## Centers of $$F$$-purity.(English)Zbl 1213.13014

Let $$R$$ be a Noetherian excellent ring of positive characteristic $$p$$, where $$p$$ is prime. The paper under review introduces, in positive characteristic, the notion of a center of $$F$$-purity for a pair $$(X=\text{Spec}(R), \Delta)$$ that is analogous to the zero characteristic notion of a center of log canonicity for a pair $$(X, \Delta)$$, where $$X$$ is a normal affine scheme and $$\Delta$$ an effective $$\mathbb{Q}$$-divisor on it. In fact, the author develops the theory for the context of triples $$(R, \Delta, \mathfrak{a}_{\bullet})$$, where $$R$$ is a ring as above (normal when $$\Delta \neq 0$$), $$\Delta$$ an effective $$\mathbb{R}$$-divisor and $$\mathfrak{a}_{\bullet}$$ a graded system of ideals that satisfy some natural conditions with regard to $$R$$. The author defines the concept of sharply $$F$$-pure and strong $$F$$-regular triples, which extends his earlier work [Math. Res. Lett. 15, No. 5-6, 1251–1261 (2008; Zbl 1185.13010)], the notion of big test ideal for triples, and proves basic results on them.
The core of the paper consists of the theory of ideals of $$R$$ that are $$(\Delta, \mathfrak{a}_{\bullet}, F)$$-compatible. This notion is related to that of $$F$$-compatible ideals of V. B. Mehta and A. Ramanathan [Ann. Math. (2) 122(1), 27–40 (1985; Zbl 0601.14043)] and leads naturally to the concept of centers of (sharp) $$F$$-purity which are the characteristic zero analog of centers of log canonicity. Important results connected to the work of G. Lyubeznik and K. E. Smith [Trans. Am. Math. Soc. 353, No. 8, 3149–3180 (2001; Zbl 0977.13002) (electronic)] on one hand and to work of I. M. Aberbach and F. Enescu [Math. Z. 250, No. 4, 791–806 (2005; Zbl 1102.13001)] on the other hand are developed. Also, finiteness results on uniformly $$F$$-compatible ideals and centers of $$F$$-purity inspired by work of F. Enescu and M. Hochster [Algebra Number Theory 2, No. 7, 721–754 (2008; Zbl 1190.13003)], and R. Y. Sharp [Trans. Am. Math. Soc. 359, No. 9, 4237–4258 (2007; Zbl 1130.13002) (electronic)] are obtained. Relations to big test ideals, multiplier ideals, and local geometric properties of the centers of $$F$$-purity are pursued in the final two sections of the paper.

### MSC:

 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14B05 Singularities in algebraic geometry
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### References:

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