Seminar Series in Mathematics: Algebra

2003, 1–9

TREES AND QUASI-TREES ∗ Let ∆ be a simplicial complex on the vertex set [n] = {1, . . . , n}. A facet of ∆ is a maximal face of ∆. We denote by F (∆) the set of facets of ∆. Let K be a field, S = K[x1 , . . . , xn ] the polynomial ring over K in n indeterminates. With ∆ there are attached naturally two monomial ideals: • the Stanley-Reisner ideal I∆ = (xF )F6∈∆ , and • the facet ideal I(∆) = (xF )F∈F (∆) . Here xF = xi1 · · · xik for F = {i1 , . . . , ik }. If F (∆) = {F1 , . . . , Fm } then we write ∆ = hF1 , . . . , Fm i, and say that F1 , . . . , Fm generate ∆. Example 1. Let ∆ be the simplicial complex 2

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4 3 Figure 1:

Then I∆ = (x1 x4 , x1 x5 , x2 x5 , x3 x5 ) and I(∆) = (x1 x2 x3 , x2 x3 x4 , x4 x5 ). If dim ∆ = 1, then ∆ is a graph G, and I(∆) = I(G) is the edge ideal of the graph. We want to discuss the following problems: (1) When is S/I(∆) Cohen-Macaulay? (2) When is the Rees ring R (I(∆)) =

L

i≥0 I(∆)

∗ This lecture was held by J¨ urgen Herzog University of Essen, Germany e-mail: [email protected]

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it i

normal?

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For the Stanley- Reisner ring K[∆] = S/I∆ the well-known Reisner criterion tells us when K[∆] is Cohen-Macaulay. However, for S/I(∆) no such criterion is known. For the edge ideal of a tree, Villareal [17] gives a criterion when it is Cohen-Macaulay. More recently there are two extensions of this, namely in the case that: (a) (Herzog-Hibi [8]) G is a bipartite graph, or (b) (Faridi [5], [6]) ∆ is a higher dimensional tree. In this lecture we discuss mostly the work of Faridi. Definition 2. Let ∆ be a simplicial complex. A facet F ∈ F (∆) is called a leaf, if either F is the only facet of ∆, or there exists G ∈ F (∆), G 6= F such that H ∩ F ⊂ G ∩ F for each H ∈ F (∆) with H 6= F. A facet G with this property is called a branch of F. A vertex i of ∆ is called a free vertex if i belongs to precisely one facet. The simplicial complex in Figure 1 has two leaves, namely {1, 2, 3} and {4, 5}, and has one branch, namely {2, 3, 4}. The free vertices are 1 and 5. Definition 3 (Faridi). A simplicial complex is called a forest, if each subcomplex Γ of ∆ which is generated by a subset of the facets of ∆ has a leaf. A forest is called a tree if it is connected. The simplicial complex in Figure 1 is a tree. However the simplicial complex

Figure 2: is not a tree, because it has no leaves at all. Definition 4 (Zheng). A simplicial complex ∆ is called a quasi-forest, if the facets of ∆ can be ordered F1 , . . . , Fm such that Fi is a leaf of hF1 , . . . , Fi−1 i for i = 2, . . . , m. Such an order of the leaves is called a leaf order. If a quasi-forest is connected, then it is called a quasi-tree. The following simplicial complex is quasi-tree, but not a tree. Because if one removes the inside branch, then the remaining simplicial complex has no leaves anymore. One always has: ∆ tree ⇒ ∆ quasi-tree. However the converse is not true in general. We will study quasi-trees in Lecture 2.

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Figure 3: Of course a graph which is a tree is also a tree in the sense of Faridi and a quasi-tree in the sense of Zheng. Lemma 5. Localization of a tree is a forest, i.e. if ∆ is a tree, I = I(∆) and P ∈ Spec(S) with I ⊂ P. Then IP ⊂ SP is minimally generated by monomials corresponding to the facets of a forest. Proof. Let P0 ⊂ P be the prime ideal generated by all xi such that xi ∈ P. Then I ⊂ P0 , and IP0 and IP have the same minimal set of monomial generators. Thus we may replace P by P0 and assume that P is generated by a subset of {x1 , . . . , xn }. Let P = (xi1 , . . . , xir ). Then the simplicial complex corresponding to the generators of IP is obtained from ∆ by removing all vertices j 6∈ {i1 , . . . , ir }. For example if we localize the simplicial complex shown in the next picture at P = (x1 , x2 , x3 , x5 , x6 , x7 ) 6

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1

4

5 7

3 Figure 4:

The leaves that “survive” are again leaves, but the resulting simplicial may be disconnected, in other words, it may be a forest. Theorem 6 (Faridi). Let ∆ be a tree. Then the Rees ring of I(∆) is normal and Cohen-Macaulay. The proof uses the theory of approximation complexes: let (R, m) be local with maximal ideal m (or a graded with graded maximal ideal m), I ⊂ m an ideal (or graded ideal, if R is graded). Let f 1 , . . . , fm be a set of generators of I (homogeneous, if I is graded).

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then we obtain 2 5•

3 Figure 5: We denote by H(g1 , . . . , gr ; R) the Koszul homology of a sequence g1 , . . . gr ∈ R. Definition 7. The ideal I satisfies sliding depth if depth Hi ( f1 , . . . , fm ; R) ≥ dim R − m + i for all i ≥ 0. The definition does not depend on the particular choice of the generators of I. Theorem 8 (Herzog-Simis-Vasconcelos-Villareal). Suppose R is Cohen-Macaulay, with I ⊂ P. Then the I satisfies sliding depth and µ (IP ) ≤ dim RP for all P ∈ Spec(R) L Rees algebra R (I) and the associated graded ring grI (R) = i≥0 I i /I i+1 are CohenMacaulay. If moreover, I is generically a complete intersection and R/I is reduced, then R (I) is normal. Theorem 9 (Faridi). Let ∆ be a tree, and I = I(∆). Then I satisfies the conditions of the previous theorem. Proof. (1) Sliding depth is proved by induction on the number of facets of ∆. (2) Let ∆ be a forest, I = I(∆). We show that µ (IP ) ≤ dim SP . We may assume that P is generated by variables. Since IP is generated by monomials corresponding to a forest, we may well assume that P = (x1 , . . . , xn ). In that case µ (IP ) = µ (I) and dim SP = dim S. Thus we have to show that µ (I) ≤ dim S. Since a forest is a quasiforest, we may assume that ∆ = hF1 , . . . , Fm i where Fi is a leaf of hF1 , . . . , Fi−1 i for all i. Now each Fi contains a free vertex, i.e. a vertex j such that j ∈ Fi but j 6∈ Fk for k < i. Therefore, dim S ≥ number of vertices of ∆ ≥ number of leaves of ∆ = µ (I). (3) For any squarefree monomial ideal, S/I is reduced and I is generically a complete intersection. Let I ⊂ S be a graded ideal. If S/I is Cohen-Macaulay, then S/I is unmixed, that is, the associated prime ideals and in particular the minimal prime ideals of I all have

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the same height. The converse is of course in general false. For example, any prime ideal is unmixed but usually is not Cohen-Macaulay. We say ∆ is unmixed if all minimal prime ideals of I(∆) have the same height. Theorem 10 (Faridi). Let ∆ be a tree. Then the following conditions are equivalent: (a) ∆ is unmixed; (b) S/I(∆) is Cohen-Macaulay. Of course only the implication (a) ⇒ (b) needs a proof. This requires some preparation. Let ∆ be a simplicial complex. Definition 11. A subset G ⊂ [n] is a vertex cover of ∆, if G ∩ F 6= 0/ for all F ∈ F (∆). The set G is a minimal vertex cover, if G is a vertex cover and G0 ⊂ G is not a vertex cover for any proper subset of G. We denote by C (∆) the set of minimal vertex covers. For example the simplicial complex 6 5

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Figure 6: has the minimal vertex covers: {1, 5}, {3, 4}, {2, 6}, {2, 4}, {2, 5}, {4, 5}. For G ⊂ [n] let PG = (xi )i∈G . Then the minimal prime ideals of I(∆) are the prime ideals PG where G is a minimal vertex cover of ∆. Thus ∆ is unmixed if all minimal vertex covers of ∆ have the same cardinality. Definition 12. Let M be a finitely generated graded S-module. The module M is called sequentially Cohen-Macaulay, if there exists a finite filtration 0 = M0 ⊂ M1 ⊂ . . . ⊂ Mr = M of graded submodules of M such that each Mi /Mi−1 is Cohen-Macaulay, and dim M1 /M0 < dim M2 /M1 < · · · < dim Mr /Mr−1 .

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Theorem 13 (Peskine). The following conditions are equivalent: (a) M is sequentially Cohen-Macaulay; (b) Extn−i S (M, S) vanishes, or is Cohen-Macaulay of dimension i for i = 0, . . . , dim M. We say ∆ is sequentially Cohen-Macaulay (over K) if K[∆] = S/I∆ is a sequentially Cohen-Macaulay S-module. The pure subcomplex ∆(q) of ∆ which is generated all q-dimensional faces of ∆ is called the pure q-skeleton of ∆. Theorem 14 (Stanley). The following conditions are equivalent: (a) ∆ is sequentially Cohen-Macaulay; (b) for all integers q, the pure subcomplex ∆(q) is Cohen-Macaulay. Corollary 15. Let I ⊂ S be a squarefree monomial ideal. The following conditions are equivalent: (a) S/I is Cohen-Macaulay; (b) S/I is sequentially Cohen-Macaulay and unmixed. Proof. (a) ⇒ (b) is trivial. (b) ⇒ (a): Let ∆ be the simplicial complex such that I = I∆ . The minimal prime ideals correspond to the facets of ∆. Therefore, I is unmixed if and only if all facets of ∆ have the same dimension. Thus, if d = dim ∆ and I is unmixed, then ∆ = ∆(d). Since ∆ is sequentially Cohen-Macaulay, ∆(= ∆(d)) is Cohen-Macaulay, i.e. S/I is Cohen-Macaulay. Theorem 16 (Faridi). Let ∆ be a tree. Then S/I(∆) is sequentially Cohen-Macaulay. The preceding discussions show that this theorem implies Theorem 10. For the proof of Theorem 16 Alexander duality is used. Definition 17. The simplicial complex ∆∨ = {[n] \ F : F 6∈ ∆} is called the Alexander dual of ∆. One has (∆∨ )∨ = ∆. Lemma 18. Let ∆ be a simplicial complex, and let Γ be the unique simplicial complex such that I∆ = I(Γ). Then I∆ =

\ G∈C (Γ)

PG ,

and

I∆∨ = (xG : G ∈ C (Γ)).

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Example 19. Let I∆ = (x1 x4 , x1 x5 , x2 x5 , x3 x5 ). Then I∆ = (x1 , x2 , x3 )∩(x1 , x5 )∩(x4 , x5 ). Therefore I∆∨ = (x1 x2 x3 , x1 x5 , x4 x5 ). Theorem 20. Let K be a field, ∆ a simplicial complex. Then (a) (Eagon-Reiner) K[∆] is Cohen-Macaulay ⇐⇒ I∆∨ has a linear resolution; (b) ∆ is shellable ⇐⇒ I∆∨ has linear quotients; (c) (Terai) proj dimS K[∆] = reg I∆∨ ; (d) (Herzog-Hibi) ∆ is sequentially Cohen-Macaulay ⇐⇒ I∆∨ is componentwise linear. We explain some of the notions used in the previous theorem: An ideal I has linear quotients if it has set of generators f 1 , . . . , fm such that ( f1 , . . . , fi−1 ) : fi is generated by linear forms for i = 1, . . . , m. It is easy to see that if I has linear quotients then I has a linear resolution. A graded ideal I ⊂ S is said to be componentwise linear if for each integer i, the ideal generated by the ith component Ii of I has a linear resolution. Now, in order to prove Theorem 16, Faridi shows: let ∆ be tree, and write IΓ = I(∆). Then all components of IΓ∨ have linear quotients. Hence IΓ∨ is componentwise linear, and hence S/I(∆) is sequentially Cohen-Macaulay. The following question remains: when is a tree unmixed? Definition 21. A simpicial complex is grafted if ∆ = hG1 , . . . , Gs i ∪ hF1 , . . . , Fr i such that (i) V (hG1 , . . . , Gs i) ⊂ V (hF1 , . . . , Fr i); (ii) G1 , . . . , Gs are all the non-leaves of ∆; (iii) F1 , . . . , Fr are all the leaves of ∆; (iv) Fi ∩ Fj = 0/ for all i 6= j; (v) if Gi is a branch of ∆, then ∆ \ hGi i is also grafted. The answer to the above question is given in Theorem 22 (Faridi). A tree is unmixed if and only it is grafted. The tree below is a not unmixed and hence not Cohen-Macaulay. However after grafting it becomes Cohen-Macaulay.

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Figure 7:

References [1] W. Bruns and J. Herzog, Cohen–Macaulay rings, Revised Edition, Cambridge University Press, 1996. [2] W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Camb. Phil. Soc., 118 (1995), 245 – 257. [3] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 38 (1961), 71 –76. [4] S. Faridi, The facet ideal of a simplicial complex, Manuscripta Mat., 109 (2002), 159 – 174. [5] S. Faridi, Cohen-Macaulay properties of squarefree monomial ideals, Preprint 2002. [6] S. Faridi, Simplicial trees are sequentially Cohen-Macaulay, Preprint 2003. [7] J. Eagon and V. Reiner, Resolutions of Stanley–Reisner rings and Alexander duality, J. Pure Appl. Algebra, 130 (1998), 265–275. [8] J. Herzog and T. Hibi, Distributive Lattices, Bipartite Graphs and Alexander Duality, Preprint 2003. [9] R. Fr¨oberg, On Stanley–Reisner rings, in: Topics in algebra, Banach Center Publications, 26 (2), (1990), 57 – 70. [10] J. Herzog, T. Hibi and X. Zheng, Dirac’s theorem on chordal graphs and Alexander duality, preprint, 2003. [11] J. Herzog, T. Hibi and X. Zheng, Monomial ideals whose powers have a linear resolution, Preprint 2002. [12] T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in “Commutative Algebra and Combinatorics” (M. Nagata and H. Matsumura, Eds.), Advanced Studies in Pure Math., Volume 11, North–Holland, Amsterdam, 1987, pp. 93 – 109. [13] T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw, Glebe, N.S.W., Australia, 1992. [14] R. P. Stanley, Enumerative Combinatorics, Volume I, Wadsworth & Brooks/Cole, Monterey, CA, 1986. [15] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkh¨auser, Boston, MA, 1996.

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[16] B. Sturmfels, Gr¨obner Bases and Convex Polytopes, Amer. Math. Soc., Providence, RI, 1995. [17] R. H. Villareal, Monomial Algebras, Dekker, New York, NY, 2001. [18] N. Terai, Generalization of Eagon–Reiner theorem and h-vectors of graded rings, Preprint 2000. [19] X. Zheng, Resolutions of facet ideals, to appear in Comm. Alg.

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