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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 of the following themes, including many other topics. 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 applications, we prove the following results. (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 and which is stated for generalized Cohen-Macaulay local rings. (b) Let $(R,fm,k)$ be a generalized Cohen-Macaulay ring of characteristic $p>0$. If the Frobenius action is injective on the local cohomology $H_{fm}^i(R)$ for all $i < dim R$, then $R$ is Buchsbaum. This gives an answer to a question of Takagi. We consider the problem when the union of two $F$-injective closed subschemes of a Noetherian $mathbb{F}_p$-scheme is $F$-injective. Using this idea, we construct an $F$-injective local ring $R$ such that $R$ has a parameter ideal that is not Frobenius closed. This result adds a new member to the family of $F$-singularities. We give the first ideal-theoretic characterization of $F$-injectivity in terms the Frobenius closure and the limit closure. We also give an answer to the question about when the Frobenius action on the top local cohomology is injective.
The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relati
We give sufficient conditions for F-injectivity to deform. We show these conditions are met in two common geometrically interesting setting, namely when the special fiber has isolated CM-locus or is F-split.
We develop new methods to study $mathfrak{m}$-adic stability in an arbitrary Noetherian local ring. These techniques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine $mathfrak{m}$-adic perturbations.
In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenst