$q$-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 ${\mathrm{\Lambda }}_n$ 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 ${\mathrm{\Lambda }}_n$. 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 $\delta :A⟶K$ a ring morphism. By definition, an $A$-module structure on $F$ is a ring morphism $\Phi : A \longrightarrow End(F)$suchthat,forall$a\in A$,$$d({\mathrm{\Phi }}_a)=\delta (a)I{d}_{T(F)}$$where$T(F)$isthetangentspaceof$F$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 $c$, $$Tor(a,F):=\{x\in F\mid {\mathrm{\Phi }}_a(x)=0\}=(A\mathrm{/}aA{)}^{r(F)}\mathrm{.}$$