On ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ for Drinfeld modules

Let $A={\mathbb{F}}_q[t]$ be the polynomial ring over a finite field ${\mathbb{F}}_q$ and let $\phi$ and $\psi$ be $A-$Drinfeld modules. In this paper we consider the group ${\mathrm{E}\mathrm{x}\mathrm{t}}^1(\phi ,\psi )$ with the Baer addition. We show that if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\phi >\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\psi$ then $\mathrm{E}\mathrm{x}{\mathrm{t}}^1(\phi ,\psi )$ has the structure of a \tm module. We give complete algorithm describing this structure. We generalize this to the cases: $\mathrm{E}\mathrm{x}{\mathrm{t}}^1(\mathrm{\Phi },\psi )$ where $\mathrm{\Phi }$ is a \tm module and $\psi$ is a Drinfeld module and $\mathrm{E}\mathrm{x}{\mathrm{t}}^1(\mathrm{\Phi },C^{\otimes e})$ where $\mathrm{\Phi }$ is a \tm module and $C^{\otimes e}$ is the $e$-th tensor product of Carlitz module. We also establish duality between $\text{Ext}$ groups for \tm modules and the corresponding adjoint ${\mathbf{t}}^{\sigma }$-modules. Finally, we prove the existence of $\mathrm{"}\text{Hom}-\text{Ext}\mathrm{"}$ six-term exact sequences for \tm modules and dual \tm motives. As the category of \tm modules is only additive (not abelian) this result is nontrivial.