aa r X i v : . [ m a t h . A C ] A ug MULTIGRADED SHIFTS OF MATROIDAL IDEALS
SHAMILA BAYATI
Abstract.
In this paper, we show that if I is a matroidal ideal, then the idealgenerated by the i -th multigraded shifts is also a matroidal ideal for every i =0 , . . . , proj dim( I ). Introduction
Matroids, as an interesting subject at the center of combinatorics, made theirway into commutative algebra through several ways including matroidal ideals . Soit has been of interest to study free resolution of (poly)matroidal ideals and relatedinvariants, see [1, 2, 5]. See also [8, 10] where matroids and the theory of freeresolutions are applied in favor of each other. If, in particular, we consider themultigraded free resolution which reflects the combinatorial structure, then morecombinatorial information encoded in the resolution will be found out.In this paper, our goal is to investigate whether the property of being matroidal isinherited by the ideals generated by multigraded shifts; a question which first cameup about the mutligraded shifts of polymatroidal ideals during a discussion withJ¨urgen Herzog and Somayeh Bandari when we met in Essen in 2012.Let S = k [ x , . . . , x n ] be the polynomial ring in the variables x , . . . , x n over afield k . We consider this ring with its natural multigrading where x a = x a . . . x a n n isthe unique monomial with multidegree ( a , . . . a n ). Throughout, a monomial and itsmultidegree will be used interchangeably, and S ( x a ) will denote the free S -modulewith one generator of multidegree x a . A monomial ideal I ⊆ S has a (unique up toisomorphism) minimal multigraded resolution F : 0 → F p → . . . → F → F with F i = M a ∈ Z n S ( x a ) β i, a . The set of i -th multigraded shifts of I is { x a | β i, a = 0 } . The ideal generated by the i -th multigraded shifts of I will be denoted by J i ( I ). Weshow that J i ( I ) can be obtained by taking i times iterated adjacency ideals startingfrom I ; see Section 2 for relevant definitions. On the other hand, we show that the Mathematics Subject Classification.
Key words and phrases.
Adjacency ideal; Free resolutions; Matroid basis graphs; Matroidalideals; Multigraded shifts . djacency ideal of a matroidal ideal is also matroidal. Therefore, we conclude thatif I is a matroidal ideal, then J i ( I ) generated by its set of i -th multigraded shifts isalso a matroidal ideal for every i = 0 , . . . , proj dim( I ).1. Preliminaries
In this section, we explain some terminology and facts that we shall use in thispaper.
Let I be a monomial ideal in S = k [ x , . . . , x n ]. We denote its minimal set ofmonomial generators by G ( I ). The ideal I is said to have linear quotients if thereexists an ordering m , . . . , m r of the elements of G ( I ) such that for all i = 1 , . . . , r − m , . . . , m i ) : ( m i +1 ) is generated by a subset of { x , . . . , x n } . Afterhaving fixed such an ordering, the set of each m i is defined to beset( m i ) = { k ∈ [ n ] : x k ∈ ( m , . . . , m i ) : ( m i ) } . Let I have linear quotients with order m , . . . , m r of the elements of G ( I ). By[4, Lemma 1.5] a minimal multigraded free resolution F of I by mapping coneconstructions can be described as follows: the S -module F i in homological degree i of F is the multigraded free S -module whose basis is formed by monomials m x a with m ∈ G ( I ) and squarefree monomial x a such that supp( x a ) is a subsetof set( m ) of cardinality i . Recall that we use monomials and their multidegree interchangeably, as well assquarefree monomials and their support. So in the rest of this paper, we will usesome notions related to squarefree monomials for the subsets of [ n ]. In particular,if B ⊆ [ n ] is the support of a squarefree monomial m , we write set( B ) instead ofset( m ). In this subsection, we give the definition of matroids and some basic facts aboutthem. See [9] for a detailed materiel on matroid theory.We denote the union of sets B ∪ { b } by B + b , and the difference of sets B \ { b } by B − b .A matroid M on a finite set E is a nonempty collection B of subsets of E , called bases , with the following exchange property :If B , B ∈ B and b ∈ B \ B , then there exists b ∈ B \ B suchthat B − b + b ∈ B .One may refer to this matroid by M = ( E, B ). If B = B − b + b , it is saidthat B is obtained from B by a pivot step: b is pivoted out and b is pivoted in.It is known that bases of M possess the following symmetric exchange property :If B , B ∈ B and b ∈ B \ B , then there exists b ∈ B \ B suchthat B − b + b and B − b + b are both in B . t is a known fact that all the bases of a matroid have the same cardinality.Associated to a matroid M = ( E, B ), its basis graph BG( M ) is defined as follows:the vertex set of BG( M ) is B , and two vertices B and B are adjacent if | B \ B | = | B \ B | =1. If B is obtained from B by b pivoted out and b pivoted in, then anedge label b b represents this pivot step diagrammatically as follows: b b B B b b Due to the above mentioned exchange property, matroid basis graphs are connected.A graph is called a matroid basis graph if its vertices could be labeled by bases ofsome matroid to be its basis graph.Let G = ( V, E ) be a graph. Recall that the distance d ( v , v ) between two vertices v , v ∈ V is the length of a shortest path between them, if one exists. An inducedsubgraph of G is a graph whose set of vertices is a subset W of V and two verticesof W are adjacent exactly when they are adjacent in G .The following result is clear, for example by the argument applied in the proofof [6, Lemma 1.4] or directly by the symmetric exchange property for bases of amatroid. Lemma 1.1.
Let M = ( E, B ) be a matroid. Suppose that B , B ∈ B are twovertices of the associated basis graph at a distance of two, and B = B − ( e + e ) + ( f + f ) . Then B and B have at least two common neighbors C , C ∈ B which on the wayfrom B to B , e is pivoted out from one of them and e from the other, and also f is pivoted in one of them and f in the other. A squarefree monomial ideal I in S = k [ x , . . . , x n ] is said to be a matroidal ideal if { supp( m ) | m ∈ G ( I ) } is the set of bases for some matroid. Matroidal ideals are generated in a singledegree, and by [7, Theorem 1.3] they have linear quotients with respect to thelexicographical order of the generators with x > x > · · · > x n .2. Multigraded shifts of matroidal ideals
Let I be a monomial ideal in S = k [ x , . . . , x n ] generated in a single degree. Sup-pose that G ( I ) = { m , . . . m r } , and consider the distance between these monomialsin the sense of [2], that is, d ( m i , m j ) = 12 n X k =1 | ν k ( m i ) − ν k ( m j ) | , where for a monomial m = x a . . . x a n n , one has ν k ( m ) = a k . We associate a graph G I to I whose set of vertices is the set of generators G ( I ), and two vertices m i and m j are adjacent if d ( m i , m j ) = 1. In particular, G I can be considered as a matroid basisgraph when I is a matroidal ideal. We define the adjacency ideal of I , denoted by ( I ), to be the monomial ideal generated by the least common multiples of adjacentvertices in G I , that is,A( I ) = h lcm( m i , m j ) : d ( m i , m j ) = 1 i ⊆ k [ x , . . . , x n ] . One should notice that when I is an ideal generated in a single degree with linearquotients, then the ideal J ( I ), generated by its set of first multigraded shifts, isexactly its adjacency ideal. In particular, this implies that if I is a squarefreemonomial ideal, then so is J ( I ). Lemma 2.1.
Let I be a matroidal ideal. Then its adjacency ideal A( I ) is also amatroidal ideal.Proof. Let M = ([ n ] , B ) be the matroid corresponding to I . Recall that we usesquarefree monomials in k [ x , . . . , x n ] and subsets of [ n ] interchangeably. Considertwo arbitrary distinct elements B + e, C + f ∈ A( I ) with B, C ∈ B , e ∈ set( B ), and f ∈ set( C ). Since e ∈ set( B ), there exists B ′ ∈ B and e ′ ∈ B \ B ′ such that B + e = B ′ + e ′ We check the exchange property for the elements B + e and C + f , that is, for eachelement b ∈ ( B + e ) \ ( C + f ), we find an element c ∈ ( C + f ) \ ( B + e ) such that B + e − b + c ∈ A( I ) . (1)For such an element b , we may assume that b = e . Because if b = e , we can proceedby the other presentation, namely B ′ + e ′ . By this assumption, we have b ∈ B \ C .Therefore, by the exchange property for bases of B , the following set is not empty: T = { ˜ c ∈ C \ B : B − b + ˜ c ∈ B} . There exists two cases:
Case 1.
First suppose that e ∈ T . So B = B − b + e ∈ B . If B = C , then f = b because B + e and C + f are distinct elements. So on one hand, f B .On the other hand, f = e because f C . So the choice of c = f for which B + e − b + f = C + f ∈ A( I ), has the required properties mentioned in (1), and weare done. Otherwise, if B = C , there exists b ′ ∈ B \ C , and consequently by theexchange property, there exists an element c ∈ C \ B such that B = B − b ′ + c ∈ B . The distinct vertices B and B of the basis graph are adjacent. This implies that B + e − b + c = B ∪ B ∈ A( I ). Therefore, this element c is an appropriate choicefor (1). Case 2.
Next suppose that there exists an element c ∈ T such that c = e . Thus B = B − b + c ∈ B . If b = e ′ , then the vertex B becomes adjacent to B ′ by e pivoted out and c pivoted in; see the induced subgraph in Figure 1. Recall that theedge labels show the pivot steps. Since B ′ and B are adjacent, one has B + e − b + c = B + e − e ′ + c = B ′ ∪ B ∈ A( I ) , as required in (1). bb B ′ B B ee ′ b c = e ′ c ec Figure 1. If b = e ′ , then B ′ = B − ( e ′ + c ) + ( e + b ). Therefore, d ( B, B ) = 2 because { e ′ , c } and { e, b } are disjoint sets of cardinality two. So by Lemma 1.1, there exista common neighbor B of B and B ′ by e pivoted in, namely B = B − e ′ + e or B = B − c + e . In any case B + e − b + c = B ∪ B ∈ A( I ), as desired. (cid:3) Considering the lexicographical order with respect to x > · · · > x n on the square-free monomials in k [ x , . . . , x n ] , we have an induced total order on subsets of [ n ].More precisely, we set B > lex C for subsets B, C ⊆ [ n ] exactly when m B > lex m C for squarefree monomials m B and m C with supp( m B ) = B and supp( m C ) = C . Forsimplicity, we denote { i } > lex { j } by i > lex j for i, j ∈ [ n ] which is the case exactlywhen i < j with ordinary order of integers. Theorem 2.2.
Let I be a matroidal ideal. Then the ideal J ℓ ( I ) generated by the setof ℓ -th multigraded shifts of I is also a matroidal ideal for all ℓ = 0 , . . . , proj dim( I ) .Proof. Let M = ([ n ] , B ) be the matroid corresponding to I . So B = { supp( m ) : m ∈ G ( I ) } . Assume that elements of G ( I ) are ordered decreasingly in the lexicographical orderwith respect to x > · · · > x n . With respect to this order of generators we considerset( m ) for each m ∈ G ( I ).We prove the assertion by induction on ℓ . One has J ( I ) = I , and so matroidal.The ideal J ( I ) is matroidal as well by Lemma 2.1. Let J ℓ ( I ) be matroidal for ℓ ≤ k .We show that for k ≥ J k +1 ( I ) is the adjacency ideal of J k ( I ), and consequently amatroidal ideal by Lemma 2.1.First, consider an element U = B + e + · · · + e k +1 ∈ J k +1 with B ∈ B and e , . . . , e k +1 ∈ set( B ). Then U is the union of adjacent vertices B + e + · · · + e k and B + e + · · · + e k − + e k +1 of the basis graph of J k ( I ). Thus, U is in the adjacencyideal of J k ( I ).Next we show that the adjacency ideal of J k ( I ) is a subset of J k +1 ( I ). Let U = B + e + · · · + e k and V = C + f + · · · + f k b b B ′ B i B v b e i t i ve i Figure 2. be two elements of J k ( I ) with B, C ∈ B , e , . . . , e k ∈ set( B ), and f , . . . , f k ∈ set( C ).Suppose that d ( U, V ) = 1, that is, they are adjacent vertices in the basis graph of J k ( I ). We show that U ∪ V , corresponding to the monomial lcm ( x U , x V ), is indeed aunion of an element of D ∈ B and a subset of cardinality k + 1 of set( D ), so belongsto J k +1 ( I ).We have d ( U, V ) = 1. Therefore, V = C + f + · · · + f k has exactly one elementoutside of U = B + e + · · · + e k , say v . The element v belongs to some C ′ ∈ B .Hence by the symmetric exchange property there exists b ∈ B \ C ′ such that B ′ = B − b + v ∈ B . If there exists an element t ∈ B such that v > lex t and B − t + v ∈ B , then B − t + v > lex B , and consequently v ∈ set( B ). Thus U ∪ V = B + e + · · · + e k + v ∈ J k +1 ( I ), as desired. So we assume that for each t ∈ B , if B − t + v ∈ B , then t > lex v .In particular, b > lex v . This implies that B > lex B ′ and b ∈ set( B ′ ). We will showthat e i ∈ set( B ′ ) for all i = 1 , . . . , k . So U ∪ V = B ′ + b + e + · · · + e k ∈ J k +1 , and we are done.For this purpose, fix i = 1 , . . . , k . Since e i ∈ set( B ), there exists t i ∈ B such that e i > lex t i and B i = B − t i + e i ∈ B . If t i = b , then B i and B ′ are adjacent vertices inthe basis graph of M . See Figure 2. More precisely, B i = B ′ − v + e i . Notice that e i > lex t i = b > lex v . Hence B i > lex B ′ and e i ∈ set( B ′ ).If t i = b , then { t i , v } and { e i , b } are disjoint sets of cardinality two. Therefore, d ( B i , B ′ ) = 2 because B i = B ′ − ( t i + v ) + ( e i + b ). By Lemma 1.1, there exists acommon neighbor of B i and B ′ in the basis graph by t i pivoted out, namely B ′ − t i + e i or B ′ − t i + b . If we have the common neighbor B ′ − t i + e i ∈ B , then the desiredconclusion follows. Because e i > lex t i implies that B ′ − t i + e i > lex B ′ , and we obtainthat e i ∈ set( B ′ ). So assume that B ′ − t i + e i
6∈ B . Thus there exists the commonneighbor B ′ − t i + b ∈ B by t i pivoted out. See the induced subgraph in Figure 3.Notice that B ′ − t i + b = ( B − b + v ) − t i + b = B + v − t i . Recall that we assume that whenever B − t + v ∈ B for some t ∈ B , then t > lex v .Thus t i > lex v . So far we only have common neighbors of B i and B ′ by b pivotedin, namely B = B ′ − v + b and B ′ − t i + b . Therefore, we must have the common b b bb e i t i v b B = B ′ − v + bB i B ′ B ′ − t i + bB ′ − v + e i Figure 3. neighbor B ′ − v + e i in B for which e i pivoted in. On the other hand, e i > lex t i > lex v .Hence also in this case, we have e i ∈ set( B i ). (cid:3) By the argument applied in proof of Theorem 2.2, we have the following result.
Corollary 2.3.
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Faculty of Mathematics and Computer Science, Amirkabir University of Tech-nology (Tehran Polytechnic), Tehran 15914, IranSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM)P. O. Box: 19395-5746, Tehran, Iran
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