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 $\phi$ and $\psi$ be $A-$Drinfeld modules. In this paper we consider the group ${\mathrm{Ext}}^1(\phi ,\psi )$ with the Baer addition. We show that if $\mathrm{rank}\phi >\mathrm{rank}\psi$ then $\mathrm{Ext^1}(\phi,\psi)$ has the structure of a \tm module. We give complete algorithm describing this structure. We generalize this to the cases: $\mathrm{Ext^1}(\Phi,\psi)$ where $\Phi$ is a \tm module and $\psi$ is a Drinfeld module and $\mathrm{Ext^1}(\Phi, C^{\otimes e})$ where $\Phi$ 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 ${\mathbf t}^{\sigma}$-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.