We investigate finite torsors over big opens of spectra strongly $F$-regular germs that do not extend to the whole spectrum. Let $(R,\mathfrak{m},k)$ be a $k$-germ where $k$ is an algebraically closed field characteristic $p>0$. prove existence local cover $R \subset R^{\star}$ so $R^{\star}$ and: for all algebraic groups $G/k$ with solvable neutral component, every $G$-torsor open $\mathrm{Spe...

We explore the equimultiplicity theory of F-invariants Hilbert–Kunz multiplicity, F-signature, Frobenius Betti numbers, and Euler characteristic in strongly F-regular rings. Techniques introduced this paper provide a unified approach to study localization these invariants detection singularities.

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. Commutative fundamental rings and the fundamental relation α which is the smallest strongly regular relation in hyperringswere introduced by Davvaz and Vougiouklis (2007). Recently, another strongly regular relation named θ on hyperrings has been studied by Ameri and Norouzi (2013). Ameri...