$q$-Varieties and Drinfeld Modules

Let $\mathbb{F}_q$ be the finite field with $q$ elements, $K$ be an algebraically closed field containing $\mathbb{F}_q$, $K\{\tau\}$ be the Ore ring of $\mathbb{F}_q$-linear polynomials and $\Lambda_n$ be a free $K\{\tau\}$-module of rank $n$. In a first part, we prove that there is a bijection between the set of Zariski closed subsets of $K^n$ which are also $\mathbb{F}_q$-vector spaces, the so-called $q$-varities, and the set of radical $K\{\tau\}$-submodules of $\Lambda_n$. 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 $\mathbb{F}_q$-linear polynomial maps from $F$ to $K$. Let $A=\mathbb{F}_q[T]$ and choose $\delta : A \longrightarrow K$ a ring morphism. By definition, an $A$-module structure on $F$ is a ring morphism $\Phi : A \longrightarrow End(F)$ such that, for all $a\in A$, $$d(\Phi_a) = \delta(a) Id_{T(F)}$$ where $T(F)$ is the tangent space of $F$ and $d(\Phi_a)$ the differential map. We prove that $K(F) := K(T)\otimes_{K[T]}K\{F\}$ 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 $c \in A\setminus \{0\}$ such that for all $a\in A$, prime to $c$, $$Tor(a,F) := \{x\in F \mid \Phi_a(x) = 0\} = (A/aA)^{r(F)}.$$