Depth of Initial Ideals of Normal Edge Rings

Abstract

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e ₁,…, e _n and K[t] = K[t ₁,…, t _d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t ^e  = t _i t _j such that e = {i, j} is an edge of G. Let K[x] = K[x ₁,…, x _n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x _i ) = t ^{e _i} for i = 1,…, n. The toric ideal I _G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <_rev on K[x] and a lexicographic order <_lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in_<rev (I _G ) = f and K[x]/in_<lex (I _G ) is Cohen–Macaulay, where in_<rev (I _G ) (resp., in_<lex (I _G )) is the initial ideal of I _G with respect to <_rev (resp., <_lex) and where depth K[x]/in_<rev (I _G ) is the depth of K[x]/in_<rev (I _G ).

Key Words:

• Edge ring
• Gröbner basis
• Initial ideal
• Shellable complex
• Toric ideal

2010 Mathematics Subject Classification:

• 13P10

Notes

Communicated by S. Goto.