Subobjects of \(\overline I\otimes\Lambda^ n\) in the category of functors between \(\mathbb{F}_ 2\)-vector spaces
type: Item
instance of: scholarly article
title: Subobjects of \(\overline I\otimes\Lambda^ n\) in the category of functors between \(\mathbb{F}_ 2\)-vector spaces
zbMATH Open document ID: 0880.18008
author: Laurent Piriou
published in: Journal of Algebra
publication date: 1998-01-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 1053876
zbMATH Keywords: unstable modules over Steenrod algebra
zbMATH Keywords: locally noetherian category
zbMATH Keywords: polynomial functors
zbMATH Keywords: injective object
zbMATH Keywords: artinian conjecture
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: jabr.1996.7010
OpenAlex ID: W2196758450
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DOI: 10.1006/JABR.1996.7010
description: scientific article; zbMATH DE number 1053876