# WEEKLY ACTIVITIES

$F$-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic $p>0$ (cont.)
 Speaker: Pham Hung QuyTime: 9h00, Wednesday, January 20, 2016Location: Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, HanoiAbstract: The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists four main parts. We prove that if every parameter ideal of a Noetherian local ring of prime characteristic $p>0$ is Frobenius closed, then it is $F$-injective. We prove a necessary and sufficient condition for the injectivity of the Frobenius action on $H^i_{fm}(R)$ for all $i le f_{fm}(R)$, where $f_{fm}(R)$ is the finiteness dimension of $R$. As application we prove that: (a) if the ring is $F$-injective, then every ideal generated by a filter regular sequence, whose length is equal to the finiteness dimension of the ring, is Frobenius closed. It is a generalization of a recent result of Ma which stated for generalized Cohen-Macaulay local rings; (b) if a generalized Cohen-Macaulay admits the injectivity of Frobenius for all local cohomologies except the top local cohomology, then it is Buchsbaum. This gives a striking answer for a question of Takagi. We also prove a recent result of Bhatt, Ma and Schwede with an elementary proof. We consider the problem when is the union of two $F$-injective schemes $F$-injective. Using this idea we construct an $F$-injective local ring $R$ such that $R$ has a parameter ideal that is not Frobenius closed. Our results add a new member to the family of $F$-singularities. We also give the first ideal-theoretic characterization of $F$-injectivity in terms the Frobenius closure and the limit closure. We also answer the question when is the Frobenius action on the top local cohomology is injective. Many other interesting results are discussed. This is joint work with Kazuma Shimomoto

### Highlights

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