We introduce the notion of a coordinate k ${\bf k}$ -algebra scheme and corresponding B $\mathcal {B}$ -operator. This class operators includes endomorphisms derivations Frobenius map, it generalizes related to D {D}$ -rings from Moosa Scanlon (J. Math. Log. 14 (2014), no. 02, 1450009) as well. classify (coordinate) schemes for perfect field we also discuss model-theoretic properties fields wit...

Part 1 of this project is concerned with proving a relative version of the results in [1]. Consider the category of Lefschetz rings in which the objects are rings of equal characteristic zero which are realized as an ultraproduct of Noetherian rings of positive characteristic, and the morphisms between two such ultraproducts are given as ultraproducts of homomorphisms between the components. A ...

A finite Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. Such a group can be expressed as a semidirect product G = K o H, where K is a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K whose connection set is an H-orbit S on K that generates K, where H is of even order or S consists...

Given a λ-ring A and a formally etale morphism f : A→ B of commutative rings, one may ask: What are the possible λ-ring strutures on B such that f is a map of λ-rings? We give the answer: Such a lifted λ-ring structure on B is determined uniquely by a compatible choice of lifts of the Adams operations ψ from A to B for all primes p which satisfy Frobenius congruences. In other words, to extend ...

We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F |K in one ...

In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using $$\lambda$$ -circulant matrices. The constructions are derived as modifications of various well-known circulant codes. Applying these together the building-up construction, construct many best known quaternary lengths 26, 32, 36,...

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne–Lusztig characters when the parameters generic. We formulate conjecture simple perversity these complexes, and prove it in level two case (thus extend result Lusztig from function field case). then discuss induction re...

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson-...

In this paper we consider stabilizer codes over local Frobenius rings. First, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer c...