph and I ( G ) ⊆ R be its edge ideal. Then R/I is a completeintersection if and only if G is a disjoint union of edges and singletons. LASSIFYING NCI 3
Proof.
Assume that E ( G ) = { u , u , . . . , u n } and that R/I ( G ) is a complete intersection. Supposeto the contrary that there exists a vertex v of degree 2 in V ( G ). This implies that there exist edges u i , u j ∈ E ( G ) such that u i ∩ u j = { v } . Assuming i = 1 and j = 2, we have that u ∈ R/ ( u , . . . , u n )is a zero-divisor. As such there does not exist a vertex of degree two and G must be a disjoint unionof edges and singletons.Assume G is a disjoint union of edges as well as singletons. With out loss of generality, we canreduce to the case that G does not contain any singletons. Thus, the edge ideal of G is I ( G ) =( x i y i | i = 1 , . . . , n ). Notice thatSupp( x i y i ) ∩ i − (cid:91) j =1 Supp( x j y j ) = ∅ for all i = 1 , . . . , n . Therefore, R/I ( G ) is a complete intersection by Lemma 2.1. (cid:3) Classifying NCIs
As mentioned in the previous section, a graph G is an NCI if the edge ideal I ( G ) is a nearlycomplete intersection as defined in Definition 2.2. The main result, Theorem 3.7, gives a completeclassification of NCI ideals generated in degree at most 2 using the above graph correspondence.Before we can prove the result, we define a new graph operation necessary for the proof. We denotethe neighbors of a vertex v in V ( G ), N( v ), and the induced subgraph on a subset V (cid:48) ⊆ V ( G ) as G [ V (cid:48) ]. Definition 3.1.
The inversion of a vertex v in a graph G is the graph defined by I ( v, G ) = ( V (cid:48) , E (cid:48) ) , where V (cid:48) = V \ { v } and E (cid:48) = E ( G [ V (cid:48) \ N( v )]).This operation is a direct translation of the operation I ( x = 1) used in Definition 2.2 (3) and isthe main tool used in the classification of NCI graphs. With it, we can further formalize NCI graphswith the following lemma whose proof is a direct translation of definitions. Lemma 3.2.
A graph G is an NCI if and only if(1) G is not a complete intersection, and(2) for each vertex v ∈ V ( G ) , I ( v, G ) is a complete intersection. From this we have an immediate corollary.
Corollary 3.3.
NCI graphs are connected.
This corollary highlights the observations in Section 4 of [BS18]. In the next example we can useLemma 3.2 to determine if graphs are NCI or not.
Example 3.4.
Here we have a graph G and two inversions at the vertices c and f . Notice thatafter the inversions we do not have a complete intersection (Lemma 2.3), hence G is not an NCIsince every inversion must be a complete intersection (Lemma 3.2). G a b cd ef g I ( c, G ) a bd ef g I ( f, G ) a b cd eg Above shows that not all graphs are NCI. In fact the NCI property seems to be quite rare. Beloware examples of graphs that are NCI. Notice that any inversion of a vertex will create a disjointunion of edges and singletons, i.e. a complete intersection. Applying Lemma 3.2 shows they areNCI.
C. MILLER AND B. STONE C K C P S S S S S S S It’s natural to look at the families these graphs belong to. For example, the family of paths arenot all NCI. Indeed, if n >
4, then the path P n is not an NCI. To see this one only needs to inverta leaf of the graph and notice the resulting graph is not a complete intersection, but another pathconnecting at least three vertices. A similar result/argument holds for cycles, i.e. if n >
5, then acycle C n is not an NCI. However, this is not the case for complete graphs. Proposition 3.5.
Any complete graph with more than 2 vertices is an NCI.Proof.
Let G = K n be a complete graph on n ≥ v ∈ V = V ( G ), the inversion of v isgiven by I ( v, G ) = ( V (cid:48) , E ( G [ V (cid:48) \ N( v )])) , where V (cid:48) = V \ { v } . As G is complete, we have that N ( G ) = V \ { v } = V (cid:48) , and hence E ( G [ V (cid:48) \ N( v )]) = ∅ . Thus I ( v, G ) is a collection of singletons and hence a complete intersection by Lemma 2.3. (cid:3) In the above path and cycle examples, we saw that the threshold for a graph to be NCI washaving | V ( G ) | (cid:54) Proposition 3.6.
Let G be a connected graph.(1) If | V ( G ) | (cid:54) , then G is not an NCI.(2) If | V ( G ) | = 3 or , then G is an NCI.Proof. When | V ( G ) | (cid:54) | V ( G ) | = 3 or 4, G cannot be a complete intersection due to theconnected assumption, i.e. any vertex v ∈ V ( G ) must be connected to at least one other vertex.Thus I ( v, G ) has at most one edge and is a complete intersection. This forces G to be an NCI. (cid:3) We are now ready to prove the main classification theorem. In the theorem, we define the graph T as the following. Tv This graph, along with P , become the major obstructions to the NCI property. Theorem 3.7.
Let G be a connected graph with | V ( G ) | (cid:62) . The graph G is not an NCI if andonly if there exist vertices v , v , v , v , v ∈ V ( G ) such that the following conditions hold:(1) the vertex v is a leaf in G [ v , v , v , v , v ] ;(2) the path P or T is a spanning tree of G [ v , v , v , v , v ] where the neighbors of v all havedegree 2 in the spanning tree.Proof. Assume G is not an NCI. As such, there exists v ∈ V ( G ) such that I ( v, G ) is not a completeintersection. In particular, I ( v, G ) has a vertex w ∈ V (cid:48) = V \ { v } of degree two. As G is connected,there must exist a path from v to w that passes through the neighbors of v in G . So there exists v ∈ N G ( v ) such that the path(1) v −→ v −→ v −→ · · · −→ w LASSIFYING NCI 5 exists in G . Without losing generality, we can assume the vertices in the path from v to w (inclusive)avoid N G ( v ). Indeed if there was a vertex u ∈ N G ( v ) between v and w , we could replace v with u , shortening the path. As such, we may assume the path from v to w is completely contained inthe subgraph G [ V (cid:48) \ N G ( v )] ⊂ G . We now consider two cases, v = w and v (cid:54) = w , which can bevisualized in the following abstract representation of G . v v v w (cid:48) ww (cid:48)(cid:48) N G ( v ) II I G [ V (cid:48) − N G ( v )] Case I.
Assume v = w . Since w is a degree two vertex in I ( v, G ), there exist w (cid:48) , w (cid:48)(cid:48) ∈ V (cid:48) suchthat ww (cid:48) , ww (cid:48)(cid:48) ∈ E (cid:48) = E ( G [ V (cid:48) \ N G ( v )]). In particular w (cid:48) , w (cid:48)(cid:48) / ∈ N G ( v ). As such, v is a leaf in theinduced subgraph H = G [ v, v , w, w (cid:48) , w (cid:48)(cid:48) ], and by construction, T is a spanning tree of H where v is the only neighbor of v . Further, the degree of v is two in the spanning tree T , thus both of thedesired conditions are satisfied. Case II.
Assume v (cid:54) = w . As v and w are distinct, we can reduce to the case where there is a singlevertex between them on the path (1), say w (cid:48) . As v , w (cid:48) , w / ∈ N G ( v ), we have that v is a leaf in theinduced subgraph H = G [ v, v , v , w (cid:48) , w ]. In this case, by construction, P is a spanning tree of H where v is the only neighbor of v . As the degree of v is two in the spanning tree P , we have ourdesired result.Conversely, assume the conditions hold for a graph G that is NCI. In this scenario, there existsvertices v , v , v , v , v ∈ V ( G ) such that v is a leaf in the induced subgraph G [ v , v , v , v , v ]. Inthe situation where P is a spanning tree of G [ v , v , v , v , v ], we can assume the vertex labels ofthe path are as follows. P : v v v v v Since v is a leaf in the induced subgraph, we know that v , v , v / ∈ N G ( v ). Hence the degree of v is at least two in I ( v , G ). This shows that I ( v , G ) is not a complete intersection, a contradictionof Lemma 3.2. A similar argument holds for when T is a spanning tree of G [ v , v , v , v , v ]. (cid:3) Theorem 3.7, together with Proposition 3.6 give a complete classification of NCI graphs. As aresult, we have a graph theoretic classification of NCI ideals generated in degree two. A naturaldesire is to extend this result to NCIs with generators in higher degrees. One direction to consideris classifying these ideals with hypergraphs . A
Hypergraph is a pair G = ( V, E ) where V is the setof vertices of G and the set of edges E is a set of nonempty subsets of V . In this scenario, morethan two vertices can be incident to a single edge. As with graphs, a similar correspondence exitsbetween hypergraphs and ideals and can be seen in the following example. Example 3.8.
The left image below is an example of an NCI hypergraph G on a vertex set V ( G ) = { a, b, c, d, e, f, g } . Notice this hypergraph has three edges, { a, b, c } , { g } , and { d, e, f } . Further, thereis a natural correspondence between these hypergraphs and monomial ideals in k [ a, b, c, d, e, f, g ]. Inparticular I ( G ) is listed below G . C. MILLER AND B. STONE a b cgd e fI ( G ) = ( abc, def, ag, bg, cg, dg, eg, f g ) b cgd e f I ( a, G ) is CILemma 3.2 can also be extended to this scenario as well as the definition of inversion. Noticethat inverting a (or any vertex) will produce the complete intersection on the right. It is worthnoting that all the examples of NCI hypergraphs we were able to construct were related to theabove example. This hints at the possibility that all higher degree NCI ideals are related to theabove hypergraph.We end this section with an observation relating to the original result of [BS18]. Let I be a height c monomial ideal in a polynomial ring S that is not a complete intersection. A. Boocher and J.Seiner show that (cid:80) β i ( S/I ) (cid:62) c + 2 c − . In particular, equality holds if and only if the generatingfunction for β i ( S/I ) is either(1 + 3 t + 2 t )(1 + t ) c − or (1 + 5 t + 5 t + t )(1 + t ) c − . When c = 2 or 3, respectively, the generating functions are defined by ideals with the betti sequence { , , } and { , , , } , respectively. We are able to retrieve these sequences from the obstructionsnoted in Theorem 3.7. Indeed, the edge ideal I ( P ) and I ( T ) both have the betti sequence { , , , } .However, if we connect the end points of the path P to create a 5-cycle, the betti sequence becomes { , , , } . Similarly, removing a leaf of either P or T can create the path P , obtaining the bettisequence { , , } . References [BE77] David A Buchsbaum and David Eisenbud. Algebra structures for finite free resolutions, and some structuretheorems for ideals of codimension 3.
American Journal of Mathematics , 99(3):447–485, 1977.[BS18] Adam Boocher and James Seiner. Lower bounds for Betti numbers of monomial ideals.
J. Algebra , 508:445–460, 2018.[Har79] Robin Hartshorne. Algebraic vector bundles on projective spaces: a problem list.
Topology , 18(2):117–128,1979.[Wal17] Mark E. Walker. Total Betti numbers of modules of finite projective dimension.
Ann. of Math. (2) , 186(2):641–646, 2017.
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