FF-pure homomorphisms, strong FF-regularity, and FF-injectivity

We discuss Matijevic-Roberts type theorem on strong $F$-regularity, $F$-purity, and Cohen-Macaulay $F$-injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of $F$-purity of homomorphisms using Radu-Andre homomorphisms, and prove basic properties of it. We also discuss a strong version of strong $F$-regularity (very strong $F$-regularity), and compare these two versions of strong $F$-regularity. As a result, strong $F$-regularity and very strong $F$-regularity agree for local rings, $F$-finite rings, and essentially finite-type algebras over an excellent local rings. We prove the $F$-pure base change of strong $F$-regularity.Comment: 37 pages, updated the bibliography, and modified some error

One-dimensional F-definable sets in F((t))

We study definable sets in power series fields with perfect residue fields. We show that certain `one-dimensional' definable sets are in fact existentially definable. This allows us to apply results from previous work about existentially definable sets to one-dimensional definable sets. More precisely, let $F$ be a perfect field and let a be a tuple from $F((t))$ of transcendence degree 1 over $F$. Using the description of $F$-automorphisms of $F((t))$ given by Schilling, we show that the orbit of a under $F$-automorphisms is existentially definable in the ring language with parameters from $F(t)$. We deduce the following corollary. Let $X$ be an $F$-definable subset of $F((t))$ which is not contained in $F$, then the subfield generated by $X$ is equal to $F((t^{p^n}))$, for some $n<\omega$.Comment: 11 page

F-adjunction

In this paper we study singularities defined by the action of Frobenius in characteristic $p > 0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein normal variety then to every normal center of sharp $F$-purity $W \subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined \bQ-divisor $\Delta_{W}$ on $W$ satisfying (K_X)|_W \sim_{\bQ} K_{W} + \Delta_{W}. Furthermore, the singularities of $X$ near $W$ are "the same" as the singularities of $(W, \Delta_{W})$. As an application, we show that there are finitely many subschemes of a quasi-projective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications.Comment: 31 pages; to appear in Algebra and Number Theory. Typos corrected, presentation improved throughout. Section 7 subdivided into two sections (7 and 8). The proofs of 4.8, 5.8 and 9.5 improve

The fully residually F quotients of F*<x,y>

We describe the fully residually F; or limit groups relative to F; (where F is a free group) that arise from systems of equations in two variables over F that have coefficients in F.Comment: 64 pages, 2 figures. Following recommendations from a referee, the paper has been completely reorganized and many small mistakes have been corrected. There were also a few gaps in the earlier version of the paper that have been fixed. In particular much of the content of Section 8 in the previous version had to be replaced. This paper is to appear in Groups. Geom. Dy