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TitleDepth of initial ideals of normal edge rings.
Author(s) Takayuki Hibi, Akihiro Higashitani, Kyouko Kimura, Augustine B. OKeefe
TypeArticle in Journal
AbstractLet 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 &#960;: K[x] &#8594; K[G] by setting &#960;(x i ) = t e i for i = 1,…, n. The toric ideal I G of G is the kernel of &#960;. It will be proved that, given integers f and d with 6 &#8804; f &#8804; 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 ).
Keywords Edge ring, Gröbner basis, Initial ideal, Shellable complex, Toric ideal, 13P10
ISSN0092-7872; 1532-4125/e
URL http://www.tandfonline.com/doi/abs/10.1080/00927872.2012.760565#.VfA0u1ppE5s
JournalCommun. Algebra
PublisherTaylor & Francis, Philadelphia, PA
Translation No
Refereed No