IDEALS GENERATED BY DIAGONAL 2-MINORS VIVIANA ENE AND AYESHA ASLOOB QURESHI 2 1 Abstract. With a simple graph G on [n], we associate a binomial ideal P 0 G 2 generatedby diagonalminors ofann×n matrix X =(xij)ofvariables. We show that for any graph G, P is a prime complete intersection ideal and determine n G the divisorclassgroupofK[X]/P . Byusing these ideals,onemayfinda normal a G J domain with free divisor class group of any given rank. 6 2 ] Introduction C A Classically, with a simple graph G on the vertex set [n], one associates the so- . h called edge ideal I(G) in the polynomial ring K[x ,...,x ] over a field K. Re- 1 n t a cently, binomial edge ideals have been considered in [7] and, independently, in m [9]. The binomial edge ideal J of G is generated by the binomials x y − x y ∈ G i j j i [ K[x ,...,x ,y ,...,y ], where {i,j} is an edge of G. For instance, the ideal of all 1 n 1 n 2 2-minors of a 2×n matrix of variables is a special example of a binomial edge ideal. v The study of the ideals of 2-minors of matrices of variables is motivated by their 1 relevance in algebraic statistics and other fields as it was shown in [4]. For more 2 7 recent results on ideals generated by 2-minors one may consult [8], [6], [10]. 3 Inthispaper,weintroduceanewclassofidealsof2-minorsassociatedwithgraphs. . 2 Let X = (x ) be an n×n-matrix of variables and S = K[X] the polynomial ring 1 ij 1 over a field K in the variables {xij}1≤i,j≤n. Let G be a simple graph on the vertex 1 set [n]. With this graph we associate an ideal generated by diagonal 2-minors of X : v in the following way. For 1 ≤ i < j ≤ n we denote by f the diagonal 2-minor of ij i X X given by the elements at the intersections of the rows i,j and the columns i,j, r that is, fij = xiixjj −xijxji. Let PG be the ideal of S generated by the binomials fij a where {i,j} is an edge of G. With respect to the lexicographical order on S induced by the natural order of variables, namely x > x > ··· > x > x > ··· > x , the reduced 11 12 1n 21 nn Gröbner basis consists of binomials of degree at most 4 which have squarefree initial monomials,asweshowinTheorem1.2. Butifweconsiderthereverselexicographical order induced by the natural order of the variables, then the generators of P form G a Gröbner basis of P and moreover, they form even a regular sequence, therefore G height(P ) equals the number of edges of G and in (P ) is squarefree. Here ≺ G ≺ G denotes the reverse lexicographic order. We show in Proposition 1.3 that P is a prime ideal, hence the ring R = S/P G G G is a normal domain. 1991 Mathematics Subject Classification. 13C20,13P10, 13C40. Key words and phrases. Binomial ideals, Gröbner bases, Graphs, Divisor class groups. The first author was supported by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023. 1 In the last section we study the divisor class group Cl(R ). We show in Theo- G rem 2.2 that Cl(R ) is free and we express its rank in terms of the graph’s data. G Finally, in Proposition 2.3, we give sharp bounds for the possible rank of Cl(R ) G when G has a given number of edges. Every abelian group is the class group of a Krull domain as shown by Claborn [1]. By using ideals generated by diagonal 2-minors, one may find an example of a normal domain with free divisor class group of any given rank. 1. Gro¨bner bases of ideals generated by diagonal 2-minors LetX = (x ) bea squarematrixofvariablesandS = K[X]thepolynomial ij 1≤i,j≤n ring over a field K. For any 1 ≤ i < j ≤ n, let f = x x − x x . Let G be a ij ii jj ij ji simple graph on the vertex set [n] and P = (f : {i,j} ∈ E(G)) where E(G) is the G ij set of edges of G. In this section we compute the Gröbner bases of P with respect G to the lexicographic and the reverse lexicographic order on S. Usually, when we study ideals generated by minors of matrices of variables, one uses the lexicographic order induced by the natural order of variables, namely, row by row from left to right. This monomial order selects as initial monomial of each 2-minor of X the product of variables of the main diagonal. It will turn out that this order gives a rather large Gröbner basis forP . But if we consider a monomial order G on S which selects the product of the variables on the anti-diagonal of each 2-minor as initial monomial, then the generators of P form a Gröbner basis. More precisely, G let us consider the reverse lexicographic order on the ring S. With respect to this order, in f = x x , that is, the initial monomial comes from the anti-diagonal of ≺ ij ij ji the minor f for any 1 ≤ i < j ≤ n. ij Proposition 1.1. Let G be a simple graph on the vertex set [n] with the edge set E(G) and let P = (f : {i,j} ∈ E(G)) be the binomial ideal generated by the G ij diagonal 2-minors associated with G. Then the set of generators of P is the reduced G Gröbner basis of P with respect to the reverse lexicographic order. Moreover, P is G G a complete intersection of height(P ) = |E(G)|. G Proof. All claims follow immediately if we notice that the initial monomials of f ij with respect to ≺ form a regular sequence. (cid:3) Wenow consider the lexicographic order induced by thenaturalorder of variables, namely, x > x > ··· > x > x > x > ··· > x > ··· > x > x > ··· > x . 11 12 1n 21 22 2n n1 n2 nn With respect to this order, in f = x x , in other words, the initial monomial < ij ii jj comes from the main diagonal of the minor f . ij Theorem 1.2. Let G be a simple graph on the set [n] and let P be its associated G ideal. The initial ideal of P with respect to the lexicographic order induced by the G natural order of indeterminates is generated by squarefree monomials of degree at most 4. 2 Proof. We compute a Gröbner basis G of P by applying Buchberger’s criterion. lex G We first compute the S-polynomials of the generators of P . Let f and f be G ij kl two binomials in the generating set P . We consider the non trivial case when G gcd(in (f ),in (f )) 6= 1. We may have one of the following possibilities: < ij < kl (i) x = x (or x = x ), jj kk ii ll (ii) x = x , ii kk (iii) x = x . jj ll Consider the case (i) as shown in Figure 1. Then, the S-polynomial S(f ,f ) = ij jl x x x − x x x must be added to G , since none of its monomials is divisible ii jl lj ll ij ji lex by a monomial of the form x x . aa bb xii xij xii xij xil xii xij x x x jj x x x kk x ji jl ji jj kj x x x x x x x lj ll li ll ji jk jj case(i) case(ii) case(iii) Figure 1. For case (ii), see Figure 1, we may assume that, for instance, j < l, and we get S(f ,f ) = x x x − x x x that should also be added to G . Moreover, ij il jj il li ll ij ji lex in (S(f ,f )) = x x x . < ij il ll ij ji In case (iii), we may choose i < k, and get S(f ,f ) = x x x − x x x in ij kj ii jk kj kk ji ij G , with in (S(f ,f )) = x x x ; see Figure 1. lex < ij kj ii jk kj By the above computation of S-polynomials, we have got all the binomials of degree 3 which belong to the Gröbner basis of P . Next we investigate the S- G polynomials S(g,f ), where g is a binomial of degree 3 and f is a quadratic bi- ij ij nomial. We are going to discuss only those cases when the S-polynomial does not reduce to 0, and hence contributes to the Gröbner basis. Case 1: Let g = x x x − x x x with i < j < l and f = x x − x x ii jl lj ll ij ji iq ii qq qi iq with i < q. We have S(f ,g) = x x x x − x x x x . If in (S(g,f )) is the iq jl lj qi iq qq ll ij ji < iq first monomial, that is, q < j, then we add S(g,f ) to G since x x x x is not iq lex jl lj qi iq divisible by any of the previous initial monomials which we have obtained so far; see Figure 2. Otherwise, that is, for q > j, we reduce S(g,f ) modulo a binomial of degree 3, iq namely the one which appears when we take S(f ,f ), and get ij iq S(g,f ) = −x S(f ,f )−x x f . iq ll ij iq iq qi jl Therefore, S(g,f ) reduces to zero. Now it remains to consider S(g,f ), where iq qi q < i. By proceeding as before, it follows that S(g,f ) reduces to zero modulo a qi binomial of degree 3 and a binomial of degree 2. Case 2: Let g = x x x − x x x and f = x x − x x , with l < q. Then ll ij ji jj il li lq ll qq ql lq we have S(g,f ) = x x x x −x x x x . Since in (S(g,f )) is not divisible by lq ij ji lq ql qq jj il li < lq 3 xii xiq xij x qi x qq x x jj x ji jl x x lj ll Figure 2. any initial monomial obtained so far, we add this polynomial to our Gröbner basis. We get the same conclusion if we take S(g,f ) with l > q; see Figure 3. ql xii xij xil xii xij xil xji xjj xji xjj x x x ll x qq x li lq ql x x x x x ql qq li lq ll l < q q < l Figure 3. Case 3: Let g = x x x −x x x and f = x x −x x with i < q. Then, ii kj jk ij ji kk iq ii qq iq qi we have S(g,f ) = x x x x − x x x x . If q < j, then in (S(g,f )) is a iq iq qi jk kj ij ji kk qq < iq new initial monomial, hence we add S(g,f ) to our Gröbner basis; see Figure 4. iq Otherwise, that is, if q > j, then, as we did in Case 1, we observe that S(g,f ) iq reduces to zero modulo a binomial of degree 3 and a binomial of degree 2. xii xiq xij x x qi qq x kk x kj x x x ji jk jj Figure 4. If q < i, we may again reduce S(g,f ). So far, apart from the original generators qi ofP ,wehaveintheGröbnerbasiswithrespecttothelexicographicorder,binomials G of degree 3 and 4. NowwediscusstheS-polynomialofdegree3binomials. Letg = x x x −x x x ii jl lj ll ij ji and h = x x x − x x x be any two binomial of the form that we obtained pp rq qr rr pq qp by computing S-polynomials of the generators of P . If gcd(in (g),in (h)) 6= 1, G < < 4 then we either have x = x or {x ,x } = {x ,x }. In the first case, we obtain a ii pp jl lj qr rq binomial of degree 5 which reduces to 0, and in the second case, S(f,g) is a binomial of degree 4 which is also reducible. To understand this, consider the following example. Let f = x x x −x x x and g = x x x −x x x where f and g qp pq rr qq rp pr ii jl lj ll ij ji are obtained as in case (i) and case(ii) of Figure 1, respectively. x x x pp pq pr x qp x qq x x x rp rr rj x x x jr jj jl x x lj ll Figure 5. If we let x = x , then S(f,g) reduces to 0 with respect to x x x x − ii rr qp pq jr rj x x x x and x x −x x ; see Figure 5. If x = x then S(f,g) is the product qq jj rp pr ll jj jl lj jl pq of x and x x x −x x x . qq ii rp pr rr ip pi By a careful computation of S-polynomials in the cases when we consider S- polynomials of degree 4 binomials with binomials of degree 2, 3 and 4, we see that they reduce to 0 and hence G consists of binomials with squarefree initial term of lex (cid:3) degree at most 4. Proposition 1.3. P is a prime ideal, thus S/P is a domain. G G Proof. We may assume that G has no isolated vertices. Let {1,i} be an edge of G and let G′ be the subgraph of G obtained by removing this edge from G. We first claim that x is regular on S/P . Indeed, we have (P ,x ) = (P ,x x ,x ). The i1 G G i1 G′ 11 ii i1 height of the ideal (P ,x x ,x ) may be obtained by computing the height of its G′ 11 ii i1 initial ideal with respect to ≺, which is |E(G′)|+2 = |E(G)|+1 = height(P )+1. G Consequently, height(P ,x ) = height(P )+1, which shows that x is regular on G i1 G i1 S/P since P is a complete intersection. G G Now, the claim of the proposition follows if we show that (S/P ) is a domain. G xi1 We have (S/P ) ∼= (S′/P )[x−1], where S′ is the polynomial ring in the variables G xi1 G′ i1 {x : 1 ≤ i,j ≤ n}\{x ,x }. Therefore, the proofis finished by applying induction ij 1i i1 on the number of edges of G. (cid:3) Corollary 1.4. Let G be a simple graph on [n]. Then the ring R = S/P is a G G normal domain. Proof. Since in (P ) is a squarefree monomial ideal, the normality follows by ap- ≺ G (cid:3) plying a well known criterion of Sturmfels [11, Chapter 13]. 5 2. The divisor class group Let G be a simple graph on [n] and R = S/P . In the sequel, we are going G G to determine the divisor class group of R . We proceed as in the case of classical G determinantalrings, see[12],[2]. Anotherusefulreferenceoncomputing classgroups of toric varieties is [3, Chapter 4]. We first choose an element y ∈ R such that (R ) is a factorial ring. Then, by G G y Nagata’s Theorem [5, Corollary 7.2], we deduce that the divisor class group Cl(R ) G is generated by the classes of the minimal prime ideals of y. For a vertex i of G, we denote by G\{i} the subgraph of G obtained by removing the vertex i together with all the edges which are incident to i. For the next lemma we need some notation. For i ∈ V(G) we denote by N(i) the set of all the neighbors of i, that is, N(i) = {a ∈ V(G) : {a,i} ∈ E(G)}, and for each a ∈ N(i), we set Ei = {x ,x }. a ai ia Lemma 2.1. Let {i,j} with i < j be an edge of G. Then: (a) (P ,x ) is an unmixed radical ideal with height(P ,x ) = heightP +1. G ji G ji G (b) The set of the minimal primes of (P ,x ) is C (i,j)∪C (i,j), where G ji 1 2 C (i,j) = {(P ,x ,x ,T)} 1 G\{i} ii ji where T is any set of variables with T ⊂ Ei and |T ∩ Ei| = 1 for S a a a∈N(i)\{j} all a ∈ N(i)\{j}, and C (i,j) = {(P ,x ,x ,U)} 2 G\{j} jj ji where U is any set of variables with U ⊂ Ej and |U ∩ Ej| = 1 for S b b b∈N(j)\{i} all b ∈ N(j)\{i}. Proof. We have (P ,x ) = (P ,x x ,x ), where G′ is the subgraph of G obtained G ji G′ ii jj ji by removing the edge {i,j}. Since in (P ,x x ,x ) is generated by a regular ≺ G′ ii jj ji sequence of squarefree monomials of length |E(G′)|+2 = |E(G)|+1, we get (a). (b) Obviously, by a height argument, the ideals of the two classes are minimal primesof(P ,x ). Indeed, forexample, fori ∈ V(G)andanysetT which definesan G ji ideal of the set C (i,j), we have height(P ,x ,x ,T) = height(P )+|N(i)|+ 1 G\{i} ii ji G\{i} 1 = heightP +1. G Let Q be a minimal prime of (P ,x ). As x ∈ Q, we also have x x ∈ Q, hence G ji ji ii jj x ∈ Q or x ∈ Q. Let, for instance, x ∈ Q. Then ii jj ii Q ⊃ (P ,x ,x ) = (P ,x ,x ,{x x | a ∈ N(i)\{j}}). G ii ji G\{i} ii ji ia ai Now, one easily sees that Q contains one of the ideals of the class C (i,j), and since 1 heightQ = heightP + 1, Q must be equal to one of the ideals of the set C (i,j). G 1 (cid:3) Let y = x ∈ S/P , and let Q = Q/P ⊂ S/P be a minimal prime Q ji G G G {i,j}∈E(G) i d. Therefore, there are 2degi − 1 minimal primes of y which cd contain x . Consequently, we have ii |Min(y)| = X (2degi −1) = X 2degi −n. i∈V(G) i∈V(G) Theorem 2.2. The class group Cl(R ) is free of rank 2degi −n−|E(G)|. G Pi∈V(G) Proof. We first notice that in (R ) we have x¯ = x¯ x¯ x¯−1 for any edge {i,j} of G y ij ii jj ji G with i < j. Therefore, ∼ (R ) = K[{x : i,j ∈ [n]}\{x : {i,j} ∈ E(G),i < j}] , G y ij ij z where z = x , which shows that (R ) is a factorial ring. By Nagata’s Q ji G y {i,j}∈E(G) i 1 and G a G graph with m edges. We remove one edge {i,j} of G and let G′ be the new graph. It is clear that at least one of the vertices i and j has degree ≥ 2, since G is connected. Then, rankCl(RG) = X (2dega −1)−m a∈V(G) = [ X (2dega−1)+(2degi−1−1)+(2degj−1−1)−(m−1)]+2degi−1+2degj−1−1 a∈V(G), a6=i,j = rankCl(R )+2degi−1 +2degj−1−1 ≥ 2m−3+2 = 2m−1. G′ We show that the only graph with m edges for which rankCl(R ) = 2m−1 is the G line graph with m+1 vertices, by induction on m. The step m = 1 is clear. Let now m > 1 and assume that rankCl(R ) = 2m−1. Then, by the above inequalities we G get 2m−1 = rankCl(R )+2degi−1 +2degj−1 −1 ≥ 2m−3+2 = 2m−1. G′ Therefore, we must have rankCl(R ) = 2m−3, thus G′ is a line, by induction, and G′ one of the vertices i,j has degree 1 and the other one has degree 2, which yields the desired conclusion. For proving the inequality rankCl(R ) ≤ 2m−1, we proceed by induction on m. G Moreprecisely, wefirstshowthatifGhasnovertex ofdegreem,thenrankCl(R ) < G 2m −1. As in the first part of the proof, we remove an edge of G, let us say {i,j}. Then rankCl(R ) = rankCl(R )+2degi−1 +2degj−1 −1 G G′ ≤ 2m−1 −1+2degi−1 +2degj−1 −1 < 2m −1, since G has no vertex of degree m. Indeed, we get the last inequality as follows 2m−1 +2degi−1 +2degj−1 −2 ≤ 2m−1 +2·2m−2 −2 = 2m −2 < 2m −1. The upper bound for rankCl(R ) is clearly reached by the star graph, that is the G graph with the edges {1,2},{1,3},...,{1,m},{1,m + 1}, which is the only one which has a vertex of degree m. (cid:3) Remark 2.4. One may easily see that, in general, not every integer between 2m−1 and 2m −1 can be the rank of Cl(R ) for some connected graph G with m edges. G For instance, for m = 4, the possible ranks of Cl(R ) are 7,8,9,10 and 15. G Remark 2.5. Note that if G is the cycle with m edges, then rankCl(R ) = 2m. G Therefore, for each positive integer n, one may find a graph G such that Cl(R ) is G a free group of rankCl(R ) = n. Indeed, if n = 2m−1, we may take G to be the G line graph with m edges, and if n = 2m, we may take the cycle with m edges. 8 References [1] L. Claborn, A note on the class group, Pac. J. Math. 18(2) (1966), 219–222. [2] A. Conca, Ladder determinantal rings, J. of Pure and Appl. Algebra 98 (1995), 119–134. [3] D. Cox, J. Little, H. Schenck, Toric varieties, Graduate Studies in Mathematics, 124, AMS (2011). [4] P.Diaconis,D.Eisenbud,B.Sturmfels,Latticewalksandprimarydecomposition,Mathematical Essays in Honor of Gian-Carlo Rota, Birkhäuser, Boston, 1998,173–193. [5] R. Fossum, The Divisor Class of Group of a Krull Domain, Springer, (1973). [6] J. Herzog, T. Hibi, Ideals generated by adjacent 2–minors, Preprint 2010, arXiv:1012.5789v3 [7] J. Herzog, T. Hibi, F. Hreinsdóttir, T. Kahle, J. Rauh. Binomial edge ideals and conditional independence statements, Adv. Appl. Math. 45 (2010), 317–333. [8] S. Ho¸sten, S. Sullivant, Ideals of adjacent 2-minors, J. Algebra 277 (2004), 615–642. [9] M.Ohtami,Graphs andideals generatedbysome2-minors,Comm.Alg.39(3)(2011),905–917. [10] A. A. Qureshi, Ideals generated by 2-minors, collections of cells and stack polyominoes, sub- mitted. [11] B.Sturmfels,GröbnerBasesandConvexPolytopes,Amer.Math.Soc.,Providence,RI,(1995). [12] U. Vetter, W. Bruns, Determinantal rings, Lecture Notes in Mathematics, Springer, (1988). Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania E-mail address: vivian@univ-ovidius.ro Abdus Salam School of Mathematical Sciences, GC University, Lahore. 68-B, New Muslim Town, Lahore 54600, Pakistan E-mail address: ayesqi@gmail.com 9

Ideals generated by diagonal 2-minors

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