On the Singer functor R_1 and the functor Fix

TL;DR

This paper uses Lannes' T-functor to construct the Singer functor R_1 on unstable modules over the Steenrod algebra, proving that Fix R_1 is naturally equivalent to the identity and exploring its properties.

Contribution

It provides a new construction of R_1 via Lannes' T-functor and establishes its key property that Fix R_1 is the identity, with further analysis of its behavior on specific modules.

Findings

Fix R_1 is naturally equivalent to the identity functor.

Properties of R_1 on reduced and nilclosed modules are characterized.

A new construction of R_1 using Lannes' T-functor is presented.

Abstract

Lannes' T-functor is used to give a construction of the Singer functor R_1 on the category U of unstable modules over the Steenrod algebra A. This leads to a direct proof that the composite functor Fix R_1 is naturally equivalent to the identity. Further properties of the functors R_1 are deduced, especially when applied to reduced and nilclosed unstable modules.