Characteristic dependence of syzygies of random monomial ideals
aa r X i v : . [ m a t h . A C ] O c t HEURISTICS FOR ℓ -TORSION IN VERONESE SYZYGIES CAITLYN BOOMS, DANIEL ERMAN, AND JAY YANG
Abstract.
To what extent should we expect the syzygies of Veronese embeddings of pro-jective space to depend on the characteristic of the field? As computation of syzygies isimpossible for large degree Veronese embeddings, we instead develop an heuristic approachbased on random flag complexes. We prove that the corresponding Stanley–Reisner idealshave Betti numbers which almost always depend on the characteristic, and we use this toconjecture that the syzygies of the d -uple embedding of projective r -space with r ≥ shoulddepend on the characteristic for almost all d . Introduction
Imagine P embedded by the d -uple Veronese embedding, where d is some large integerlike d = 100 or d = 100000 . What should we expect about the syzygies? Such questionswere raised by Ein and Lazarsfeld in [18] and later in [16]. They focused on quantitativebehaviors that are independent of the ground field. We ask: To what extent should we expectthe syzygies to depend on the characteristic, if at all? Given the impossibility of computingdata for large d , how can we make reasonable conjectures? The central idea in this paper is the development of an heuristic—based on a randomflag complex construction—for modelling the syzygies of Veronese embeddings of projec-tive space. The resulting conjectures propose that, when it comes to dependence on thecharacteristic of the ground field, pathologies are the norm. Let us make this more precise.For any integers r, d ≥ and any field k , we may consider the d -uple embedding of P rk into P ( r + dd ) − k . The image is defined by a homogeneous ideal I ⊂ S , where S is a polynomialring in (cid:0) r + dd (cid:1) variables over k . We denote the algebraic Betti numbers of the image by β i,j ( P rk ; d ) := dim k Tor Si ( S/I, k ) j . These encode the number of degree j generators for the i ’th syzygies. A much-studied open question is to describe the Betti table β ( P rk ; d ) , which isthe collection of all these Betti numbers [2, 6, 7, 9, 17, 18, 21, 22, 26, 30–32, 35].Since each individual Betti number is invariant under flat extensions, the Betti table isdetermined by the integers r, d and the characteristic of k . For a prime ℓ , we say that β ( P r ; d ) has ℓ -torsion if β ( P r F ℓ ; d ) = β ( P r Q ; d ) , and we say that β ( P r ; d ) depends on thecharacteristic if this occurs for some ℓ . There are two cases where we know something: • For r = 1 and any d , the Betti numbers in β ( P r ; d ) do not depend on the character-istic, as any rational normal curve is resolved by an Eagon-Northcott complex. • For r = 6 and d = 2 , Andersen’s thesis [2] shows that β , ( P ; 2) has -torsion.This was later generalized—using the connection between the degree Veronese and
The authors were supported by NSF GRFP grant DGE-1747503 and by NSF grants DMS-1502553, DMS-1902123, and DMS-1745638. Support was also provided by the Graduate School and OVCRGE at UW-Madison with funding from the Wisconsin Alumni Research Foundation. This is equivalent to a certain integral Tor group having ℓ -torsion: see Remark 2.1. atching complexes (see [32])—by Jonnson’s work [25] showing that for d = 2 andvarious larger values of r , one can find ℓ -torsion for odd primes ℓ = 3 , , , , and .Little else is known or conjectured about the dependence of Veronese syzygies on the charac-teristic. There are some results of a parallel nature, such as Hashimoto’s result that the Bettitable of P × P in the Segre embedding has -torsion [23], and Bouc’s results on -torsion forwhen d = 2 [6]. For the d = 2 case, Jonsson has also conjectured the existence of ℓ -torsionfor all odd primes ℓ [25, Conjecture 1.9], as long as r ≫ . However, there are not any ofthe following: known examples of -torsion, known examples of torsion when d > , or evenany conjectures for a fixed r .One key challenge in this area is the difficulty of generating good data. For instance, thesyzygies of P under the -uple embedding were only recently computed [7, 9]. For largervalues of d and r , computation is essentially impossible: in the case of P and d = 100 mentioned above, the computation would involve ≈ . × variables.Heuristics can provide an alternate route for generating conjectures, especially when com-putation is infeasible. (Such an approach is quite common for predicting properties of howthe prime numbers are distributed, for instance.) In this paper, we use an heuristic modelto motivate conjectures about ℓ -torsion in β ( P r ; d ) . Our first conjecture is that dependenceon the characteristic should be commonplace as d → ∞ . Conjecture 1.1.
Fix r ≥ . For any d ≫ , the Betti table of P r under the d -uple embeddingdepends on the characteristic. This conjecture is based upon corresponding properties of the following model for Veronesesyzygies. We let ∆ ∼ ∆( n, p ) denote a random flag complex on n vertices with attachingprobability p . (See §2 for details.) For a given field k , we let I ∆ be the corresponding Stanley–Reisner ideal in S = k [ x , . . . , x n ] . Ein and Lazarsfeld showed that if d ≫ , then almost all ofthe Betti numbers in rows , . . . , r of β ( P rk ; d ) are nonzero (see for instance [19, Theorem 1.1]).Theorem 1.3 of [19] gives that a similar result holds for I ∆ as long as n − / ( r − ≪ p ≪ n − /r and n ≫ . Thus, if p is in the specified range, then the Betti table β ( S/I ∆ ) as n → ∞ satisfies similar nonvanishing properties as β ( P rk ; d ) as d → ∞ ; in this sense, the Betti tables β ( S/I ∆ ) determined by ∆( n, p ) can act as a random model for Veronese syzygies.To predict how β ( P r ; d ) depends on the characteristic, we will therefore consider the cor-responding questions for β ( S/I ∆ ) for various fields k . As with Veronese syzygies, we saythat the Betti table of the Stanley–Reisner ideal of ∆ has ℓ -torsion if this Betti table isdifferent when defined over a field of characteristic ℓ than it is over Q , and we say that thisBetti table depends on the characteristic if this occurs for some ℓ . We prove: Theorem 1.2.
Fix r ≥ , and let ∆ ∼ ∆( n, p ) be a random flag complex with n − / ( r − ≪ p ≪ n − /r . With high probability as n → ∞ , the Betti table of the Stanley–Reisner ideal of ∆ depends on the characteristic. In other words, if p is in the range where the Betti table of the Stanley–Reisner idealof ∆ behaves like P r —in the sense of [19, Theorem 1.3]—then this Betti table will almostalways depend on the characteristic for n ≫ . This theorem is the basis of Conjecture 1.1.Our proof of Theorem 1.2 requires the r ≥ hypothesis, which is why we have the samehypothesis in Conjecture 1.1. The fact that this value of r is close to Andersen’s exampleappears to be a coincidence; see Remarks 1.4 and 7.4 for more discussion. Note also that, See [17, 18] for more on these nonvanishing properties. ased on [2, 24, 25], we might even find ℓ -torsion in β ( P r ; d ) for small values of d as well.However, Theorem 1.2 is asymptotic in nature, which motivates the d ≫ hypothesis inConjecture 1.1.We also prove the following sharper result: Theorem 1.3.
Fix m ≥ , and let ∆ ∼ ∆( n, p ) be a random flag complex with n − / ≪ p ≤ − ǫ for some ǫ > . With high probability as n → ∞ , the Betti table of the Stanley–Reisner ideal of ∆ has ℓ -torsion for every ℓ dividing m . In particular, this holds when n − / ( r − ≪ p ≪ n − /r for any r ≥ . The proof of Theorem 1.3 (which implies Theorem 1.2) proceeds as follows. By Hochster’sformula [8, Theorem 5.5.1], it suffices to show that some induced subcomplex of ∆ has m -torsion in its homology. So, for each m , we construct a flag complex X m with a smallnumber of vertices and with m -torsion in H ( X m ) . This complex is derived from Newman’sconstruction of a two-dimensional simplicial complex X where H ( X ) has m -torsion [29, §3],though we modify his work to ensure that X m is a flag complex and to lower the maximalvertex degree. We then apply Bollobás’s theorem on subgraphs of a random graph [5,Theorem 8]—or rather a minor variant of that result for induced subgraphs—to prove that X m appears as an induced subcomplex of ∆ with high probability as n → ∞ , yieldingTheorem 1.3.Theorem 1.3 fits into an emerging literature on random monomial ideals. Our currentwork seems to be the first application of random monomial ideal methods to generate newconjectures outside of the world of monomial ideals. Random monomial ideals first appearedin the work of De Loera–Petrović–Silverstein–Stasi–Wilburne [15], which outlined an arrayof frameworks for studying random monomial ideals, including the model used in this paper,as well as models related to other types of random simplicial complexes such as [12, 27]; theyalso proved threshold results for dimension and other invariants of these ideals. In [14], sim-ilar methods were applied to study the average behavior of Betti tables of random monomialideals and to compare these with certain resolutions of generic monomial ideals. Recent workof Banerjee and Yogeshwaran analyzes homological properties of the edge ideals of Erdős–Rényi random graphs [3]. The forthcoming [34] looks more closely at threshold phenomenain the phase transitions of the random models from [15]. There is also the previously refer-enced [19], which uses random monomial methods to demonstrate some asymptotic syzygyphenomena observed/conjectured in [16, 18].Additionally, there is a great deal of literature on the study of ℓ -torsion arising in randomconstructions. The most relevant such study is perhaps the recent work by Kahle–Lutz–Newman–Parsons on ℓ -torsion in the homology of random simplicial complexes [28], whichconjectures the existence of bursts of torsion homology at specific thresholds. For comparison,those authors are interested in ℓ -torsion in the global homology of a complex like ∆( n, p ) ,whereas, due to Hochster’s formula, we analyze the simpler question of finding ℓ -torsion inthe homology of any induced subcomplex of ∆( n, p ) . Remark . The bound r ≥ in Theorem 1.3 is not necessarily sharp. In fact, a moredetailed investigation of the -torsion of the Betti table of the Stanley–Reisner ideal of ∆ yields a bound of r ≥ ; see §5 and Proposition 6.1. See also Remarks 4.4 and 7.4 for furtherdiscussion on restrictions on r in both Conjecture 1.1 and Theorem 1.3.Theorem 1.3 also leads us to a stronger conjecture on Veronese ℓ -torsion: onjecture 1.5. Let r ≥ . As d → ∞ , the number of primes ℓ such that β ( P r ; d ) has ℓ -torsion is unbounded. Regarding Conjectures 1.1 and 1.5, it is worth emphasizing the total lack of direct evidence.As noted above, [2] and [25] appear to provide the only known instances of ℓ -torsion forVeronese embeddings. Our conjectures are based primarily upon the heuristic model and, toa lesser extent, upon the nonvanishing results of [17, 18], both of which rely on an inductivestructure where pathologies in β ( P r ; d ) tend to propagate as d → ∞ , and both of which showthat the asymptotic behavior of syzygies exhibits a strong uniformity.However, we do not expect our random flag complex model to be a perfect predictor of allproperties of Veronese syzygies. In fact, while the results in [19] imply that the Betti tablesassociated to ∆ have similar overarching nonvanishing properties as Veronese embeddings,those results also imply that these Betti tables do not demonstrate more nuanced proper-ties. For instance, the random flag complex model would not give correct predictions aboutGreen’s N p -property [22] for Veronese embeddings. This is why Conjectures 1.1 and 1.5echo certain qualitative aspects of Theorem 1.3 as opposed to more specific and quantitativepredictions about ℓ -torsion in β ( P r ; d ) .In a rather different direction, an alternate heuristic model for Veronese syzygies is con-sidered in [16]. That model is based on Boij-Söderberg theory and is used to generatequantitative conjectures about the entries of β ( P rk ; d ) for d ≫ . However, since that modeldoes not take into account the characteristic of the field, it cannot be used to generate con-jectures such as those above. See also the results of [11], which provided a combinatorialparallel of the asymptotic results of [18]. Remark . Ein and Lazarsfeld’s asymptotic nonvanishing results are more-or-less uniformfor any smooth variety of dimension r [18, Theorem A], and these were even expanded tointegral varieties by Zhou [36]. In this paper, we restrict attention to P r for concreteness,but we would expect that Conjecture 1.1 would likely apply to the d -uple embeddings of any r -dimensional integral variety which is flat over Z , including products of projective spaces,toric varieties, hypersurfaces, Grassmanians, and more. (cid:3) This paper is organized as follows. In §2, we review notation and background, includingon Betti numbers, Hochster’s formula, and random flag complexes. §3 contains our mainconstruction in which we construct an explicit flag complex X m with m -torsion in homology;see Theorem 3.1. In §4, we apply a minor variant of Bollobás’s theorem on subgraphs of arandom graph to show that, with high probability, X m appears as an induced subgraph of ∆( n, p ) for any n − / ≪ p ≤ − ǫ where ǫ > and m ≥ . In §5, we analyze the case of2-torsion more closely, using the techniques from §4 to expand known results from [13]. In §6,we then combine this result with Hochster’s formula to prove Theorem 1.3. Finally, §7 returnsto the geometric setting where we use our results to produce heuristics and conjectures about ℓ -torsion in Veronese syzygies. Acknowledgments.
We thank Kevin Kristensen, Rob Lazarsfeld, Andrew Newman, VictorReiner, Gregory G. Smith, and Melanie Matchett Wood for helpful conversations. We thankClaudiu Raicu and Steven Sam for thoughtful comments on an early draft. . Background and Notation
Betti tables for Veronese embeddings.
For a given r, d ≥ and field k , we havethe d -uple Veronese embedding P rk → P ( r + dd ) − k . The image is determined by a homogeneousideal I ⊂ S where S is a polynomial ring with coefficients in k and (cid:0) r + dd (cid:1) variables. Thehomogeneous coordinate ring S/I of the image is a graded S -module. We can thus take aminimal free resolution F ← F ← · · · of S/I , where each F i is a graded free S -module, F i = M j ∈ Z S ( − j ) β i,j ( S/I ) . This provides one way to define the algebraic Betti numbers. Analternate definition is β i,j ( S/I ) := dim k Tor Si ( S/I, k ) j . To emphasize the dependence on r and d (and to avoid referencing the ambient ring S and the homogeneous coordinate ring S/I ,both of which change with d ), we will denote these Betti numbers by β i,j ( P rk ; d ) instead of themore standard β i,j ( S/I ) . Further, we write β ( P rk ; d ) for the Betti table of this embedding,which is the collection of all β i,j ( P rk ; d ) .2.2. Torsion in Betti tables.
Throughout this paper we will analyze graded algebras, allof which have the following form: there is an ideal I in a polynomial ring T with coefficientsin Z , where T /I is flat over Z , and we are interested in specializations ( T /I ) ⊗ Z k to variousfields k . We consider such graded algebras that arise in two ways: as the coordinate ringsof Veronese embeddings of projective space and as the Stanley–Reisner rings of simplicialcomplexes. The central questions of this paper are concerned with when the Betti numbersof such algebras depend on the choice of the characteristic of k .First, we consider the Veronese embeddings. For any positive integers r and d , we canembed P r Z → P ( r + dd ) − Z via the d -uple Veronese embedding. If T is the polynomial ring for thelarger projective space, then there is an ideal I ⊂ T defining the image of this map. Since T /I is flat over Z , the coordinate ring of the Veronese embedding over a field k is given by ( T /I ) ⊗ Z k . As noted in the previous subsection (with S = T ⊗ Z k ), the algebraic Bettinumbers are defined as β i,j ( P rk ; d ) := dim k Tor T ⊗ Z ki (( T /I ) ⊗ Z k, k ) j . Since field extensions are flat, algebraic Betti numbers are invariant under field extensions,and thus, β ( P rk ; d ) only depends on r, d and the characteristic of k . Moreover, by semi-continuity, we have an inequality β i,j ( P r Q ; d ) ≤ β i,j ( P r F ℓ ; d ) for any prime ℓ (with equalityfor all but finitely many ℓ ). As noted in the introduction, we will say that β ( P r ; d ) has ℓ -torsion if this inequality is strict for some i, j , and we will say that β ( P r ; d ) depends onthe characteristic if this inequality is strict for some i, j and some ℓ . Remark . Let I be an ideal in T = Z [ x , . . . , x n ] which is flat over Z . Let S ′ = T ⊗ Z F ℓ = F ℓ [ x , . . . , x n ] and I ′ = IS ′ . By a standard argument, it follows that dim F ℓ Tor S ′ i ( S ′ /I ′ , F ℓ ) j = dim F ℓ (Tor Ti ( T /I, Z ) j ⊗ Z F ℓ ) + dim F ℓ (Tor Z (Tor Ti +1 ( T /I, Z ) j , F ℓ )) . In particular, the Betti table of such an ideal has ℓ -torsion in the sense of the introductionif and only if one of the Tor Ti +1 ( T /I, Z ) j has ℓ -torsion as an abelian group. (cid:3) We next consider notation for monomial ideals since Stanley–Reisner ideals of simplicialcomplexes are monomial ideals. Let J be a monomial ideal in T = Z [ x , . . . , x n ] . For a field k , the algebraic Betti numbers of ( T /J ) ⊗ Z k are given by β i,j (( T /J ) ⊗ Z k ) := dim k Tor T ⊗ Z ki (( T /J ) ⊗ Z k, k ) j .
12 34 H
12 3
Figure 1.
In the graphs shown above, H is a subgraph of G , but it is not theinduced subgraph on the vertex set { , , } since H is missing the diagonaledge connecting vertices and .As in the Veronese case, these only depend on the characteristic of the field, and we havethe same inequality β i,j (( T /J ) ⊗ Z Q ) ≤ β i,j (( T /J ) ⊗ Z F ℓ ) . As in the introduction, we saythat β ( T /J ) has ℓ -torsion if this inequality is strict for some i, j , and we say that β ( T /J ) depends on the characteristic if it has ℓ -torsion for some prime ℓ .2.3. Graphs and simplicial complexes.
For a simplicial complex X , we write V ( X ) ,E ( X ) , and F ( X ) for the set of vertices, edges, and (2-dimensional) faces of X , respectively.We use | ∗ | to denote the number of elements in these sets. The degree of a vertex v (denoted deg( v ) ) is the number of edges in X containing v . We write maxdeg( X ) for the maximumdegree of any vertex of X , and we write avg( X ) for the average degree of a vertex in X .For a pair of graphs H, G , we write H ⊂ G if H is a subgraph of G . We write H ind ⊂ G if H is an induced subgraph of G , that is, if the vertices of H are a subset of the vertices of G and the edges of H are precisely the edges connecting those vertices within G (see Figure 1).We use similar definitions and notations for a simplicial complex ∆ ′ to be a subcomplex (oran induced subcomplex) of another complex ∆ . If α ⊂ V (∆) , then we let ∆ | α denote theinduced subcomplex of ∆ on α .The following definitions, adapted from [5] and [10], will be used in sections 4, 5, and 6. Definition 2.2.
The essential density of a graph G is m ( G ) := max (cid:26) | E ( H ) || V ( H ) | : H ⊂ G, | V ( H ) | > (cid:27) , and G is strictly balanced if m ( H ) < m ( G ) for all proper subgraphs H ⊂ G .From a simplicial complex ∆ on n vertices, there is a corresponding Stanley–Reisnerideal I ∆ ⊂ S = k [ x , . . . , x n ] . Since these I ∆ are squarefree monomial ideals, Hochster’sformula [8, Theorem 5.5.1] relates the Betti table of S/I ∆ to topological properties of ∆ ,providing our key tool for studying β ( S/I ∆ ) for various fields k . An immediate consequenceof Hochster’s formula is the following fact, which characterizes when these Betti tables aredifferent over a field of characteristic ℓ than over Q . Fact 2.3.
For a simplicial complex ∆ , the Betti table of the Stanley–Reisner ideal I ∆ has ℓ -torsion if and only if there exists a subset α ⊂ V (∆) such that ∆ | α has ℓ -torsion in one ofits homology groups. Monomial ideals from random flag complexes.
Our monomial ideals are Stanley–Reisner ideals associated to random flag complexes. Recall that a flag complex is a simplicialcomplex obtained from a graph by adjoining a k -simplex to every ( k + 1) -clique in the graph.In particular, a flag complex is entirely determined by its underlying graph, and the processof obtaining a flag complex from its underlying graph is called taking the clique complex. e write ∆ ∼ ∆( n, p ) to denote the flag complex which is the clique complex of an Erdős–Rényi random graph G ( n, p ) on n vertices, where each edge is attached with probability p . If α ⊂ V (∆) , then we note that ∆ | α is also flag. The properties of random flag complexes havebeen analyzed extensively, with [27] providing an overview. As discussed in the introduction,the syzygies of Stanley–Reisner ideals of random flag complexes were first studied in [19].2.5. Probability.
We use the notation P [ ∗ ] for the probability of an event. For a randomvariable X , we use E [ X ] for the expected value of X and Var( X ) for the variance of X .For functions f ( x ) and g ( x ) , we write f ≪ g if lim x →∞ f /g → . We use f ∈ O ( g ) if there is aconstant N where | f ( x ) | ≤ N | g ( x ) | for all sufficiently large values of x , and we use f ∈ Ω( g ) if there is a constant N ′ where | f ( x ) | ≥ N ′ | g ( x ) | for all sufficiently large values of x .3. Constructing a flag complex with m -torsion homology The goal of this section is to prove the following result:
Theorem 3.1.
For every m ≥ , there exists a two-dimensional flag complex X m such thatthe torsion subgroup of H ( X m ) is isomorphic to Z /m Z and maxdeg( X m ) ≤ . This result is the foundation of our proof of Theorem 1.3 as we will show that this specificcomplex X m appears as an induced subcomplex of ∆( n, p ) with high probability under thehypotheses of that theorem.Here is an overview of our proof of Theorem 3.1, which is largely based on ideas from [29].Given an integer m ≥ , we write its binary expansion as m = 2 n + · · · + 2 n k with ≤ n < · · · < n k . Note that k is the Hamming weight of m and n k = ⌊ log ( m ) ⌋ . With this setup,the “repeated squares presentation” of Z /m Z is given by Z /m Z = h γ , γ , . . . , γ n k | γ = γ , γ = γ , . . . , γ n k − = γ n k , γ n + · · · + γ n k = 0 i . We will construct a two-dimensional flag complex X m such that the torsion subgroup of H ( X m ) has this presentation. To do so, we follow Newman’s “telescope and sphere” con-struction in [29], where Y is the telescope satisfying H ( Y ) ∼ = h γ , γ , . . . , γ n k | γ = γ , γ = γ , . . . , γ n k − = γ n k i ,Y is the sphere satisfying H ( Y ) ∼ = h τ , . . . , τ k | τ + · · · + τ k = 0 i , and X m is created by gluing Y and Y together to yield a complex with the desired H -group.Because we want our construction to be a flag complex with maxdeg( X m ) ≤ , we cannotsimply quote Newman’s results. Instead, we must alter the triangulations to ensure that Y , Y , and X m are flag complexes. Then, we must further alter the construction to reduce maxdeg( X m ) . However, each of our constructions is homeomorphic to each of Newman’sconstructions. Notation 3.2.
Throughout the remainder of this section we assume that m ≥ is given.We write m = 2 n + · · · + 2 n k with ≤ n < · · · < n k . To simplify notation, we also denote X m by X for the remainder of this section. i v i v i +1 v i +1 v i +2 v i +2 v i +3 v i +3 v i +4 v i +5 v i +6 v i +7 v ′ i v ′ i +1 v ′ i +2 v ′ i +3 v ′ i +4 v ′ i +5 v ′ i +6 v ′ i +7 Figure 2.
Building block for the telescope construction with i = 0 , , . . . , ( n k − .3.1. The telescope construction.
The telescope Y that we construct will be homeomor-phic to the Y that Newman constructs in [29, Proof of Lemma 3.1] for the d = 2 case. Westart with building blocks which are punctured projective planes; in contrast with [29], ourblocks are triangulated so that each is a flag complex. Explicitly, for each i = 0 , . . . , ( n k − ,we produce a building block which is a triangulated projective plane with a square faceremoved, with vertices, edges, and faces as illustrated in Figure 2. Our building blocks dif-fer from Newman’s in order to ensure that Y and the final simplicial complex X are flagcomplexes; for instance, we need to add extra vertices v ′ i , . . . , v ′ i +7 .We construct Y by identifying edges and vertices of these n k building blocks as labeled.The underlying vertex set is V ( Y ) = { v , v , v , . . . , v n k +3 , v ′ , v ′ , . . . , v ′ n k − } , so we have | V ( Y ) | = (4 n k + 4) + 8 n k = 12 n k + 4 . Since each building block has edges, of which areglued to the next building block, and faces, a similar computation yields | E ( Y ) | = 40 n k +4 and | F ( Y ) | = 28 n k . In addition, observe that the vertices of highest degree are those in thesquares in the “middle” of the telescope, such as vertex v when n k ≥ . In this case, v isadjacent to v , v , v ′ , v ′ , v ′ , v ′ , v ′ , v ′ , and v ′ , so deg( v ) = 9 . By the symmetry of Y , wehave that maxdeg( Y ) = 9 when n k ≥ , and maxdeg( Y ) = 6 when n k = 1 (when m = 2 , ).To compute H ( Y ) , we simply apply the identical argument from [29]. We order thevertices in the natural way, where v j > v k if j > k , similarly for the v ′ ℓ , and where v ′ ℓ > v j for all ℓ, j . We let these vertex orderings induce orientations on the edges and faces of Y . For each i = 0 , . . . , n k , denote by γ i the 1-cycle of Y represented by [ v i , v i +1 ] +[ v i +1 , v i +2 ] + [ v i +2 , v i +3 ] − [ v i , v i +3 ] . Then γ i − γ i +1 is a 1-boundary of Y for each i = 0 , . . . , ( n k − , and, as in Newman’s construction, we have that H ( Y ) can be presentedas h γ , γ , . . . , γ n k | γ = γ , γ = γ , . . . , γ n k − = γ n k i .3.2. The sphere construction.
The sphere part Y is a flag triangulation of the sphere S that has k square holes such that the squares are all vertex disjoint and nonadjacent. Our Y will be homeomorphic to the Y that Newman constructs in [29] for the d = 2 case, but ourconstruction involves a few different steps. First, we will show that for any integer k ≥ ,there exists a flag triangulation T i of S (here i = ⌊ k − ⌋ ) with at least k faces such that axdeg( T i ) ≤ . Then, we will insert square holes on k of the faces of T i , while subdividingthe edges, and call the resulting flag complex e T i . Finally, we describe a process to replaceeach vertex of degree 14 in e T i with two degree 9 vertices so that the resulting complex, Y , has maxdeg( Y ) ≤ . Throughout these constructions, we will have four cases correspondingto the value of k mod 4 , and we carefully keep track of the degrees of each vertex in T i , e T i ,and Y for each case.3.2.1. T i and flag bistellar 0-moves. We begin by constructing an infinite sequence T , T , . . . of flag triangulations of S such that maxdeg( T i ) ≤ for all i . To do so, we adapt the bistellar0-moves used in [29, Lemma 5.6]. Let T be the -simplex boundary on the vertex set { w , w , w , w } . Note that each vertex of T has degree 3. We will construct the remaining T i inductively. To build T , first remove the face [ w , w , w ] and edge [ w , w ] . Then,add two new vertices w and w as well as new edges [ w , w ] , [ w , w ] , [ w , w ] , [ w , w ] , [ w , w ] , [ w , w ] , and [ w , w ] . Taking the clique complex will then give T . See Figure 3.Essentially, this process is the same as making the face [ w , w , w ] into a square face [ w , w , w , w ] , removing that square face, taking the cone over it, and then ensuring thatthe resulting complex is a flag triangulation of S . We will call such a move a flag bistellar0-move . Each T i +1 for i ≥ will be obtained from T i by performing a flag bistellar 0-move on the face [ w i +1 , w i +2 , w i +3 ] of T i . Explicitly, to construct T i +1 , remove the face [ w i +1 , w i +2 , w i +3 ] and the edge [ w i +1 , w i +3 ] . Then, add new vertices w i +4 and w i +5 andnew edges [ w i , w i +4 ] , [ w i +1 , w i +4 ] , [ w i +3 , w i +4 ] , [ w i +1 , w i +5 ] , [ w i +2 , w i +5 ] , [ w i +3 , w i +5 ] , [ w i +4 , w i +5 ] , and take the clique complex to get T i +1 . Note that each flag bistellar 0-moveadds 2 vertices, 6 edges, and 4 faces. Since | V ( T ) | = 4 , | E ( T ) | = 6 , and | F ( T ) | = 4 , thismeans that | V ( T i ) | = 2 i + 4 , | E ( T i ) | = 6 i + 6 , and | F ( T i ) | = 4 i + 4 .Further, Table 1 summarizes the degrees of the vertices in each T i . To compute the degrees T i Degree Vertices T w , w , w , w T w , w , w , w , w , w T w , w , w , w w , w , w , w T i w , w , w i +2 , w i +3 i ≥ w , w , w i , w i +1 w , . . . , w i − Table 1.
Degrees of the vertices in T i .of vertices in T i for i ≥ , observe that when the new vertices w i +2 and w i +3 are added,they have degree in T i . For each of the next two iterations of the flag bistellar-0 move,the degree of these vertices increases by one, resulting in degree 6 in T i +2 . In the remainingtriangulations T j with j ≥ i + 3 , these vertices are not affected. Therefore, maxdeg( T i ) ≤ for each i .From this infinite sequence of flag triangulations of S with bounded degree, we are in-terested in the particular T i with i = ⌊ k − ⌋ to use in our construction of Y , where k is theHamming weight of m as in Notation 3.2. Note that this T i has vertex set { w , . . . , w i +3 } and has ⌊ k − ⌋ + 4 faces. Let δ be the integer ≤ δ ≤ where δ ≡ − k mod 4 . Then T i has exactly k + δ faces. w w w T w w w w w w T w w w w w w w w T Figure 3.
The first few flag triangulations of S using flag bistellar 0-moves.3.2.2. Constructing e T i . Next, we insert square holes in the first k faces of T i and subdividethe remaining faces in such a way that the squares will be vertex disjoint and nonadjacent.First, we will insert square holes in k of the faces of T i , making sure to triangulate theresulting faces and take the clique complex so that our simplicial complex remains flag. Let [ w r , w s , w t ] with r < s < t be the j th of these k faces with respect to a fixed ordering ofthe faces (where j ranges from 1 to k ). We remove this face and subdivide the edges byadding new vertices w ′ r,s , w ′ r,t , and w ′ s,t and new edges [ w r , w ′ r,s ] , [ w s , w ′ r,s ] , [ w r , w ′ r,t ] , [ w t , w ′ r,t ] , [ w s , w ′ s,t ] , and [ w t , w ′ s,t ] . Then, we add vertices u j − , u j − , u j − , and u j − to form a squareinside the original face with indices increasing counterclockwise. Moreover, we add edges [ w r , u j − ] , [ w r , u j − ] , [ u j − , w ′ r,s ] , [ u j − , w ′ r,s ] , [ w s , u j − ][ u j − , w ′ s,t ] , [ u j − , w ′ s,t ] , [ w t , u j − ] , [ u j − , w ′ r,t ] , [ u j − , w ′ r,t ] . After applying this process, we take the clique complex. The result of this operation on face [ w r , w s , w t ] is depicted in Figure 4 (left).The remaining δ faces of T i will simply be subdivided and triangulated before taking theclique complex. Explicitly, this means that after removing the face [ w i +1 , w i +2 , w i +3 ] andits edges, we add vertices w ′ i +1 , i +2 , w ′ i +1 , i +3 , and w ′ i +2 , i +3 and edges [ w i +1 , w ′ i +1 , i +2 ] , [ w i +2 , w ′ i +1 , i +2 ] , [ w i +1 , w ′ i +1 , i +3 ] , [ w i +3 , w ′ i +1 , i +3 ] , [ w ′ i +1 , i +2 , w ′ i +1 , i +3 ] , [ w i +2 , w ′ i +2 , i +3 ] , [ w i +3 , w ′ i +2 , i +3 ] , [ w ′ i +1 , i +2 , w ′ i +2 , i +2 ] , [ w ′ i +1 , i +3 , w ′ i +2 , i +3 ] . This subdivision of face [ w i +1 , w i +2 , w i +3 ] is shown in Figure 4 (right). We do similarly forthe faces [ w i − , w i +2 , w i +3 ] and [ w i , w i +1 , w i +3 ] , if necessary. The clique complex of thisconstruction is a flag complex which is homeomorphic to S with k distinct points removed.Call this complex e T i .Let’s consider the degrees of the vertices of e T i . We have that deg( w ′ m,n ) = 6 for all m, n and deg( u ℓ ) ∈ { , } for all ℓ , where the “top” u ℓ have degree 4 and the “bottom” u ℓ havedegree 5. To determine the degrees of the w j vertices, we need to consider their degrees in T i and how their degrees increase during the subdivision and square face removal processes.As we are interested in bounding the maximum degree of the vertices of e T i , we need onlyconsider the case when δ = 0 and all k faces of T i have a square removed from them. Table2 gives the degrees of each of the w j vertices in e T i when δ = 0 .To verify the degrees of the w j in e T i when i ≥ , we consider how the degrees of thevertices change as i increases. Between e T i − and e T i (with δ = 0 for both), the only vertices r w s w t w ′ r,s w ′ s,t w ′ r,tu j − u j − u j − u j − w i +1 w i +2 w i +3 w ′ i +1 , i +2 w ′ i +2 , i +3 w ′ i +1 , i +3 Figure 4.
Example of square insertion done on k faces of T i (left), and sub-divided triangulation on remaining faces (right). e T i Degree Vertices6 w , w e T w ( k = 4) w w , w e T w , w ( k = 8) w w w , w e T w ( k = 12) w , w w , w , w w i +2 , w i +3 e T i w i ≥ w i , w i +1 ( k = 4 i + 4) w , w , w w , . . . , w i − Table 2.
Degrees of the vertices in e T i when k ≡ .that change degree are w i − , w i − , w i , w i +1 , each of which increase degree by 3. This isbecause they each get one new edge from the T i flag bistellar 0-move and two new edgesfrom the square removal triangulation process (since each vertex is the smallest indexed andhence the “top” vertex of one new triangular face). Further, the new vertices w i +2 , w i +3 in e T i have degree 8, and they increase degree by 3 in the next two iterations, resulting in degree14 in e T i +2 and all future iterations.The above argument shows that regardless of m and k , maxdeg( e T i ) ≤ , where i = ⌊ k − ⌋ .Furthermore, the only vertices that could have degree 14 are w , . . . , w i − , each of which is eparated from the others by a w ′ m,n vertex, which only has degree 6. We want to know exactlywhich vertices in e T i have degree 14, for all possible k with i ≥ , because we plan to alter thesevertices to decrease maxdeg( e T i ) . Note that as δ increases from 0 to 3, the degree of each w j vertex is nonincreasing. When k = 4 i + 4 and δ = 0 , the above table gives that w , . . . , w i − have degree 14. When k = 4 i + 3 and δ = 1 , the face [ w i +1 , w i +2 , w i +3 ] is subdividedinstead of having a square removed, but this does not change the degrees of w , . . . , w i − ,so these all still have degree 14. When k = 4 i + 2 and δ = 2 , the faces [ w i +1 , w i +2 , w i +3 ] and [ w i − , w i +2 , w i +3 ] are subdivided. Therefore, w i − has two fewer edges than in theprevious case since w i − is the smallest indexed vertex in [ w i − , w i +2 , w i +3 ] and so wouldhave two “top” u ℓ adjacent to it if this face had a square removed from it. So, in this case, w , . . . , w i − have degree 14 and w , w , w , w i − have degree 12 in e T i . Finally, if k = 4 i + 1 and δ = 3 , then additionally the face [ w i , w i +1 , w i +3 ] is subdivided, which means that thedegree 12 and 14 vertices are the same as in the previous cases.3.2.3. Replacing degree 14 vertices to construct Y . Having identified the vertices of e T i of thehighest degree, we now describe a process by which we will replace each vertex of degree 14by two vertices of degree 9 in order to ensure that maxdeg( e T i ) ≤ for all k and i . Theresulting flag complex, given by taking the clique complex of this construction, will be thefinal Y , and it will be homeomorphic to e T i . The process is summarized by Figure 5 anddescribed in detail in the following paragraphs.Suppose w j is a vertex of degree 14 in e T i . Locally, on a small neighborhood of w j , e T i ishomeomorphic to a -manifold. Since deg( w j ) = 14 , w j is surrounded by six triangular facescoming from T i , all of which have had a square removed. By our construction, two of thesesquares (which are in adjacent triangular faces) have both of their “top” u ℓ vertices connectedto w j , but the other four squares just have a single edge connecting one of their “bottom” u ℓ vertices to w j . So, w j has six w ′ m,n neighbors and eight u ℓ neighbors, which form a 14-sidedpolygon with w j as its “star” point. Choose two w ′ m,n vertices which are across from each otherin this 14-sided polygon, say w ′ a,b and w ′ c,d . Next, we will remove w j and all of the 14 faces thatit is contained in. Then, we add vertices w j and w j in place of w j and add edges in such away that deg( w j ) = deg( w j ) = 9 , there are edges [ w j , w j ] , [ w j , w ′ a,b ] , [ w j , w ′ c,d ] , [ w j , w ′ a,b ] , and [ w j , w ′ c,d ] , and the 14-sided polygon is triangulated with 16 triangles. This processonly changes the degree of w ′ a,b and w ′ c,d , each of which now have degree 7. Therefore, themaximum degree of w j , w j , and the 14 vertices in the polygon is 9 (since deg( u ℓ ) ∈ { , } and deg( w ′ m,n ) = 6 ). To illustrate this construction, we consider the case when k = 20 . Then i = 4 , δ = 0 , and deg( w ) = 14 in e T . Figure 5 depicts this process when w ′ a,b = w ′ , and w ′ c,d = w ′ , .After repeating the above process for each degree 14 vertex in e T i , we take the cliquecomplex and call the resulting flag complex Y . Observe that this process increases thenumber of vertices by 1, the number of edges by 3, and the number of faces by 2 each timea degree 14 vertex in e T i is replaced. Also, note that maxdeg( Y ) ≤ for all m .Now, we give the w j , w ′ m,n , and u ℓ vertices their natural orderings and say that w ′ m,n > w j and w ′ m,n > u ℓ for all ℓ, m, n, and j , and then let these vertex orderings induce orientationson the edges and faces of Y (as shown in Figure 3). Counting the vertices, edges, andfaces of Y we have that if ≤ k ≤ , then there were no degree 14 vertices to remove, so | V ( Y ) | = 6 k + 2 δ + 2 , | E ( Y ) | = 17 k + 6 δ , and | F ( Y ) | = 10 k + 4 δ . If k ≥ , then i ≥ w w w w w w w ′ , w ′ , w ′ , w ′ , w ′ , w ′ , w w w w w w w w w ′ , w ′ , w ′ , w ′ , w ′ , w ′ , Figure 5.
Replacing a degree 14 vertex in e T when k = 20 .and at least one degree 14 vertex was removed to construct Y from e T i . Table 3 gives thenumber of vertices, edges, and faces of Y for all values of k ≥ . k δ | V ( Y ) | | E ( Y ) | | F ( Y ) | i + 4 k − k −
18 11 k − i + 3 k −
32 372 k − k − i + 2 k k − k − i + 1 k +
52 372 k + k + 1 Table 3.
Number of vertices, edges, and faces in Y when k ≥ .3.2.4. Homology of Y . Since Y is an oriented flag triangulation of S with k square holes,each of which are vertex disjoint and nonadjacent, our Y is homeomorphic to Newman’s Y in the d = 2 case of [29, Lemma 5.7], and we can apply the same argument to compute thehomology of Y . We denote the 1-cycles that are the boundaries of the k square holes by τ , . . . , τ k . Explicitly, for j = 1 , . . . , k , we define τ j := [ u j − , u j − ] + [ u j − , u j − ] + [ u j − , u j − ] − [ u j − , u j − ] . Then, by our construction, each τ j is a positively-oriented 1-cycle in H ( Y ) , and exactly asin [29, Proof of Lemma 5.7], we have that H ( Y ) = h τ , . . . , τ k | τ + · · · + τ k = 0 i . Construction of X and proof of Theorem 3.1. Now we attach Y and Y togetherto form the two-dimensional flag complex X such that the torsion subgroup of H ( X ) isisomorphic to Z /m Z . This part essentially follows [29, §3], though we must confirm that theresulting complex is flag and satisfies the desired bound of vertex degree. Proof of Theorem 3.1.
For a given m , let Y and Y be the complexes constructed in the previ-ous subsections. Let S denote the subcomplex of Y induced by the k vertices u , . . . , u k − . ince the square holes in Y are vertex-disjoint and have no edges between any two of them, S is a disjoint union of k square boundaries. Let f : S → Y be the simplicial map defined,for j = 1 , . . . , k , by u j − v n j , u j − v n j +1 , u j − v n j +2 , u j − v n j +3 . Following [29, §3], let X = Y ⊔ f Y and observe that this is a simplicial complex by thesame argument as Newman gives. In addition, X is a flag complex because Y and Y areflag, and we subdivided the edges of Y and Y to avoid the possibility that X might containa 3-cycle which doesn’t have a face. Furthermore, in X the squares τ j and γ n j are identifiedby f for j = 1 , . . . , k , and, as in [29], H ( X ) ∼ = Z k − ⊕ Z /m Z , where Z /m Z has the repeated squares representation given by h γ , γ , . . . , γ n k | γ = γ , γ = γ , . . . , γ n k − = γ n k , γ n + · · · + γ n k = 0 i . Finally, using our counts for the number of vertices, edges, and faces of Y and Y and with δ defined as above, we have | V ( X ) | = 2 k + 12 n k + 6 + 2 δ, | E ( X ) | = 13 k + 40 n k + 4 + 6 δ, and | F ( X ) | = 10 k + 28 n k + 4 δ. If k ≥ , then Table 4 gives the number of vertices, edges, and faces in X (where i = ⌊ k − ⌋ ). k δ | V ( X ) | | E ( X ) | | F ( X ) | i + 4 k + 12 n k k + 40 n k −
14 11 k + 28 n k − i + 3 k + 12 n k +
52 292 k + 40 n k − k + 28 n k − i + 2 k + 12 n k + 4 k + 40 n k − k + 28 n k − i + 1 k + 12 n k +
132 292 k + 40 n k + k + 28 n k + 1 Table 4.
Number of vertices, edges, and faces in X when k ≥ .Additionally, recall that maxdeg( Y ) ≤ and maxdeg( Y ) ≤ . Since in X we are onlyidentifying the squares of Y with k of the squares of Y , to find the maximum degree of anyvertex of X , we need only check the degrees of the identified vertices. In Y , we know that deg( v j ) ≤ for each j , and in Y , we know that deg( u ℓ ) ∈ { , } for each ℓ . Let v j and u ℓ be vertices that are identified in X . Since two of their adjacent edges in the squares areidentified as well, in X we see that deg( v j ) = deg( u ℓ ) ≤ . Thus, maxdeg( X ) ≤ . (cid:3) We also note the following corollary:
Corollary 3.3.
For every finite abelian group G there is a two-dimensional flag complex X such that the torsion subgroup of H ( X ) is isomorphic to G and maxdeg( X ) ≤ .Proof. Let G = Z /m Z ⊕ Z /m Z ⊕ · · · ⊕ Z /m r Z with m | m | · · · | m r be an arbitrary finiteabelian group. By Theorem 3.1, there exist two-dimensional flag complexes X m i such thatthe torsion subgroup of H ( X m i ) is isomorphic to Z /m i Z and maxdeg( X m i ) ≤ . If X isthe disjoint union of all the X m i , then X satisfies the hypotheses of the corollary. (cid:3) . Appearance of subcomplexes in ∆( n, p ) The goal of this section is to show that, for attaching probabilities p in an appropriaterange, the flag complex X m from Theorem 3.1 will appear with high probability as an inducedsubcomplex of ∆( n, p ) . See §2 for the relevant definitions and notation used throughout thissection. Here is our main result: Proposition 4.1.
Fix m ≥ , and let X m be as in Theorem 3.1. If ∆ ∼ ∆( n, p ) is a randomflag complex with n − / ≪ p ≤ − ǫ for some ǫ > , then P (cid:20) X m ind ⊂ ∆( n, p ) (cid:21) → as n → ∞ . Our proof of this result will rely on Bollobás’s theorem on the appearance of subgraphs ofa random graph, which we state here for reference.
Theorem 4.2 (Bollobás [5]) . Let G ′ be a fixed graph, let m ( G ′ ) be the essential density of G ′ defined in Definition 2.2, and let G ( n, p ) be the Erdős-Rényi random graph on n verticeswith attaching probability p . As n → ∞ , we have P [ G ′ ⊂ G ( n, p )] → ( if p ≪ n − /m ( G ′ ) if p ≫ n − /m ( G ′ ) . Since any flag complex is determined by its underlying graph, we can almost apply thisto prove Proposition 4.1. However, Proposition 4.1 (and our eventual application of itvia Hochster’s formula to Theorem 1.3) requires X m to appear as an induced subcomplex,whereas Bollobás’s result is for not necessarily induced subgraphs. The following proposition,which is likely known to experts, shows that so long as p is bounded away from , thisdistinction is immaterial in the limit. Proposition 4.3.
Let G ′ be a fixed graph, let m ( G ′ ) be the essential density of G ′ defined inDefinition 2.2, and let G ( n, p ) be the Erdős-Rényi random graph on n vertices with attachingprobability p . Suppose p = p ( n ) ≤ − ǫ for some ǫ > . Then as n → ∞ , we have P (cid:20) G ′ ind ⊂ G ( n, p ) (cid:21) → ( if p ≪ n − /m ( G ′ ) if p ≫ n − /m ( G ′ ) . Proof.
Since an induced subgraph is a subgraph, if P [ G ′ ⊂ G ( n, p )] → , then P (cid:20) G ′ ind ⊂ G ( n, p ) (cid:21) → . Thus, the first half of the threshold is a direct consequence ofTheorem 4.2, and all that needs to be shown is the second half of the threshold.Suppose that p ≫ n − /m ( G ′ ) . We will mirror the proof of Bollobàs’s theorem from [20,Theorem 5.3] (originally due to [33]), which relies on the second moment method. Let Λ( G ′ , n ) be the set containing all of the possible ways that G ′ can appear as a inducedsubgraph of G ( n, p ) . Thus, an element H ∈ Λ( G ′ , n ) corresponds to a subset of the n vertices and specified edges among those vertices such that the resulting graph is a copy of G ′ . We want to count the number of times G ′ appears as an induced subgraph of G ( n, p ) .For each H ∈ Λ( G ′ , n ) , we let H be the corresponding indicator random variable, where H = 1 occurs in the event that restricting G ( n, p ) to the vertices of H is precisely thecopy of G ′ indicated by H . Note that the random variables H are not independent, as twodistinct elements from Λ( G ′ , n ) might have overlapping vertex sets. If we let N G ′ be the andom variable for the number of copies of G ′ appearing as induced subgraphs in G ( n, p ) ,then we have N G ′ = X H ∈ Λ( G ′ ,n ) H . Our goal is to show that P [ N G ′ ≥ → , or equivalently that P [ N G ′ = 0] → . Since N G ′ is non-negative, the second moment method as seen in [1, Theorem 4.3.1] states that P [ N G ′ = 0] ≤ Var( N G ′ ) E [ N G ′ ] , so it suffices to show that Var( N G ′ ) E [ N G ′ ] → . To start, we will boundthe expected value. To simplify notation throughout the following computation, we let v = | V ( G ′ ) | and e = | E ( G ′ ) | denote the number of vertices and edges of G ′ . E [ N G ′ ] = X H ∈ Λ( G ′ ,n ) E [ H ]= X H ∈ Λ( G ′ ,n ) p e (1 − p )( v ) − e = Ω( n v ) · p e (1 − p )( v ) − e . Now let us repeat this with the variance instead.
Var( N G ′ ) = X H,H ′ ∈ Λ( G ′ ,n ) E [ H H ′ ] − E [ H ] E [ H ′ ]= X H,H ′ ∈ Λ( G ′ ,n ) P [ H = 1 and H ′ = 1] − P [ H = 1] P [ H ′ = 1]= X H,H ′ ∈ Λ( G ′ ,n ) P [ H = 1] ( P [ H ′ = 1 | H = 1] − P [ H ′ = 1])= p e (1 − p )( v ) − e X H,H ′ ∈ Λ( G ′ ,n ) P [ H ′ = 1 | H = 1] − P [ H ′ = 1] If H and H ′ don’t share at least two vertices, H and H ′ are independent of each other. Sowe can restrict to the case where they share at least two vertices, which gives = p e (1 − p )( v ) − e v X i =2 X H,H ′ ∈ Λ( G ′ ,n ) | V ( H ) ∩ V ( H ′ ) | = i P [ H ′ = 1 | H = 1] − P [ H ′ = 1] . We now come to the key observation, which is also at the heart of the proof in [20, Theo-rem 5.3]: P [ H ′ = 1 | H = 1] is maximized if those edges and non-edges in H are exactlythose that are required by H ′ . Thus, by applying the fact that any subgraph of G ′ with i vertices, has at most i · m ( G ′ ) edges and at most (cid:0) i (cid:1) non-edges we get the following boundfor H, H ′ ∈ Λ( G ′ , n ) sharing i vertices: P [ H ′ = 1 | H = 1] ≤ P [ H ′ = 1] · p − i · m ( G ′ ) (1 − p ) − ( i ) rom here, it is a standard computation. Substituting this back into the previous equationand simplifying, we get Var( N G ′ ) ≤ p e (1 − p )( v ) − e v X i =2 X H,H ′ ∈ Λ( G ′ ,n ) | V ( H ) ∩ V ( H ′ ) | = i P [ H ′ = 1] (cid:16) p − i · m ( G ′ ) (1 − p ) − ( i ) − (cid:17) ≤ (cid:16) p e (1 − p )( v ) − e (cid:17) v X i =2 O (cid:0) n v − i (cid:1) (cid:16) p − i · m ( G ′ ) (1 − p ) − ( i ) − (cid:17) . And since p is bounded away from and − p is bounded away from , we get ≤ (cid:16) p e (1 − p )( v ) − e (cid:17) v X i =2 O (cid:16) n v − i p − i · m ( G ′ ) (cid:17) . Finally, applying the second moment method gives P [ N G ′ ≤ ≤ Var( N G ′ ) E [ N G ′ ] = v X i =2 O (cid:16) n v − i p − i · m ( G ′ ) (cid:17) Ω( n v ) = v X i =2 O (cid:16) n − i p − i · m ( G ′ ) (cid:17) . Since p ≫ n − /m ( G ′ ) , we conclude that np m ( G ′ ) → ∞ , and therefore, P [ N G ′ = 0] → . Itfollows that P (cid:20) G ′ ind ⊂ G ( n, p ) (cid:21) → . (cid:3) We now turn to the proof of Proposition 4.1.
Proof of Proposition 4.1.
Recall that X m is the complex from Theorem 3.1, and let H m beits underlying graph. Moreover, the underlying graph of ∆( n, p ) is the Erdős-Rényi randomgraph G ( n, p ) . Since a flag complex is uniquely determined by its 1-skeleton, it suffices toshow that P (cid:20) H m ind ⊂ G ( n, p ) (cid:21) → .Since maxdeg( H m ) ≤ , every subgraph has average degree at most . Thus, theessential density m ( H m ) satisfies m ( H m ) ≤ . Since p ≫ n − / , we have p ≫ n − /m ( H m ) .Applying Proposition 4.3 gives P (cid:20) H m ind ⊂ G ( n, p ) (cid:21) → ; thus, P (cid:20) X m ind ⊂ ∆( n, p ) (cid:21) → . (cid:3) Remark . Explicitly computing the essential density m ( H m ) seems difficult in general,and our chosen bound m ( H m ) ≤ , which is determined by the fact that maxdeg( X m ) ,is likely too coarse. It would be interesting to see a sharper result on m ( H m ) , as this couldpotentially provide an heuristic for decreasing the bound on r in Conjecture 1.1. Might iteven be the case that m ( H m ) is half the average degree, avg( H m ) ?In any case, avg( H m ) at least provides a lower bound on m ( H m ) . Due to the detailednature of the constructions in §3, we can estimate this value. Let k ≥ and m ≫ sothat n k = ⌊ log ( m ) ⌋ will be much larger than δ . By Table 4, the number of vertices will beapproximately k + 12 n k and the number of edges will be approximately k + 40 n k . Thesmallest the ratio of edges to vertices can be is when n k ≫ k , in which case the ratio will beapproximately . A similar computation holds for k ≤ and for m ≫ . We can concludethat m ( H m ) ≥ − ǫ , where ǫ is a positive constant that goes to as m → ∞ . (cid:3) v v v v v v v v v v v v v v v Figure 6.
A minimal flag triangulation of R P , denoted by ∆( G ) .5. A detailed analysis of 2-torsion
The goal of this section is to provide a more detailed analysis of what happens in the caseof 2-torsion (when m = 2 in Proposition 4.1). In [13], Costa, Farber, and Horak analyzethe -torsion of the fundamental group of ∆( n, p ) . Their results, specifically Theorem 7.2,give that if n − / ≪ p ≪ n − / − ǫ where < ǫ < is fixed, then the probability that H (∆( n, p )) has -torsion tends to 1 as n → ∞ . Since our aim is to show that there is -torsion with high probability in the homology of an induced subcomplex of ∆( n, p ) , ratherthan in the global homology, we are able to extend the threshold to n − / ≪ p ≤ − ǫ where ǫ > . We use the same techniques as in §4, but instead of using X from Theorem3.1, we use a known flag triangulation of R P that minimizes the number of vertices andwhere we can easily compute its essential density. This gives the less restrictive threshold of p ≫ n − / in the -torsion case as opposed to p ≫ n − / in the general case. In [4, Figure1], the authors find two (nonisomorphic) minimal flag triangulations of R P , each of whichhave 11 vertices and 30 edges and differ by a single bistellar 0-move; one of these is usedin [13], and the other, which we use in this section, is depicted in Figure 6.For the remainder of this section, let G denote the underlying graph of this flag triangu-lation of R P , which we denote by ∆( G ) as it is the clique complex of G . To understandthe probability that this particular triangulation of R P appears as an induced subcomplexof ∆( n, p ) , we need to compute the essential density m ( G ) . Lemma 5.1.
For the graph G underlying the flag triangulation of R P exhibited in Figure 6,the essential density m ( G ) is / .Proof. This amounts to an exhaustive computation, which is summarized in Table 5. Inparticular, Table 5 identifies the maximal number of edges that a subgraph H ⊂ G on | V ( H ) | vertices can have, for each | V ( H ) | ≤ . One can see from the table that m ( G ) ismaximized by the entire graph, and thus m ( G ) = | E ( G ) | / | V ( G ) | = 30 / . (cid:3) Lemma 5.1 shows that the graph G is strongly balanced in the sense of Definition 2.2.While we expect the essential density of our complexes X m to be lower than the coarse bound V ( H ) | max {| E ( H ) |} V ( H ) max n | E ( H ) || V ( H ) | o { v }
02 1 { v , v } { v , v , v }
14 5 { v , v , v , v } { v , v , v , v , v } { v , v , v , v , v , v } { v , v , v , v , v , v , v } { v , v , v , v , v , v , v , v } { v , v , v , v , v , v , v , v , v }
10 25 { v , v , v , v , v , v , v , v , v , v }
11 30 { v , . . . , v } Table 5.
With G as the underlying graph of the complex in Figure 6, thistable computes the maximal number of edges of subgraphs H ⊂ G with varyingnumber of vertices.of maxdeg( X m ) (see Remark 4.4), we note that in the case of the graph G , this differenceis not very large. In fact, we have maxdeg( G ) = 3 and m ( G ) = 30 / ≈ . .Combining Lemma 5.1 and Theorem 4.2 we obtain an analogue of Proposition 4.1. Proposition 5.2. If ∆ ∼ ∆( n, p ) is a random flag complex with n − / ≪ p ≤ − ǫ forsome ǫ > , then P (cid:20) ∆( G ) ind ⊂ ∆( n, p ) (cid:21) → as n → ∞ .Proof. The proof is nearly identical to that of Proposition 4.1, so we omit the details. (cid:3)
Question 5.3.
It would be interesting to know whether p ≪ n − / is a sharp threshold forthe appearance of 2-torsion in the homology of any induced subcomplex of ∆( n, p ) . WhileCosta, Farber, and Horak show in [13, Theorem 7.1] that the global homology has no torsionif p ≪ n − / , it is possible that some induced subcomplex of ∆( n, p ) has -torsion. Aclosely related question is whether there exists a flag complex X with 2-torsion homology anda smaller essential density. Torsion in the Betti tables associated to ∆ We now prove Theorem 1.3. The hard work was done in the previous sections.
Proof of Theorem 1.3.
Assume n − / ≪ p ≤ − ǫ and let ∆ ∼ ∆( n, p ) . Let X m be as inTheorem 3.1. By Proposition 4.1, ∆ contains X m as an induced subcomplex, with highprobability. Since H ( X m ) has m -torsion, Hochster’s Formula (see Fact 2.3) gives that theBetti table of the Stanley–Reisner ideal of ∆ has ℓ -torsion for every ℓ dividing m . (cid:3) e can also apply the more detailed study of -torsion from §5 to obtain a result on theappearance of -torsion in the Betti tables of random flag complexes. Proposition 6.1.
Let ∆ ∼ ∆( n, p ) be a random flag complex with n − / ≪ p ≤ − ǫ forsome ǫ > . With high probability as n → ∞ , the Betti table of the Stanley–Reisner ideal of ∆ has -torsion. In particular, this holds where n − / ( r − ≪ p ≪ n − /r for any r ≥ .Proof. The proof is the same as the proof of Theorem 1.3, but utilizing Proposition 5.2 inplace of Proposition 4.1. Further, n − / ≪ p gives that n − / ( r − ≪ p for any r ≥ . (cid:3) Note that the bound on r for the appearance of -torsion in Proposition 6.1 is lower thanin Theorem 1.3. See Question 7.3 and Remark 7.4 for more on the possibility of loweringthe bound on r in Theorem 1.3. It would be interesting to understand a precise thresholdon the attaching probability p such that the Betti table of the Stanley–Reisner ideal of ∆ does not depend on the characteristic. A related question is posed in Question 7.3. Remark . Our constructions are based entirely on torsion in the H -groups, and thus weobtain Betti tables where the entries in the second row of the Betti table (that is the row ofentries of the form β i,i +2 ) depend on the characteristic. Since Newman’s work also producessmall simplicial complexes where the H i -groups have torsion, for any i ≥ [29, Theorem 1],one could likely apply the methods of §3 to produce thresholds for where the other rows ofthe Betti table would depend on the characteristic, and it might be interesting to explorethe resulting thresholds. 7. ℓ -torsion in Veronese syzygies Finally, we return to the question of ℓ -torsion in Veronese syzygies. Since there is verylittle computational evidence either in favor or in opposition to Conjecture 1.1, we base theconjecture upon an heuristic model.As noted in the introduction, one of the central results of [19] is that for ∆ ∼ ∆( n, p ) with n − / ( r − ≪ p ≪ n − /r and S = k [ x , . . . , x n ] , the Betti table β ( S/I ∆ ) as n → ∞ will exhibitcertain known nonvanishing properties of the Betti table of the Veronese embeddings β ( P rk ; d ) as d → ∞ . Based on this connection, we use Theorem 1.3 as an heuristic for understandingthe behavior of β ( P r ; d ) : in particular, when these Betti tables depend on the characteristic.For Conjecture 1.1, we set r ≥ based on the corresponding hypothesis of Theorem 1.2.With these hypotheses, as n → ∞ , the Betti table associated to ∆ will depend on the char-acteristic with high probability. We thus conjecture a corresponding statement for β ( P r ; d ) with r ≥ as d → ∞ . While we conjecture that this dependence on characteristic shouldbe quite widespread, the only known examples of such behavior come from [2] and [25]. Itwould thus be very interesting to produce any new examples (or non-examples!) of torsionin Veronese syzygies. Question 7.1.
Can one find any new examples of Veronese embeddings whose Betti tablesdepend on the characteristic? For a given ℓ , can one produce a Betti table with ℓ -torsion? We find it especially surprising that there are no known examples of -torsion.Conjecture 1.5 represents one way to sharpen Conjecture 1.1. In particular, since Theo-rem 1.3 shows that, with r ≥ and within the given framework, m -torsion appears with highprobability as n → ∞ in the Betti table of the Stanley–Reisner ideal of ∆ , we conjecture that m -torsion should appear frequently in the Betti tables of the d -uple Veronese embeddings or P r as d → ∞ . There are many follow-up questions one might ask, and we assemble someof these below. Question 7.2.
What is the minimal value of r such that β ( P r ; d ) depends on the character-istic for some d ? (It is known that < r ≤ .) To develop an heuristic for this question, along the lines of this paper, one would need toconsider the following question, which seeks to sharpen Theorem 1.3.
Question 7.3.
Fix m ≥ . For a random flag complex ∆ ∼ ∆( n, p ) , what is the thresholdon p such that the Betti table of the Stanley–Reisner ideal of ∆ has m -torsion with highprobability as n → ∞ ? A closely related result is [13, Theorem 8.1], which implies that for any given odd prime ℓ , the Betti table of the Stanley–Reisner ideal of ∆ (with high probability as n → ∞ ) hasno ℓ -torsion when p ≪ n − / − ǫ where ǫ > is fixed. Following the philosophy of this paper,this suggests that finding ℓ -torsion for Veronese embeddings might be more difficult for P r when r ≤ . Remark . We know of two natural ways that one could improve the bound on r inTheorem 1.3. First, one could perform a more detailed study of the essential density m ( H m ) ,as that value is surely lower than our chosen bound maxdeg( X m ) . Second, one could aim toproduce flag complexes X ′ m with torsion homology (not necessarily in H ) whose underlyinggraphs have a lower essential density than that of X m . Of course, following the heuristicat the heart of this paper, any such improvement of the bound on r in Theorem 1.3 wouldsuggest a corresponding improvement of the bound on r in Conjectures 1.1 and 1.5.In a different direction, one might ask about how large n needs to be before we expect tosee that the Betti table associated to ∆ has ℓ -torsion. Question 7.5.
Fix a prime ℓ and integer r ≥ . Let ∆ ∼ ∆( n, p ) be a random flag complexwith n − / ( r − ≪ p ≪ n − /r . For a constant < ǫ < , approximately how large does n needto be to guarantee that P [ Betti table associated to ∆ has ℓ -torsion ] ≥ − ǫ ? It would be interesting to even answer this question for -torsion, where the thresholdsfrom [13, Theorems 7.1 and 7.2] make the question seemingly quite tractable. The corre-sponding question for Veronese embeddings would be the following: Question 7.6.
Fix a prime ℓ and integer r ≥ . Can one provide lower/upper bounds onthe minimal value of d such that β ( P r ; d ) has ℓ -torsion? We could turn to even more quantitative questions related to Conjecture 1.5 as well.
Question 7.7.
Fix a prime ℓ and an integer r ≥ . Can one describe the set of d ∈ Z suchthat β ( P r ; d ) has ℓ -torsion? Can one bound or estimate the density of that set? Question 7.8.
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