Subobjects of \(\overline I\otimes\Lambda^ n\) in the category of functors between \(\mathbb{F}_ 2\)-vector spaces
The abelian category \({\mathcal F}\) of functors from finite-dimensional vector spaces over the field \(\mathbb{F}_q\) with \(q\) elements to arbitrary vector spaces over the same field recently has been investigated for several reasons. In representation theory, this category \({\mathcal F}\) (and its subcategories of polynomial and of strictly polynomial functors) contains much information about cohomology of both the infinite and the finite general linear group [see, e.g., a recent paper by \textit{E. M. Friedlander} and \textit{A. Suslin}, Invent. Math. 127, No. 2, 209-270 (1997)]. In algebraic topology, especially in the work of \textit{N. Kuhn}, \({\mathcal F}\) is discussed because of its relation to unstable modules over the Steenrod algebra. The paper under review considers the special case of \({\mathcal F}\) over \(\mathbb{F}_q= \mathbb{F}_2\), the field of two elements. It is well-known how to associate to each finite-dimensional \(\mathbb{F}_2\)-vector space, say \(E\), an injective object \(I_E\) in \({\mathcal F}\). The so-called artinian conjecture, which is due to Kuhn, Lannes and Schwartz, asserts that these objects \(I_E\) are artinian objects. In other words, the category \({\mathcal F}\) is conjectured to be locally noetherian. A recent preprint by \textit{G. Powell} proves this conjecture in the case of \(E\) having dimension two (the case of dimension one is known). The paper under review contributes one of the main ingredients for Powell's proof. The results are interesting, but too technical to be stated here. The proof involves looking at filtrations by polynomial subfunctors and determining certain subfunctors and extension groups in a rather precise way.
scientific article; zbMATH DE number 1053876