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Presheaves on VI, nil-closed unstable algebras and their centres


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 G_F 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.
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hal-03589376 , version 1 (25-02-2022)


  • HAL Id : hal-03589376 , version 1


Ouriel Bloede. Presheaves on VI, nil-closed unstable algebras and their centres. 2022. ⟨hal-03589376v1⟩
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