Proceedings of the 44th Symposium on Ring Theory and ...

By comparing this <strong>the</strong>orem and Proposition 8, we get <strong>the</strong> following corollary. A graph
G is said to be complete r-partite if <strong>the</strong>re exists a partition V (G) = ∐ r
i=1 V i such that
{i, j} ∈ E(G) if and only <strong>of</strong> a ≠ b for all i ∈ V a and j ∈ V b .
Corollary 10. Closed graphs and complete r-partite graphs are weakly closed.
Pro<strong>of</strong>. Assume that G is complete r-partite and V (G) = ∐ r
i=1 V i. Let {i, j} ∈ E(G) with
i ∈ V a and j ∈ V b . Then a ≠ b. Hence for all i < k < j, k ∉ V a or k /∈ V b . This implies
that {i, k} ∈ E(G) or {k, j} ∈ E(G).
□
3. F -purity <strong>of</strong> binomial edge ideals
In this section, we study about F -purity <strong>of</strong> binomial edge ideals. Firstly, we recall that
<strong>the</strong> definition <strong>of</strong> F -purity <strong>of</strong> a ring R.
Definition 11 (See [5]). Let R be an F -finite reduced Noe<strong>the</strong>rian ring <strong>of</strong> characteristic
p > 0. R is said to be F-pure if <strong>the</strong> Frobenius map R → R, x ↦→ x p is pure, equivalently,
<strong>the</strong> natural inclusion τ : R ↩→ R 1/p , (x ↦→ (x p ) 1/p ) is pure, that is, M → M ⊗ R R 1/p ,
m ↦→ m ⊗ 1 is injective for every R-module M.
The following proposition, which is called <strong>the</strong> Fedder’s criterion, is useful to determine
<strong>the</strong> F -purity <strong>of</strong> a ring R.
Proposition 12 (See [3]). Let (S, m) be a regular local ring <strong>of</strong> characteristic p > 0. Let
I be an ideal <strong>of</strong> S. Put R = S/I. Then R is F -pure if and only if I [p] : I ⊈ m [p] , where
J [p] = (x p | x ∈ J) for an ideal J <strong>of</strong> S.
In this section, we consider <strong>the</strong> following question:
Question. When is S/J G F -pure ?
In [8], Ohtani proved that if G is complete r-partite graph <strong>the</strong>n S/J G is F -pure. Moreover,
it is easy to show that if G is closed <strong>the</strong>n S/J G is F -pure. However, <strong>the</strong>re are many
examples <strong>of</strong> G such that G is nei<strong>the</strong>r complete r-partite nor closed but S/J G is F -pure.
Namely, <strong>the</strong>re is room for improvement about <strong>the</strong> above studies.
The second main <strong>the</strong>orem <strong>of</strong> this chapter is as follows:
Theorem 13. If G is weakly closed, <strong>the</strong>n S/J G is F -pure.
Pro<strong>of</strong>. For a sequence v 1 , v 2 , . . . , v s , we put
Y v1 (v 1 , v 2 , . . . , v s )X vs := (Y v1 [v 1 , v 2 ][v 2 , v 3 ] · · · [v s−1 , v s ]X vs ) p−1 .
Let m = (X 1 , . . . , X n , Y 1 , . . . , Y n )S. By taking completion and using Proposition 2.2, it
is enough to show that Y 1 (1, 2, . . . , n)X n ∈ (J [p]
G : J G) \ m [p] . It is easy to show that
Y 1 (1, 2, . . . , n)X n ∉ m [p] by considering its initial monomial.
Next, we use <strong>the</strong> following lemmas (see [8]):
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