Componentwise Linearity of Stanley-Reisner Ring of Broken Circuit Complexes
aa r X i v : . [ m a t h . A C ] D ec Componentwise Linearity of Stanley-Reisner Ring of Broken CircuitComplexes
Mohammad Reza Rahmati ∗†‡§
December 29, 2020
Abstract
We present new conditions that characterize the componentwise linearity of Stanley-Reisnerideal of broken circuit complexes of simple matroids. We employ the notion of graded linearity(called also quasilinearity in the literature), which is more general than having linear resolutionand show that componentwise linearity is equivalent to have graded linear resolution. As a resultgeneralizing the one in [32], we prove that the componentwise linearilty of Stanley-Reisner idealof broken circuit complex translates to the decomposition of the matroid into specific matroids.An application to the Koszul property of Orlik-Terao ideal of hyperplane arrangements is given.We end up with some partial results on Koszul property and complete intersection of Orlik-Teraoalgebra of central arrangements.
We provide a criteria for componentwise linearity of Stanley-Reisner ideal of broken circuit complexof matroids. Our strategy generalizes that in [32] where similar criteria is given for linear resolu-tions. We begin by an argument on graded linear resolutions (called quasi-linear resolution in theliterature) and prove that this property is equivalent to componentwise linearity [19], [9], [15], [28].This notion is more general than having linear resolution. In fact having linear resolution for agraded module M over a polynomial ring corresponds to having the nonzero graded Betti numbersjust in one horizontal line in the graded Betti diagram. In the case of graded linear resolution thenonzero graded Betti numbers line in several horizontal lines, and in a connected way on each line.As an application we provide a similar assertions for componentwise linearity of Orlik-Terao idealof hyperplane arrangements.We shall consider finitely generated modules M over the polynomial ring, A = k [ x , . . . , x n ]where k is an infinite field. A relevant notion to linear resolution property of M is that of havinga linear quotients, [3], [17], [23]. This notion is defined via the successive colon ideals constructedfrom a minimal set of generators. It is well known that the latter implies the linearity, [[17] section8]. A generalization is a graded version of the two concepts. These concepts are replacements of the ∗ Mathematics Subject classification: 05E40, 05E45 † Keywords: Linear resolution, Componentwise Linearity, Stanley-Reisner ideal, Broken Circuit Complex, Orlik-Terao Ideal, Hyperplane Arrangements. ‡ Universidad De La Salle Baj´ıo, Campestre - Le´on, Guanajuato, Mexico. § email: [email protected] f -vectors, see [6]. In fact the decomposition characterizes the f -vector to be minimal.We shall consider the broken circuit complex of simple matroid. The broken circuit complex ofan ordered matroid M is a Shellable complex where it shows the Stanley-Reisner ring A/I < ( M ) isCohen-Macaulay [23]. The Stanley-Reisner ideal I < ( M ) is generated by monomials on the brokencircuits of M . They are crucial in combinatorial algebra, [8], [26], [7]. The theory of matroidsprovide a systematic mechanism to understand hyperplane arrangements, [29], [30], [21], [13]. Wespecifically check out different criterias equivalent to the linearity and also the graded linearity ofStanley-Reisner ring of broken circuit complexes.We also discuss Koszul property of Stanley-Reisner ring of the broken circuit complex, as wellas the complete intersection property of Orlik-Terao ideal of hyperplane arrangements, [9], [32],[15], [27]. We define an extension of Koszulness namely graded Koszulness and characterize thisconcept via the decomposition of the associated matroid. The reader may consider it for furtherinvestigations, see [32], [9].Finally we present an application to graphs and their cycle matroids. Specifically we discussthe complete intersection property of facet ideal of r -cyclic graph G n,r . The properties of the facetideal I F (∆) has been discussed in [4], [14], [24], [31], [5]. Our method gives a simple proof of theCohen-Macaulayness of the facet ideal of G n,r . This property was proved in [4] by direct methods.Our proof is based on the criterias characterizing the complete intersection properties on the cyclematroid of a graph. Let A = K [ x , . . . , x n ] be the polynomial ring over an infinite field k and m = ( x , . . . , x n ) themaximal ideal. A finitely generated graded A -module M is said to have s -linear resolution if thegraded minimal free resolution of M is of the form0 −! A ( − s − p ) β p −! . . . −! A ( − s − β −! A ( − s ) β −! M −! β ij of a graded module M are the exponents appearing in a gradedminimal free resolution 2 . . −! M j A ( − j ) β p,j −! . . . −! M j A ( − j ) β j −! M −! β ij = dim k T or i ( M, k ) j . For instance M has s -linear resolution if β i,i + j = 0 for j = s . A relevant notion is that of s -linear quotient (see [17, 3]). The next theoremcharacterizes linear resolution property. Theorem 2.1. [32] Assume I = L k I k is a graded Cohen-Macaulay ideal in A of codimension q and s is the smallest index that I s = 0 . The followings are equivalent • I has s -linear resolution. • If a , . . . , a n − q is a maximal ( A/I ) -regular sequence of linear forms in A we have I = m s in A/ ( a , . . . , a n − q ) . • The Hilbert function of I has the form H ( I, s ) = (cid:0) s + q − s (cid:1) . • Suppose < is a monomial order on A . If in < ( I ) (the initial ideal of I ) is Cohen-Macaulay,then in < ( I ) has s -linear resolution. Fix a monomial order < on A . If f = P v a v v where v ∈ A is a monomial, is a nonzeropolynomial of A , the initial monomial of f with respect to < is the biggest monomial with respectto < among the monomials belonging to supp ( f ). We write in < ( f ) for the initial monomial of f with respect to < . The leading coefficient of f is the coefficient of in < ( f ). If I is a nonzero idealof A , the initial ideal of I with respect to < is the monomial ideal of A which is generated by { in < ( f ) : f ∈ I } . We write in < ( I ) for the initial ideal of I . If I is a graded ideal of A and in < ( I )is Cohen-Macaulay, then from the above Theorem follows that I has linear resolution if and onlyif in < ( I ) has linear resolution, cf. [32] loc. cit. Definition 2.2.
Say M has linear quotient, if there a exists minimal system of generators a , . . . , a m of M such that the colon ideals J l = ( a , . . . , a l − ) : a l (2.3)are generated by linear forms of all l . Theorem 2.3. ([17] prop. 8.2.1) Suppose that I is a graded ideal of A generated in degree s , If I has linear quotients, then it has a s -linear resolution. Let (
M, < ) be an ordered simple matroid of rank r on [ n ]. The minimal dependent subsets of M are called circuits. Denote by BC < ( M ) the broken circuit complex of M , i.e the collection of allsubsets of M not containing any broken circuit. It is known that, BC < ( M ) is an ( r − BC < ( M ) denoted I < ( M ) is the ideal generated byall monomials on broken circuits. We denote a broken circuits by bc < ( C ) = C \ min < ( C ), [32]. Bydefinition I < ( M ) = (cid:10) x bc < ( C ) := Y j ∈ bc < ( C ) x j (cid:11) (2.4)The shellability of BC < ( M ) implies that A/I < ( M ) is Cohen-Macaulay. The linear resolution ofthe ideal I < ( M ) is characterized by 3 heorem 2.4. [32] The following are equivalent • I < ( M ) has s -linear resolution. • M = U s,n − r + s L U r − s,r − s . • All powers of I < ( M ) have linear resolutions. The matroid U p,n is the uniform matroid on n -element, whose independent sets are subsets havingat most p element. U p,n has rank p and its circuits have p + 1 elements. U n,n has no dependentsets, called free. The proof of the Theorem uses the assertions of the Theorem 2.1 and an extermalbound for the f -vectors, see [32]. Theorem 2.5. [6](1) Let ∆ be an r − dimensional shellable complex on the vector set S of size | S | = n . If forsome s ∈ N , all s -element subsets of S belong to ∆ , then the inequality f k − ≥ s − X i =0 (cid:18) n − r + i − i (cid:19)(cid:18) r − ik − i (cid:19) , k = 0 , . . . , r (2.5) holds for the f -vector.(2) In case ∆ = In ( M ) the complex of independent subsets of a loopless matroid, The sameinequality holds for f k replaced with the I k the number of independent k -element subsets of M . The equality holds if and only if M = U s,n − r + s M U r − s,r − s , s + 1 = min {| C | ; C is a circuit } (2.6)We sketch some explanations from [6]. The condition that, all s -element subsets of S are in ∆,implies that f k = (cid:18) nk (cid:19) , k = 0 , , , , . . . , s (2.7)Then by using formal combinatorial identities, h k = k X i =0 ( − i + k (cid:18) r − ik − i (cid:19)(cid:18) ni (cid:19) = (cid:18) n − r + k − k (cid:19) , k ≤ s (2.8)Therefore the inequality in (2.3) follows by singling out the first s + 1 terms in f k = k X i =0 h i (cid:18) r − ik − i (cid:19) , k = 1 , . . . , r (2.9)In case ∆ = In ( M ), let b ( M ) = I r , then b ( M ) = (cid:18) n − r + ss (cid:19) (2.10)4s the number of bases of M . The h -vector is given by h = (cid:18) , n − r, . . . , (cid:18) n − r + s − s (cid:19) (cid:19) (2.11)and the Tutte polynomial of M is T M ( x,
1) = h ∆ ( x ) = x r + ( n − r ) x r − + · · · + (cid:18) n − r + s − s − (cid:19) x r − s (2.12)The form of the Tutte polynomial shows that, M has exactly n − r + s coloops. This forces M = N ⊕ U r − s,r − s where N is a matroid of rank s on n − r + s points. Now b ( N ) = b ( M ) = (cid:18) n − r + ss (cid:19) (2.13)This forces that N = U s,n − r + s . The equality in (2.3) holds, if h c = h c +1 = · · · = 0 (2.14)cf. [6] loc. cit. In this section we extend some of the main results of [32] to the corresponding graded notions.Some of the methods of the proofs are also generalizing those in the reference also. Thus thesimilarity to the linear case should not make confusions. In fact we stress that having linearproperty for the graded modules under consideration is a strictly special case.Assume M = L j M j is a finitely generated graded A -module and let( F , d ) = (( F n ) , d n : F n ! F n − ) (3.1)be the minimal graded free resolution of M . Define subcomplexes F ( i ) := (cid:0) m i − n F n , d n (cid:1) (3.2)The linear part of F is the complex F lin = (cid:0) M i F ( i ) / F ( i +1) , gr ( i ) ( d ) (cid:1) (3.3) Definition 3.1.
The graded module M = ⊕ j M j is said to be componentwise linear if M h s i haslinear resolution for all s , where by definition M h s i is the submodule generated by M s . Theorem 3.2. [19], [9] The followings are equivalent • M is componentwise linear. • M h q i has a linear resolution for every q . F lin is acyclic. In fact one can choose homogeneous basis for the A -graded modules F ( i ) so that the differentialsare presented by matrices with homogeneous entries. Then the differential of the complex F lin isobtained by killing all the monomials of degrees bigger than one in the enteries. The resultingmatrix is called the linearization of differentials. Definition 3.3.
We say M has graded linear quotient, if there exists a minimal system of generators a , . . . , a m of M such that the colon ideals satisfy indeg ( J l ) = 1.We call the least degree of a homogeneous generator of a graded A -module M , the initial degree of M , denoted indeg ( M ). Definition 3.4.
Say M has graded linear resolution if for each integer i if there exists j such that β ij = 0 then β i +1 ,j +1 = 0.An example of having graded linear resolution is that of having linear resolution. However theformer is more general. The concept of graded linearity was called quasilinearity in [3]. Along theabove definitions we prove the following results. Theorem 3.5.
Assume M is a graded A -module minimally generated by a regular sequence ( a , . . . , a n ) in m and J l is defined as in (2.3). If M has graded linear quotients, then J l has graded linear res-olution for any l .Proof. We use induction on l . Consider the short exact sequence0 ! A ( J l : z l +1 ) × z l +1 −! AJ l −! AJ l +1 ! . (3.4)The first module has quasi-linear resolution, and the second by induction assumption. Through thedefinition go to the stage that for the middle module β i − ,j − = 0 implies β ij = 0. In the associatedlong graded Tor exact sequence −! T or i (cid:16) A ( J l : z l +1 ) , k (cid:17) j × z l +1 −! T or i (cid:16) AJ l , k (cid:17) j + d −! T or i (cid:16) AJ l +1 , k (cid:17) j + d −! (3.5) T or i − (cid:16) A ( J l : z l +1 ) , k (cid:17) j + d − × z l +1 −! T or i − (cid:16) AJ l , k (cid:17) j + d − −! (3.6)where d = deg( a l +1 ). One notes that multiplication by z l +1 is in the annihilator of k . This impliesthe sequence0 ! T or i (cid:16) AJ l , k (cid:17) j + d −! T or i (cid:16) AJ l +1 , k (cid:17) j + d −! T or i − (cid:16) A ( J l : a l +1 ) , k (cid:17) j + d − ! T or i (cid:16) AJ l , k (cid:17) j + d embedds into T or i (cid:16) AJ l +1 , k (cid:17) j + d . Therefore the non-triviality of the first one implies nontriviality of the second. Therefore in order to check inductionstep it suffices to prove 6 or i (cid:0) AJ l +1 , k (cid:1) j + d = 0 ⇐⇒ T or i − (cid:0) A ( J l : a l +1 ) , k (cid:1) j + d − = 0 (3.8)The last assertion is true for the implication T or i − (cid:16) A ( J l : a l +1 ) , k (cid:17) j + d − = 0 ⇒ T or i (cid:16) AJ l , k (cid:17) j + d = 0 (3.9)Now the assertion follows by induction. We are done. Theorem 3.6.
Assume M is a finitely generated graded A -module. Then the following are equiv-alent.(1) M has graded linear resolution.(2) M h s i have graded linear resolutions for all s .(3) M is componentwise linear.Proof. : One has (3) ⇒ (2) ⇒ (1), [Compare to Theorem 3.1]. To prove (1) ⇒ (3) it suffice toshow that F lin is acyclic. By the definition 3.3 the graded linearity is understood as a naturalgeneralization of linearity. The linear resolution of a module M over the polynomial ring impliesthat the non-zero graded Betti numbers all lie successively in a horizontal line in the Betti table.In the graded linear case the non-zero ones give several horizontal (connected) lines. It follows; itsuffice to check the acyclicity of F lin just in the linear case. Following the explanation after theTheorem 3.1 the entries of the matrices defining the differentials in F lin all are linear forms in m .Applying this to a minimal resolution of the form 2.1, the minimality shows that F lin is exact. Thiscriteria simply extends to the graded linear case. Theorem 3.7.
Let ( M, < ) be an ordered simple matroid of rank r on [ n ] . The followings areequivalent(1) I < ( M ) has graded linear resolution.(2) I < ( M ) is componentwise linear.(3) If a , . . . , a n − q is a maximal ( A/I < ( M )) -regular sequence of linear forms in A we have AI < ( M ) = d M i =0 m s i m s i +1 (3.10) in A/ ( a , . . . , a n − s ) .(4) The Hilbert function of I < ( M ) has the form H ( AI < ( M ) , s ) = c (cid:18) s + q − s (cid:19) + c (cid:18) s + q − s (cid:19) + · · · + c d − (cid:18) s + q − ds (cid:19) (3.11) for some d .
5) If ≺ is a monomial order on A and if in ≺ ( I < ( M )) is Cohen-Macaulay in ≺ ( I < ( M )) hasgraded linear resolution (componentwise linear).Proof. The proof for equivalence of (1) and (2) follows from Theorem 3.5 and the definition 3.1.(4) follows from (3) and also (5) follows from (4) and the equality of the Hilbert functions H ( A/I < ( M ) , s ) = H ( A/I < ( M ) , s ) (3.12)where bar means the image in A/ ( a , . . . , a n − q ). One has the stronger statement H ( A/I < ( M ) , s ) = H ( A/I < ( M ) , s ) ≤ H ( d M i =0 m s i m s i +1 , s ) = d − X i =0 c i (cid:18) s + q − i − s (cid:19) (3.13)with equality when (3.10) holds. In fact factoring out the regular sequence ( a , . . . , a n − q ) onecan assume A is Artinian. Then it is easy to establish that it has the form (3.10), see also [32]proposition 3.1. This also proves (3).The coefficients c i = c i ( I < ( M )) can be easily calculated, (called Hilbert coefficient of I < ( M )). Thesum in the Hilbert polynomial is the sum of the Hilbert function of each summands where can becalculated by hand. The following theorem is a generalization of proposition 7.5.2 in [6]. Theorem 3.8.
Assume M is a rank r loopless matroid on [ n ] such that the Stanley-Reisner idealof the broken circuits of M has graded linear resolution (componentwise). Then I j − ≥ c − X i =0 d X l =1 c ( S ) l (cid:18) n − r + l − l (cid:19)(cid:18) r − ij − i (cid:19) (3.14) and the equality holds if and only if there exists a stratification M = M ⊃ M ⊃ · · · ⊃ M d bysubmatroids M j for some d such that on each strata M j (cid:31) M j +1 = U r j − s j ,r j − s j M U s j ,n j − r j + s j (3.15) where s j ≤ r j are positive integers and U l,k denotes the uniform matroid. We have n = Σ j n j , r =Σ j r j .Proof. (sketch) Following the argument of Theorem 2.5 of previous section [or theorem 7.2.5 [6]],letting ∆ = In ( M ), then we have h k = H ( A/I < ( M ) , k ) = X l c l (cid:18) k + q − l − k (cid:19) , k = 0 , , , . . . , c − c l are Hilbert coefficients of I < ( M ). Then b ( M ) = X l c l (cid:18) q − l + pp (cid:19) (3.17)is the number of bases of M . The h -vector is given by8 = , X l c l (cid:18) q − l (cid:19) , . . . , X l c l (cid:18) c − q − l − c − (cid:19) ! (3.18)and the Tutte polynomial of M is T M ( x,
1) = h ∆ ( x ) = x r + h x r − + · · · + h p x r − p (3.19)The form of the Tutte polynomial shows the existence of stratification M = M ⊃ M ⊃ · · · ⊃ M d by submatroids M j such that M j (cid:31) M j +1 = U r j − s j ,r j − s j M N j (3.20)where N j is a matroid of rank s j on n j − r j + s j points. In fact the h -vector of the submatroids M j is given by the trucations of the sums appearing in the h -vector of M . Now b ( N j ) = b ( M j (cid:31) M j +1 ) = c j (cid:18) c − j + q − lc − (cid:19) (3.21)This shows that N j has the desired form. The equality in (3.14) holds when h c = h c +1 = · · · = 0 (3.22) Remark . Some basic operations on matroids are as follows.Given a matroid ([ n ] , I ) we can construct other matroids: If X ⊂ [ n ] then the deletion M (cid:31) X isthe matroid with ground set [ n ] (cid:31) X and independent sets { J ⊂ [ n ] (cid:31) X : J ∈ I } . The matroid dual M ∗ of M is the matroid on [ n ] where I is a basis of M iff I c is a basis of M ∗ . If X ⊂ [ n ] then thecontraction is defined M/X = ( M ∗ (cid:31) X ) ∗ .Let M and N be matroids on the same ground set E . We say that N is a quotient of M if one ofthe following equivalent statements holds: • Every circuit of M is a union of circuits of N . • If X ⊂ Y ⊂ E , then r M ( Y ) − r M ( X ) ≥ r N ( Y ) − r N ( X ), ( r is the rank function, see [10]). • There exists a matroid R and a subset X of E ( R ) such that M = R \ X and N = R/X . • For all bases B of M , and x ∈ B , there is a basis B of N with B ⊂ B and such that { y : ( B − y ) ∪ x ∈ B ( N ) } ⊂ { y : ( B − y ) ∪ x ∈ B ( M ) } .For the equivalences, we refer to [[22], Proposition 7.3.6]. If N be a quotient of M . • Every basis of N is contained in a basis of M , and every basis of M contains a basis of N . • rk ( N ) ≤ rk ( M ) and in case of equality, N = M .cf. [10] loc. cit. The stratification of the matroid in Theorem 3.8 may be thought of as thecorresponding operation in the category of matroid to the operation of making quotients in thecategory of modules. 9 heorem 3.10. Let ( M, < ) be an ordered simple matroid of rank r on [ n ] , and I < ( M ) the Stanley-Reisner ideal of broken circuit complex of M . Then the followings are equivalent.(1) I < ( M ) is componentwise linear.(2) There exists a stratification M = M ⊃ M ⊃ · · · ⊃ M d for some d such that on each strata M j (cid:31) M j +1 = U r j − s j ,r j − s j M U s j ,n j − r j + s j (3.23) where s j ≤ r j are positive integers. We have n = Σ j n j , r = Σ j r j .Proof. Assume (2) holds. Then by re-arrangement of variables if necessary I < ( M ) = (cid:10) x j · · · x j s , . . . , x l · · · x l p | j < · · · < j s < n − r + s (3.24) , . . . , l < · · · < l p < n − r + p (cid:11) (3.25)This ideal has clearly graded linear resolution.Now assume I < ( M ) has graded linear resolution. The property (3) implies that if a = ( a , . . . , a r )is a maximal S/I < ( M )-regular sequence of linear forms, then the quotient S = A/ ( I < ( M ) + a ) = d M j =0 m s j / m s j +1 , ( m = A ) (3.26)Thus the Hilbert function of A/I < ( M ) is given by H ( A/I < ( M ) , t ) = H ( S, t ) / Y i (1 − t s i ) (3.27)So that the f -vector of Stanley-Reisner ring is given by f j − = r X i =0 c i (cid:18) r − ij − i (cid:19) = r X i =0 d X l =1 c l (cid:18) n − r + l − l (cid:19)(cid:18) r − ij − i (cid:19) (3.28)where d = deg( H ( S, t )) and c l = c ( S ) l are the coefficients of H ( S, t ). The Theorem 3.8 implies thedecomposition in item (2).
Remark . Let M be the graphic matroid of parallel connection of a 3-circuit with a 4-circuit.Then M is a non-uniform 2-connected maroid, and therefore can not be a direct sum of uniformmatroids. We can order the elements of M from 1 to 6 in a way that the Stanley-Reisner ideal of thebroken circuit complex of M gets equal I ( M ) = ( x x , x x x , x x x x ). Then I ( M ) has linearquotient and therefore componentwise linear. This example simply shows that the stratificationmentioned in the above theorem is essential. 10 Application to Orlik-Terao ideal of hyperplane arrangements
Assume that the matroid M ( A ) is associated to an essential central hyperplane arrangement of n hyperplanes A = { H , . . . , H n } in a vector space V of dimension r over the infinite field k . When k = C , then the integral cohomology of the complement V \ S i H i is isomorphic to the Orlik-Solomon algebra of A ( A ) which is the quotient of standard graded exterior algebra on C h e , . . . , e n i by the Orlik-Solomon ideal J ( A ) generated by the elements ∂e i ...i p = X r ( − r − e i . . . c e i r . . . e i p (4.1)when { H i , . . . , H i p } are dependent subsets of A . i.e the reflections α i j through H i j are independent.Therefore A ( A ) = AJ ( A ) = V Z h e , . . . , e n ih ∂e i ...i p i = H ∗ ( V \ [ i H i , Z ) (4.2)The Orlik-Terao algebra is a commutative analogue of Orlik-Solomon algebra. That is insteadof exterior algebra on n vectors we work on A = k [ x , . . . , x n ] and whenever we have a relation P j a j α i j = 0 we assume a relation r = P j a j α i j in a relation space F ( A ). The Orlik-Terao idealis generated by ∂r = X j ( − j − a j x i . . . c x i j . . . x i p , ⇐ r = X j a j α i j (4.3)The quotient C ( A ) = A/I ( A ) is called the Orlik-Terao algebra of the hyperplane arrangement. Westate the analogous generalization of our Theorems for that of [32] theorem 3.10. Theorem 4.1.
For an essential central arrangement of A of n hyperplanes in a vector space ofdimension r , the following are equivalent. • I ( A ) has graded linear resolution. • There exists a stratification M = M ⊃ M ⊃ · · · ⊃ M d by submatroids M j for some d suchthat M j ( A ) (cid:31) M j +1 ( A ) = U r j − s j ,r j − s j L U s j ,n j − r j + s j where s j ≤ r j are positive integers. Wehave n = Σ j n j , r = Σ j r j . • A = A × A × · · · × A k where A is boolian and the other factors A j for j > are genericarrangement of n j − r j + s j hyperplanes in s j -dimensional vector space.Proof. The proof is analogous to the 2-decomposition case in [32], and is just an interpretation ofthe Theorems in the previous section.The Orlik-Solomon algebras and Orlik-Terao algebras of hyperplane arrangements are related toeach other by an idea of deformation [17]. A relevant notion in this context is that of a Koszulalgebra.
Definition 4.2. [32] Suppose L is either an exterior or a polynomial algebra over a field k . Agraded algebra B = L/I , is called Koszul if k has linear resolution over B .11 heorem 4.3. [32] The Orlik-Terao ideal I ( A ) of an essential central arrangement has 2-linearresolution if A is obtained by successively coning a central arrangement of lines in the plane. Inthis case The Orlik-Terao algebra is Koszul. We can generalize this notion to that of graded Koszul.
Definition 4.4.
In the settings of Definition 4.2 we call B graded Koszul if k has graded linearresolution, or by what we said in the previous sections if k is componentwise linear. Theorem 4.5.
The Orlik-Terao ideal I ( A ) has 2-graded linear resolution if A is obtained by suc-cessively coning central arrangements of lines in the plane. In this case the Orlik-Terao algebra isgraded Koszul.Proof. The proof is a direct application of Theorem 4.1 in this case.
For a commutative Noetherian local ring A , the depth of A (the maximum length of a regularsequence in the maximal ideal of A ) is at most the Krull dimension of A . The ring A is calledCohen–Macaulay if its depth is equal to its dimension. A regular sequence is a sequence a , . . . a n ofelements of m such that, for all i , the element a i is not a zero divisor in A/ ( a , . . . a n ). A local ring A is a Cohen–Macaulay ring if there exists a regular sequence a , . . . a n such that the quotient ring A/ ( a , . . . a n ) is Artinian. In that case n = dimA . More generally, a commutative ring is calledCohen–Macaulay if it is Noetherian and all of its localizations at prime ideals are Cohen–Macaulay.Say that a ring A is a complete intersection if there exists some surjection R ! A with R a regularlocal ring such that the kernel is generated by a regular sequence.Assume ( M, < ) is an ordered matroid on [ n ], and C ( M ) set of broken circuits of M . Let I < ( M )be the Stanley-Reisner ideal of the broken circuit complex BC < ( M ). By definition I < ( M ) = (cid:10) x bc < ( C ) := Y j ∈ bc < ( C ) x j (cid:11) (5.1)A subset D ∈ C ( M ) is called generating set if h x bc < ( C ) ; C ∈ D i generates I < ( M ). It is clear fromdefinition that D is a generating set of C ( M ) iff bc < ( C ) for C ∈ D contains minimal broken circuitsof M . Denote by L ( D ) the intersection graph of D , that is the vertex set of L ( D ) is D and theedges are pairs ( C, C ′ ) such that C ∩ C ′ = ∅ . Say D is simple if whenever C, C ′ ∈ D then either C ∩ C ′ = min < ( C ) or min < ( C ′ ), and otherwise C ∩ C ′ = ∅ . Theorem 5.1. [32] Assume ( M, < ) be and ordered simple matroid. The following are equivalent. • I < ( M ) is complete intersection. • The minimal broken circuits of M are pairwise disjoint. • There exists a simple subset D ⊂ C ( M ) such that C ( M ) = { L ( D ′ ); D ′ ⊂ D } . (5.2)12 similar statement can be stated for the Orlik-Terao ideal of essential central arrangement, cf.[32] loc. cit. Theorem 5.2.
Let A be an essential central arrangement of n hyperplanes in dimension r , andassume I ( A ) is complete intersection. Then the following are equivalent.(1) A ( A ) is Koszul.(2) C ( A ) is Koszul.(3) M ( A ) = U ,n − r +2 L U r − ,r − .(4) A = A × A where A is generic central arrangement of n − r + 2 hyperplanes in dimension p and A is Boolian in dimension r − .(5) I ( A ) has -linear resolution.(6) J ( A ) has -linear resolution.Proof. The proof is a comparison of theorem 2.2 and corollary 4.18 in [32].We generalize this theorem for the graded case as follows.
Theorem 5.3.
With the same set up of Theorem 5.2 the following are equivalent(1) A ( A ) is graded Koszul.(2) C ( A ) is graded Koszul.(3) There exists a stratification M = M ⊃ M ⊃ · · · ⊃ M d by submatroids M j for some d suchthat M j ( A ) (cid:31) M j +1 ( A ) = U r j − s j ,r j − s j L U s j ,n j − r j + t j where s j ≤ r j are positive integers. Wehave n = Σ j n j , r = Σ j r j .(4) A = A × A × · · · × A k where A is boolian and the other factors A j for j > are genericarrangement of n j − r j + s j hyperplanes in s j -dimensional vector space.(5) I ( A ) has -graded linear resolution.(6) J ( A ) has -graded linear resolution.Proof. The equivalence of 2-5 were already established. The equivalence of 1-2 follows from Theo-rem 5.2 and the argument in Theorem 3.10.We give another application of Theorem 5.1 to graphs and cycle matroids. Let (
G, V, E ) be agraph, V the vertices and E the edges. Set C the edge set of cycles in G . It is the set of circuitsof a matroid namely cycle matroid. We assume the graph and this matroid are simple, i.e. everycircuit has at least 3 vertices. In [4] the Cohen-Macaulayness of the facet ideal of r -cyclic graph G n,r has been discussed. In fact we can show that the facet ideal of G n,r is a complete intersection.13 efinition 5.4. [4] Let ∆ be a simplicial complex over [ n ]. Let I F (∆) be the monomial idealminimally generated by square-free monomials f F (1) , . . . , f F ( s ) such that f F ( i ) = x i x i . . . x i r , ⇔ F ( i ) = { i , . . . , i r } is a facet of ∆ for all i (5.3)known as the facet ideal of complex ∆ and denoted I F (∆).The graph G n,r has n edges and r distinct cycles. One naturally associates a simplicial complex∆( G n,r ) = h E \ { e i , . . . , e ri r }i (5.4)to G n,r , where C i = { e , . . . , e i } (5.5)is the i -th cycle, [4]. . Proposition 1.
The facet ideal I F (∆( G n,r )) is a complete intersection. Proof.
By definition G n,r is a graph on n edges with exactly r distinct cycles. Therefore the brokencircuits of the cycle matroid M ( G n,r ) are pairwise disjoint. On the other hand the Stanley-Reisnerideal of the broken circuit complex of cycle matroid M ( G n,r ) is exactly the facet ideal of the complex∆( G n,r ). Now the theorem follows from 5.1. Corollary 5.4.1.