The structure of the tensor product of \(\mathbb{F}_2[-]\) with a finite functor between \(\mathbb{F}_2\)-vector spaces
The paper under review is the continuation of the work of the author related to the artinian conjecture [see \textit{G. M. L. Powell}, J. Pure Appl. Algebra 128, No. 3, 291-310 (1998; Zbl 0928.18004), Trans. Am. Math. Soc. 350, No. 10, 4167-4193 (1998; Zbl 0903.18006), J. Algebra 194, No. 2, 455-466 (1997; Zbl 0893.55011)]. Let \(\mathcal F\) be the category of functors from the category of finite-dimensional \({\mathbb F}_2\)-vector spaces to the category of all \({\mathbb F}_2\)-spaces, and let \(\overline{I}\) denotes the injective envelope (in \(\mathcal F\)) of the first exterior power functor \(\bigwedge^1\). Then the main result of the paper is that for any finite functor \(F\) of \({\mathcal F}\), the functor \(\overline{I}\otimes F\) is artinian (of type one). The strategy of the proof is as follows: first the author gives a criterion for showing that a functor is artinian by using some knowledge on an artinian sub-functor of it. Then he reduces to the case where \(F\) is simple. Finally he constructs a sub-functor of \(\overline{I}\otimes F\), shows that it is artinian by using a detection result (theorem 4 in the paper), and applies the criterion to this setting in order to show that \(\overline{I}\otimes F\) is artinian.
scientific article; zbMATH DE number 1478804