On ${\mathrm{Ext}}^1$ for Drinfeld modules

D. E. Kedzierski, P. Krasoń

Published 2022-10-15Version 1

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 formulas describing this structure. We also establish duality between Ext groups for t-modules and the corresponding adjoint ${t}^{\sigma}$-modules. Finally, we prove the existence of "Hom-Ext"$ six-term exact sequences for \tm modules. As the category of t- modules is only additive (not abelian) this result is nontrivial.