Strict polynomial functors and unstable modules over the Steenrod algebra
type: Item
instance of: scholarly article
title: Strict polynomial functors and unstable modules over the Steenrod algebra
zbMATH Open document ID: 1201.55005
author: J. Martínez
published in: Journal of Algebra
publication date: 2010-09-17T00:00:00Z
review text: Let \(p\geq 2\) be a prime number. The starting point of the paper is the diagram \[ {\mathcal U} \overset{f}{\to} {\mathcal F} \overset{O}{\leftarrow} {\mathcal P} \] Here \({\mathcal U}\) denotes the category of unstable \({\mathcal A}_p\)-modules and \({\mathcal F}\) the category of functors from the category \({\mathcal V}^f\) of finite dimensional \(\mathbb F_p\)-vector spaces to the category of \(\mathbb F_p\)-vector spaces. Denote now by \(\Gamma^d(-)\) the \(d\)th divided power functor and by \(\Gamma^d{\mathcal V}^f\) the category whose objects are finite dimensional \(\mathbb F_p\)-vector spaces and maps are defined by \(\text{Hom}_{\Gamma^d{\mathcal V}^f} (V,W) = \Gamma^d(\text{Hom}(V,W))\). Following Friedlander and Suslin, an \(\mathbb F_p\)-linear functor from \(\Gamma^d{\mathcal V}^f \) into \({\mathcal V}^f\) is called a strict homogeneous polynomial functor of degree \(d\). Then \({\mathcal P}= \bigoplus_{d\geq 0} {\mathcal P}_d\) is the category of strict homogeneous functors. The functor \(f\) in the diagram is standard and associates to an unstable module \(M\) the functor \(f(M)(V) = \text{Hom}_{\mathcal U}(M, S(V^*))\), where \(S(V^*)\) is the symmetric algebra on the dual of \(V\). In this paper Hai constructs a functor \(\widehat{m} : {\mathcal P}\to {\mathcal U}\) that is exact, that commutes with tensor product and Frobenius torsion, and satisfies \(f\circ \widehat{m} = O\).
reviewed by: Yves Félix
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 5787078
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: unstable module
zbMATH Keywords: polynomial functor
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.04.016
OpenAlex ID: W1999655549
cites work: The Categories of Unstable Modules and Unstable Algebras Over the Steenrod Algebra Modulo Nilpotent Objects
cites work: Generic Representations of the Finite General Linear Groups and the Steenrod Algebra: I
cites work: Cohomology of finite group schemes over a field
cites work: Q4875306
cites work: Koszul duality and extensions of exponential functors
cites work: Q4831101
cites work: Sur les ${\scr U}$-injectifs
cites work: General linear and functor cohomology over finite fields
cites work: Q3952291
cites work: A stratification of generic representation theory and generalized Schur algebras.
cites work: Subalgebras of Steenrod algebra and the action of matrices on truncated polynomial algebras
cites work: Q4707486
cites work: Polynomial representations of \(GL_n\)
DOI: 10.1016/J.JALGEBRA.2010.04.016
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description: scientific article; zbMATH DE number 5787078