Edge rings satisfying Serre's condition R_1

A combinatorial criterion for the edge ring of a finite connected graph to satisfy Serre's condition R_1 is studied.


Introduction
The study on edge polytopes and edge rings of finite connected graphs has been achieved from the viewpoints of both combinatorics and computational commutative algebra ( [3], [4]). Especially, a combinatorial characterization for the edge ring to be normal is obtained by both [3] and [6] independently. It follows immediately from [2, Theorem 6.4.2] that a normal edge ring is Cohen-Macaulay. However, in general it seems unclear when the edge ring is Cohen-Macaulay. Recall that a noetherian ring is normal if and only if it satisfies Serre's conditions (R 1 ) and (S 2 ). Thus in particular an edge ring satisfying Serre's condition (R 1 ) is normal if and only if it is Cohen-Macaulay. In the present paper the problem when a given edge ring satisfies Serre's condition (R 1 ) is investigated.

Edge rings and edge polytopes of finite connected graphs
First, we recall from [3] what edge rings and edge polytopes of finite connected graphs are. Let G be a finite connected graph on the vertex set [ d ] = {1, . . . , d} with E(G) = {e 1 , . . . , e n } its edge set. We always assume that G is simple, i.e., G has no loop and no multiple edge. Let e 1 , . . . , e d denote the ith unit coordinate vectors of R d . We associate each edge e = {i, j} ∈ E(G) with the vector ρ(e) = e i + e j ∈ R d . The edge polytope is the convex polytope P G ⊂ R d which is the convex hull of the finite set {ρ(e 1 ), . . . , ρ(e n )}. Let K[t] = K[t 1 , . . . , t d ] be the polynomial ring in d variables over a field K. We associate each edge e = {i, j} ∈ E(G) with the quadratic monomial t e = t i t j ∈ K[t]. The edge ring is the affine semigroup ring Let, in general, P ⊂ R d be an integral convex polytope, i.e., a convex polytope all of whose vertices have integer coordinates, which lies on a hyperplane H ⊂ R d with 0 ∈ H, where 0 is the origin of R d . We assume that P ⊂ R d ≥0 , where R ≥0 is the set of nonnegative real numbers. Then for each integer point a = (a 1 , . . . , a d ) belonging to P, we associate the monomial t a = t a 1 1 · · · t a d d ∈ K[t]. The toric ring of P is the affine semigroup ring K[P] = K[{t a : a ∈ P ∩ Z d }]. Thus in particular the edge ring K[G] of a finite connected graph G is the toric ring of the edge polytope P G of G.
We say that an integral convex polytope P is normal if its toric ring K[P] is normal. It is shown [3] and [6] that the edge ring of a finite connected graph is normal if and only if G satisfies the odd cycle condition [3, p. 410]. Recall that a toric ring is normal if and only if it satisfies Serre's condition (R 1 ) and (S 2 ) and that every normal toric ring is Cohen-Macaulay. Thus in particular a toric ring satisfying Serre's condition (R 1 ) is normal if and only if it is Cohen-Macaulay. In the present paper the problem when a given edge ring satisfies Serre's condition (R 1 ) is investigated.
Let G be a finite connected graph on the vertex set is normal and satisfies Serre's condition (R 1 ). Thus in what follows we assume that G is nonbipartite, i.e., G possesses at least one odd cycle. If then the bipartite graph induced by T is defined to be the bipartite graph having the vertex set T ∪ N(G; T ) and consisting of all edges {i, j} ∈ E(G) with i ∈ T and j ∈ N(G; T ). Finally, we say that a nonempty subset ) has at least one odd cycle.
We are now in the position to state our criterion for an edge ring to satisfy Serre's condition (R 1 ). First recall the description of the facets of P G . To every regular vertex i we associate the linear form σ i : R d → R which projects onto the i-th component. Moreover, we set H i = x ∈ R d : σ i (x) = 0 and F i = P G ∩ H i . Similarly, to every fundamental set T we associate the linear form  We apply Proposition 3.2 to the affine monoid generated by the integer points in P G . Note that the support hyperplanes H i and H T of P G are also the support hyperplanes of M G . We start proving Theorem 2.1 by the following To check the second condition of Proposition 3.2 we need to compute the lattice generated by M G . The following Lemma 3.4 appears in [3, p. 426] without an explicit proof. However, for the sake of completeness, we give its detailed proof. Proof. Since every generator of M G has an even coordinate sum, it follows that the lattice gp(M G ) is contained in the set of all integer vectors in Z d with an even coordinate sum.
To prove the converse, assume the edges e 1 , . . . , e ℓ form an odd cycle of G and let i be the common vertex of e 1 and e ℓ . Then Now consider a spanning tree G ′ of G. The set { ρ(e) e ∈ E(G ′ ) } together with 2e i forms a Z-basis for the space of all integer vectors in Z d with an even coordinate sum. Now, we can prove two propositions, which complete our proof of Theorem 2.1. Proof. We denote the connected components of  Proof. Again, we denote the connected components of For this we consider a spanning tree of the induced bipartite graph on T ∪ N(G; T ). Its edges form a Z-basis for the left set, hence it is contained in gp(M G ∩ F T ). Next, we note that Now the reasoning is completely analogous to the proof of Proposition 3.5.