Consider the (abelian) category \(\mathcal{F}\) of functors from \(\mathcal{E}_f\) to \(\mathcal{E}\), where \(\mathcal{E}\) is the category of \(\mathbb F\)-vector spaces, with \(\mathbb F=\mathbb F_2\) the field with two elements, and \(\mathcal{E}_f\) the sub-category of finite-dimensional spaces. The injective object \(I_V\), for \(V\in \mathcal{E}_f\), is defined by the co-representing property: \(\operatorname{Hom}_{\mathcal{F}}(F,I_V) \cong F(V^*)^*\), where \((\;)^*\) denotes the vector space dual. Recall that an object \(F\in\mathcal{F}\) is said to be Artinian if every descending sequence of sub-objects of \(F\) stabilizes. The “Artinian conjecture” of Kuhn, Lannes and Schwartz, states that: the functors \(I_V\) are Artinian. The main result of the paper under review is that the functor \(I_{{\mathbb F}^2}\cong I^{\otimes 2}\) is Artinian (where \(I=I_{\mathbb F}\)). This result is the first non-trivial case of the Artinian conjecture.
In fact the author shows a stronger result, namely: the functor \(I^{\otimes 2}\) is Artinian of type 2 (a functor is said simple Artinian of type \(0\) if it is a finite simple functor. Then a functor \(F\) is simple Artinian of type \(n\) if every proper subobject of \(F\) has Artinian type strictly less than \(n\) and \(F\) is said to be Artinian of type \(n\) if it admits a finite filtration of which the sub-quotients are simple Artinian of type \(\leq n\)). The proof of the conjecture in this case follows by reducing the problem to showing that certain “smaller functors”, related to the decomposition of \(I^{\otimes 2}\), are Artinian (and more precisely simple Artinian of type 2). The method of proof exploits the polynomial filtration of the difference functor \(\Delta : \mathcal F \to \mathcal F\) and the study of co-Weyl objects in the category \(\mathcal F\).