## The Artinian conjecture for $$I^{\otimes 2}$$. With an appendix by Lionel Schwartz.(English)Zbl 0928.18004

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 $$\mathcal F$$.

### MSC:

 18G05 Projectives and injectives (category-theoretic aspects) 20G05 Representation theory for linear algebraic groups 55S10 Steenrod algebra
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### References:

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