L. Piriou, L. Schwartz

A Property of the Degree Filtration of Polynomial Functors

This paper is a detailed version of the note with the same title in Note to CRAS (2002). It treats a result related to what is commonly referred to as the artinian conjecture (or finiteness conjecture). This conjecture can be stated in the following way. Consider the category F of functors from the category of finite dimensional vector spaces over the two element field to that of all vector spaces. Consider its full subcategory of functors whose injective envelopes are finite direct sums of indecomposable injectives. The conjecture is that this subcategory is abelian. In our circumstances the only point to prove is that it is stable under quotients (that this formulation is equivalent to the usual one is easy but not formal).
The result proved in this paper shows that the subobject lattices of standard injectives of the category are "as simple as possible" in what concerns the weight filtrations of unstable modules. It is shown that the filtrations by weights and socles are compatible in an appropriate sense. In addition to the recalled notions and facts, the appendix contains a result showing that certain unstable modules are cyclic.