Centers of F-purity

By Karl Schwede. 2008

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In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair (R, Δ). We call these analogues centers of F-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of F-purity which we call uniformly F-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that Δ = 0, we show that uniformly F-compatible ideals coincide with the annihilators of the $${\mathcal{F}(E_R(k))}$$ -submodules of ER(k) as defined by Lyubeznik and Smith.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cc94d0c2b2a224e124b3e1dd1d33304190be96cf

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Citations in the corpus

Test ideals in non-Q-Gorenstein rings (2009, background): “…pairs ( R, T ) (respectively triples ( R, T , a t )) are generalizations of triples ( R, Δ , b • ) (respectively triples ( R, T , b • · a t )) as studied by several authors (here b • is a graded system of ideals 1 ); see for example [Tak04b], [HY03], [Har05], [BMS08], [Sch08a], [Sch09, Remark 2.8].”
F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, background): “t follows that ϕ(Fe ∗ x) ∈ PD and so PD is D-compatible. Since any proper D-compatible ideal of Ris automatically contained in PD, we see that PD is in fact the largest proper D-compatible ideal (cf. [Sch10a]). Lemma 2.13 (cf. [Sch09a, Main Theorem, part (v)]). Suppose D is a Cartier subalgebra on a”
On the three dimensional minimal model program in positive characteristic (2013, result): “For a pair (X,∆), we can define F -pure centers as in [Schwede10].”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, unknown): “See [16,35] for additional details.”
A survey of test ideals (2011, background): “t;0 if and only if Ris strongly F-regular. Deﬁnition 8.24. [AE05] Suppose Ris F-pure. If Ie is as in Proposition 8.18, the ideal P= ∩e∈NIeis called the F-splitting prime of R. *Exercise 8.25. [AE05], [Sch10a] Check that Pis a prime ideal, and that φ(P1/pe) ⊆Pfor all φ∈HomR(R1/p e,R). In particular, ”
Test ideals via algebras of $p^{-e}$-linear maps (2009, background): “onsidered in [FST10] under the name of maximal non-F-pure ideal. Question 5.7. More a suggested investigation than a question: Develop Schwede’s theory of Fadjunction [Sch09a] and centers of F-purity [Sch08] in the general context of this paper. With the centers of F-purity we essentailly do this wi”
F-adjunction (2009, methodology): “Here a• is a graded system of ideals [Hara 2005; Schwede 2008a].”