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Register a new userIntegral closure of ideals
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Authors
Douglas A. Leonard
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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 relative to a local monomial ordering. A non-homogeneous version of the standard (homogeneous) Rees algebra is introduced as well.
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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 of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.
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.
Let $(R,mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial allows to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $bz^noplus T$ with no invertible elements, where $T$ is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.
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