Purity and almost strict purity of Anderson \(t\)-modules (Q6607451)

In [Duke Math. J. 53, 457--502 (1986; Zbl 0679.14001)] \textit{G. W. Anderson} defined \(t\)-modules and the notion of purity. \textit{C. Namoijam} and \textit{M. A. Papanikolas} defined in [Hyperderivatives of periods and quasi-periods for Anderson \(t\)-modules. Providence, RI: American Mathematical Society (AMS) (2024; Zbl 07946547)] the notion of almost strict purity. In fact, this concept was already mentioned by Anderson and by D. Goss. Namoijam and Papanikolas proved that an almost strictly pure \(t\)-module is pure.\N\NThe main objective of this paper is to study the reciprocal. It is shown that these two notions are not equivalent. The main result, Theorem 3, establishes that for any integer \(d\geq 2\), there exists a pure but not almost strictly pure \(t\)-module of dimension \(d\).