Singularities via the Frobenius homomorphism

Resolution of singularities in algebraic varieties has been a topic of interest in commutative algebra and algebraic geometry during the last decades. Altough we have a theory of resolution of singularities for rings of characteristic zero, the case of positive characteristic is not as developed. However, it was discovered that the Frobenius homomorphism in rings of positive characteristic has some properties that play a similar role to the resolution of singularities in characteristic zero. Kunz's theorem characterizes the regularity of a ring in terms of the flatness of its Frobenius homomorphism, and it was the first result that hinted that we could study singularities in a ring via its Frobenius morphism. This is what led to the introduction of F-singularities, which are singularities of a ring of positive characteristic that are related to properties of its Frobenius morphism. What it is surprising is that, when reducing modulo p, the already known singularities in characteristic zero coincide with the F-singularities defined through the Frobenius homomorphism. In this thesis we will present some of the main results in the theory of F-singularities for rings of positive characteristic. We will introduce three types of F-singularities: F-splitness, F-purity and strongly F-regularity. We will study relations between them and methods to test these properties for a given ring. In particular, Fedder's criterion for F-purity and Glassbrenner's criterion for strongly F-regularity will be presented. We will also review some invariants that characterise these singularities, such as test ideals and F-thresholds