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- W2951279002 abstract "Let $G$ be a finite graph on the vertex set $[d] = {1, ..., d }$ with the edges $e_1, ..., e_n$ and $K[tb] = K[t_1, ..., t_d]$ 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 $tb^e = t_it_j$ such that $e = {i, j}$ is an edge of $G$. Let $K[xb] = K[x_1, ..., x_n]$ be the polynomial ring in $n$ variables over $K$ and define the surjective homomorphism $pi : K[xb] to K[G]$ by setting $pi(x_i) = tb^{e_i}$ for $i = 1, ..., n$. The toric ideal $I_G$ of $G$ is the kernel of $pi$. It will be proved that, given integers $f$ and $d$ with $6 leq f leq d$, there exist a finite connected nonbipartite graph $G$ on $[d]$ together with a reverse lexicographic order $<_{rev}$ on $K[xb]$ and a lexicographic order $<_{lex}$ on $K[xb]$ such that (i) $K[G]$ is normal, (ii) $depth K[xb]/ini_{<_{rev}}(I_G) = f$ and (iii) $K[xb]/ini_{<_{lex}}(I_G)$ is Cohen--Macaulay, where $ini_{<_{rev}}(I_G)$ (resp. $ini_{<_{lex}}(I_G)$) is the initial ideal of $I_G$ with respect to $<_{rev}$ (resp. $<_{lex}$) and where $depth K[xb]/ini_{<_{rev}}(I_G)$ is the depth of $K[xb]/ini_{<_{rev}}(I_G)$." @default.
- W2951279002 created "2019-06-27" @default.
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- W2951279002 date "2011-01-20" @default.
- W2951279002 modified "2023-09-24" @default.
- W2951279002 title "Depth of initial ideals of normal edge rings" @default.
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- W2951279002 doi "https://doi.org/10.48550/arxiv.1101.4058" @default.
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