Depth of Initial Ideals of Normal Edge Rings |
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| Authors: | Takayuki Hibi Kyouko Kimura Augustine B O'keefe |
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| Institution: | 1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology , Osaka University , Toyonaka , Osaka , Japan;2. Department of Mathematics, Graduate School of Science , Shizuoka University , Suruga-ku , Shizuoka , Japan;3. Mathematics Department , University of Kentucky , Lexington , KY , USA |
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| Abstract: | Let G be a finite graph on the vertex set d] = {1,…, d} with the edges e 1,…, e n and Kt] = Kt 1,…, t d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring KG] which is generated by those monomials t e = t i t j such that e = {i, j} is an edge of G. Let Kx] = Kx 1,…, x n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: Kx] → KG] 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 Kx] and a lexicographic order <lex on Kx] such that (i) KG] is normal with Krull-dim KG] = d, (ii) depth Kx]/in<rev (I G ) = f and Kx]/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 Kx]/in<rev (I G ) is the depth of Kx]/in<rev (I G ). |
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| Keywords: | Edge ring Gröbner basis Initial ideal Shellable complex Toric ideal |
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