**
L. Piriou, L. Schwartz**

##
A Property of the Degree Filtration of Polynomial Functors

**abstract:**

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.