Strict polynomial functors and unstable modules over the Steenrod algebra
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 \(f\circ \widehat{m} = O\).
scientific article; zbMATH DE number 5787078
scientific article