A Property of the Degree Filtration of Polynomial Functors

Abstract

This paper is a detailed version of the note with the same title ([Piriou and Schwartz, 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).