On Ext1 for Drinfeld modules

Abstract

Let A=Fq[t] be the polynomial ring over a finite field Fq and let ϕ and ψ be A-Drinfeld modules. In this paper we consider the group Ext1(ϕ,ψ) with the Baer addition. We show that if rankϕ>rankψ then Ext1(ϕ,ψ) has the structure of a t-module. We give complete algorithm describing this structure. We generalize this to the cases: Ext1(Φ,ψ) where Φ is a t-module and ψ is a Drinfeld module and Ext1(Φ,C⊗e) where Φ is a t-module and C⊗e is the e-th tensor product of Carlitz module. We also establish duality between Ext groups for t-modules and the corresponding adjoint tσ-modules. Finally, we prove the existence of “Hom−Ext” six-term exact sequences for t-modules and dual t-motives. As the category of t-modules is only additive (not abelian) this result is nontrivial.