WEAKLY CLOSED GRAPH KAZUNORI MATSUDA (NAGOYA UNIVERSITY)

Let k be an F -finite field of characteristic p > 0, and S = k[X1 , . . . , Xn , Y1 , . . . , Yn ] a polynomial ring in 2n variables. Let G be a connected simple graph on the vertex set V (G) = [n] with edge set E(G), and JG = (Xi Yj − Xj Yi | {i, j} ∈ E(G)) ⊂ S a binomial edge ideal of G (see [HeHiHrKR], [O]). The purpose of this talk is to study the following question: Question 1. When is S/JG F -pure ? About the definition of F -purity, see [HoR]. In order to answer this question, we introduce the notion of weakly closed graph. This notion is a generalization of closedness (see [HeHiHrKR], [EHeHi]). We will state the definition of weakly closedness in this talk. The first main theorem of this talk is a characterization of weakly closed graphs. Theorem 2. Let G be a connected simple graph. Then the following conditions are equivalent: 1. G is weakly closed. 2. For all i, j such that {i, j} ∈ E(G) and j > i + 1, the following assertion holds: for all i < k < j, {i, k} ∈ E(G) or {k, j} ∈ E(G). Corollary 3. Closed graphs and complete r-partite graphs are weakly closed. The second main theorem of this talk is an answer of Question 1. Theorem 4. If G is weakly closed, then S/JG is F -pure. At the 32th Symposium on Commutative Algebra, Ohtani proved that if G is complete r-partite graph then S/JG is F -pure. Theorem 4 is a generalization of his theorem. References [EHeHi] V. Ene, J. Herzog and T. Hibi, Cohen-Macaulay binomial edge ideals, arXiv:1004.0143. [HeHiHrKR] J. Herzog, T. Hibi, F. Hreind´ottir, T. Kahle and J. Rauh, Binomial edge ideals and conditional independence statements, Adv. Appl. Math., 45 (2010), 317–333. [HoR] M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math., 21 (1976), 117–172. [O] M. Ohtani, Graphs and ideals generated by some 2-minors, Comm. Alg., 39 (2011), 905–917. Graduate School of Mathematics, Nagoya University, Nagoya 464–8602, Japan E-mail address: [email protected]

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