COMBINATORIAL COMMUTATIVE ALGEBRA. Fatemeh Mohammadi (University of Bristol)

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1 COMBINATORIAL COMMUTATIVE ALGEBRA Fatemeh Mohammadi (University of Bristol) Abstract. In this lecture we focus on the ideals associated to graphs. We see many interesting examples in which the Betti numbers of the ideals are given in terms of combinatorics of some graphs. We give a purely combinatorial interpretation of ideals with 2-linear linear resolution (Fröberg Theorem). We see some classic results in literature like Reisner s criterion and Eagon-Reiner s Theorem. 1. Graph ideals As we have seen in the previous lectures, an important topic in commutative algebra is the study of graded Betti numbers of an ideal. We are in particular interested to give a combinatorial description of the Betti numbers of Stanley-Reisner ideals. The next natural step after studying the Koszul complex which gives a nice combinatorial formula for the Betti numbers of the maximal ideal (x 1,..., x n ) is to study the monomial ideals generated in degree 2. We recall that by polarization it is enough to study the squarefree monomial ideals generated in degree 2. Such an ideal corresponds to a graph Edge ideal. Let G be any finite simple graph. We denote the vertex set of G with V(G) and its edges with E(G). Let K be a field and consider the variable x i corresponding to the vertex i in G for every i V(G). Let R = K[x 1,..., x n ] be the polynomial ring on the variables corresponding to the vertices of G. The edge ideal I(G) associated with G is generated by all degree 2 square-free monomials uv for which (u, v) E(G). Note that every ideal in a polynomial ring generated by degree 2 square-free monomials is indeed an edge ideal of some graph G. Date: March 6,

2 As we have seen in the first lecture, there is a simplicial complex denoted by (G) whose Stanley-Reisner ideal is I G. More precisely, (G) is the simplicial complex on V(G) whose faces consist of a subset of vertices, no two joined by an edge. The simplicial compelx (G) is also known as the independence complex of G. It can be also defined as a simplicial complex on [n] whose faces are (G) = {W [n] : W is an independent set of G}. Definition 1.1. A simplicial complex is called flag if all its minimal non-faces are of size 2. Exercise 1.2. A simplicial complex is flag if and only if it can be written as (G) for some finite graph G A minimal edge ideal with characteristic-dependent Betti numbers. Example 1.3 (Katzman). Let be the following simplicial complex which is a 12 point triangulation of the real projective plane. There exists a graph G whose corresponding simplicial complex is. The Betti numbers of I(G) depend on the characteristic of the field K. Let G be the graph on the vertex set [12] with the following edge set E(G): E(G) = {x 1 x 2, x 1 x 3, x 1 x 7, x 1 x 8, x 1 x 10, x 2 x 3, x 2 x 8, x 2 x 9, x 2 x 12, x 3 x 7, x 3 x 9, x 3 x 11, x 4 x 5, x 4 x 6, x 4 x 8, x 4 x 11, x 5 x 6, x 5 x 7, x 5 x 12, x 6 x 9, x 6 x 10, x 7 x 10, x 7 x 11, x 7 x 12, x 8 x 10, x 8 x 11, x 8 x 12, x 9 x 10, x 9 x 11, x 9 x 12, x 10 x 11, x 10 x 12, x 11 x 12 } The Betti numbers of (G) when K has characteristic 0 are total: : : :

3 and when K has characteristic 2 are total: : : : : Here the 9th Betti number depends on the characteristic of K. There is a subgraph H of G on 11 vertices such that the Betti numbers of I H depend on the characteristic of the field. However this is the smallest example. Theorem 1.4. For any graph with at most 10 vertices, the Betti numbers of I(G) do not depend on the characteristic of K Complete graphs. The Complete Graph K n is the graph on n vertices with 2 n edges. The edge ideal of K n is I = I(K n ) = x i x j 1 i < j n. The Stanley-Reisner simplicial complex of I G is (K n ) = {{1},..., {n}}. β i,j (R/I(K n )) = W [n] dim K H W i 1 ( W ; K). As (K n ) is just n isolated W =j vertices, for any nonempty subset W [n] the simplicial subcomplex W is just W isolated vertices. This implies that the only possibly non-zero 3

4 reduced homology groups for such complexes are those in the 0th position: dim K H0 ( W ; k) = the number of connected components of W minus 1 = W 1 So the only contribution of W to β i,j (R/I(K n )) is whenever W = j = i + 1. There are ( ) ( n j = n i+1) subsets of cardinality j. Hence, ( ) n β i,i+1 (R/I(K n )) = i. i + 1 β i,j (R/I(K n )) = 0 if j i + 1. Theorem 1.5. The minimal free resolution of I(K n ) is linear. 2. Monomial ideals with 2-linear resolutions Question 2.1. Give a combinatorial characterization of monomial ideals generated in degree 2 whose minimal free resolution is linear. Question 2.2. Which edge ideals have linear resolutions? The complete answer is given by Fröberg. To state Fröberg s Theorem we need to first have the following definition: Definition 2.3. A graph G is chordal if every cycle of four or more vertices of G has a chord. A chord is an edge that is not part of the cycle but connects two vertices of the cycle Figure 1. A chordal graph and a non-chordal graph Theorem 2.4 (Fröberg s Theorem). Let G be a graph. resolution if and only if G c is a chordal graph. Then I(G) has a linear Theorem 2.5. Let G be a connected simple graph. Then the following are equivalent: 4

5 Figure 2. The complement of the above chordal graph (1) G c is chordal. (2) The simplex complex whose facets are the complete subgraphs of G c is contractible. (3) The edge ideal has a linear resolution. Theorem 2.6 (Herzog, Hibi, Zheng). If G c is chordal, then all the powers of the ideal I(G) has linear resolutions. This is not true in general. Example 2.7. The monomial ideal I = (def, cef, cdf, cde, bef, bcd, acf, ade) K[a, b, c, d, e, f] has a linear resolution. However, I 2 does not have a linear resolution. Example 2.8 (Conca). Let K be a field of characteristic 0. The ideal I = (a 2, ab, ac, ad, b 2, ae + bd, d 2 ) K[a, b, c, d, e] has a linear resolution, but the ideal I 2 does not have a linear resolution Monomial ideals with d-linear resolutions. Question 2.9. Which ideals have linear resolutions? Question Give a combinatorial characterization of monomial ideals generated in degree d whose minimal free resolution is linear. 5

6 3. Vertex cover ideal of G We define the ideal I G = I(G) to be the Alexander dual of the graph ideal I(G). Definition 3.1. Let G be a graph on the vertex set [n]. A vertex cover of a graph G on [n] is a subset C [n] such that every edge {u, v} of G has at least one vertex in C. A vertex cover C is called minimal if no proper subset of C is a vertex cover of G. An independent set of G is a set S [n] such that no edge is a subset of S. Note that we have: S is an independent set of G if and only if [n]\s is a vertex cover of G. The maximal independent sets of G correspond to its minimal vertex covers. Exercise 3.2. Let G be a graph on [n]. Then I G is minimally generated by the monomials x C = i C x i corresponding to the minimal vertex covers of G. Moreover, I G is the Stanley-Reisner ideal of the simplicial complex (G) = (Ḡ). ring. 4. Cohen-Macaulay simplicial complexes A simplicial complex is called Cohen-Macaulay if R/I is a Cohen-Macaulay The characterization of when a simplicial complex is Cohen-Macaulay is known as Reisner s criterion. First we recall the definition of the link of a face. Given a face F of a simplicial complex, we define the simplicial complex link (F ) as a subcomplex of whose faces are link (F ) = {G : F G, F G = }. Theorem 4.1 (Reisner s criterion). A simplicial complex is Cohen-Macaulay over K if and only if for any face F of we have dim K ( H i (link (F ); K)) = 0 for i < dim(link (F )). Reisner s criterion implies that is Cohen-Macaulay over K if and only if the homology of each face s link vanishes below its top dimension. The following theorem connects Cohen-Macaulayness and minimal free resolutions. 6

7 Theorem 4.2 (Eagon-Reiner). A simplicial complex is Cohen-Macaulay over K if and only if the ideal I has a linear resolution over R = K[x 1,,..., x n ]. The characterization of CohenMacaulay graphs have been studied a lot and a complete characterization is given for the following families of graphs: Chordal graphs Bipartite graphs Cactus graphs. We recall that G is called a cactus graph if every two cycles in G have at most one vertex in common. Figure 3. A cactus graph We only mention the characterization of Cohen-Macaulay bipartite graphs here: Theorem 4.3 (Herzog-Hibi). Let G be a bipartite graph with vertex partition V V. Then the following conditions are equivalent: (1) G is a CohenMacaulay graph. (2) V = V and the vertices V = {x 1,..., x n } and V = {y 1,..., y n } can be labelled such that: {x i, y i } are edges for i = 1,..., n. if {x i, y j } is an edge, then i j. if {x i, y j } and {x j, y k } are edges, then {x i, y k } is an edge. Example 4.4. It is easy to check the combinatorial conditions on the following bipartite graphs to see if they are Cohen-Macaulay or not. An immediate corollary of Eagon-Reiner s Theorem is that: 7

8 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 y 1 y 2 y 3 y 4 y 1 y 2 y 3 y 4 Figure 4. (left) A C.M. and (right) a non-c.m. bipartite graph Corollary 4.5. Let G be a Cohen-Macaulay graph. Then the vertex cover ideal I G of G has a linear resolution. As we have seen before, there are ideals with a linear resolution whose second power does not have the same property. However applying the combinatorial characterization of Cohen-Macaulay graphs in special families of graphs one obtains that all powers of the vertex cover ideal of a Cohen-Macaulay graph have this property. Theorem 4.6 (Herzog-Hibi). Let G be a Cohen-Macaulay chordal graph. Then all powers of I G have a linear resolution. Theorem 4.7 (Mohammadi). All powers of the vertex cover ideal of a Cohen-Macaulay cactus graph have linear resolutions. Here is another example which is not a cactus graph, but Cohen-Macaulay and all the powers of I G have linear resolutions. Figure 5. A Cohen-Macaulay non-cactus graph such that all powers of I G have linear resolutions. Question 4.8. Do all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear resolutions? 8