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

TL;DR

This paper investigates the structure of Ext^1 groups for Drinfeld modules over polynomial rings, providing algorithms, generalizations, duality results, and exact sequences in the context of t-motives.

Contribution

It introduces a detailed structure for Ext^1 groups of Drinfeld modules, generalizes to t-motives, and establishes duality and exact sequences in a non-abelian setting.

Findings

Ext^1(, \psi) has a t-module structure when rank conditions are met

Provides an explicit algorithm for the structure of Ext^1 groups

Establishes duality and six-term exact sequences for t-motives

Abstract

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.