2014/09/18 by Alain Thiéry, Thiéry, Alain
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1409.5281
arxiv created 2014/09/18 · openalex publication_date 2014/09/18 · arxiv updated 2014/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbFq be the finite field with q elements, K be an algebraically closed field containing \mathbbFq, K\τ\ be the Ore ring of \mathbbFq-linear polynomials and Λn be a free K\τ\-module of rank n. In a first part, we prove that there is a bijection between the set of Zariski closed subsets of Kn which are also \mathbbFq-vector spaces, the so-called q-varities, and the set of radical K\τ\-submodules of Λn. We also study the dimension of q-varieties and their tangent spaces. Let F be a q-variety, K\F\ := Mor(F,K) be the set of \mathbbFq-linear polynomial maps from F to K. Let A=\mathbbFq[T] and choose δ: A \longrightarrow K a ring morphism. By definition, an A-module structure on F is a ring morphism Φ: A \longrightarrow End(F) such that, for all a∈ A, d(Φa) = δ(a) IdT(F) where T(F) is the tangent space of F and d(Φa) the differential map. We prove that K(F) := K(T)⊗K[T]K\F\ has finite dimension over K(T). This dimension is called the rank of the A-module and is denoted by r(F). We then prove that there exists c ∈ A∖ \0\ such that for all a∈ A, prime to c, Tor(a,F) := \x∈ F | Φa(x) = 0\ = (A/aA)r(F).