Homological Methods in the Generalization of Drinfeld Modules

TL;DR

This paper introduces triangular t-modules, a homological generalization of Drinfeld modules, exploring their structure, morphisms, endomorphisms, and duality, with implications for arithmetic and Taelman's conjecture.

Contribution

It develops a comprehensive theory of triangular t-modules, including criteria for purity, reduction procedures, morphism characterizations, and duality, extending the understanding of Anderson t-modules.

Findings

Triangular t-modules are uniformizable.

Almost strictly pure modules become strictly pure after finite extension.

Established duality and analogues of classical theorems for these modules.

Abstract

We introduce and study a natural class of Anderson t- modules, called triangular t-modules, characterized by having Drinfeld modules as their $\tau$-composition factors. They form a homologically meaningful generalization of Drinfeld modules and exhibit rich arithmetic structure.\smallskip We establish criteria for purity, strict and almost strict, and develop a reduction procedure that lowers the degrees of the defining biderivations. As a consequence, every almost strictly pure triangular t-module becomes strictly pure after a finite base extension. We then investigate morphisms and isogenies between triangular t-modules, provide a characterization of triangular isogenies, and describe the algebra of endomorphisms, including a criterion for commutativity. On the analytic side, we show that all triangular t- modules are uniformizable and establish finiteness and purity criteria with consequences for Taelman's conjecture. Finally, we develop a duality theory for triangular t- modules and their biderivations, proving compatibility with $\tau$-composition series and establishing analogues of the Cartier-Nishi theorem and the Weil-Barsotti formula.