2022/02/25 by Bloede, Ouriel
Mathematics · Medicine · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2202.12878
openalex publication_date 2022/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A nil-closed, noetherian, unstable algebra K over the Steenrod Algebra is determined, up to isomorphism, by the functor HomKf.g.(K,H^*(_)), which is a presheaf on the category VI of finite dimensional vector spaces and injections, by the theory of Henn-Lannes-Schwartz. In this article, we use this theory to study the centre, in the sense of Heard, of a nil-closed noetherian unstable algebra. For F a presheaf on VI, we construct a groupoid GF which encodes F. Then, taking F:=HomKf.g.(K,H^*(_)), we show how the centre of K is determined by the associated groupoid. We also give a generalisation of the second theorem of Adams-Wilkerson, defining sub-algebras H^*(W)G of H^*(W) for appropriate groupoids G. There is a H^*(C)-comodule structure on K that is associated with the centre. For K integral, we explain how the algebra of primitive elements of this H^*(C)-comodule structure is also determined by the groupoid associated with HomKf.g.(K,H^*(_)). Along the way, we prove that this algebra of primitive elements is also noetherian.