On ${\mathrm{Ext}}^1$ for Drinfeld modules
Let $A={\mathbb F}_q[t]$ be the polynomial ring over a finite field ${\mathbb F}_q$ and let $ϕ$ and $ψ$ be $A-$Drinfeld modules. In this paper we consider the group ${\mathrm{Ext}}^1(ϕ,ψ)$ with the Baer addition. We show that if $\mathrm{rank}ϕ>\mathrm{rank}ψ$ then $\mathrm{Ext^1}(ϕ,ψ)$ has the structure of a \tm module. We give complete algorithm describing this structure. We generalize this to the cases: $\mathrm{Ext^1}(Φ,ψ)$ where $Φ$ is a \tm module and $ψ$ is a Drinfeld module and $\mathrm{Ext^1}(Φ, C^{\otimes e})$ where $Φ$ is a \tm module and $C^{\otimes e}$ is the $e$-th tensor product of Carlitz module. We also establish duality between $\Ext$ groups for \tm modules and the corresponding adjoint ${\mathbf t}^σ$-modules. Finally, we prove the existence of $"\Hom-\Ext"$ 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.