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Filtration nilpotente, catégories de foncteurs et étude du centre d’une algèbre instable noethérienne sur l’algèbre de Steenrod

Abstract : This thesis is concerned with the study of the centre of unstable algebra over the Steenrod algebra. We use tools that come from an equivalence of categories between the category of unstable modules over the Steenrod algebra localized by its n-nilpotent objects and some functor categories.We start by giving an alternative description of this already classical equivalence of categories, and we explain its behaviour regarding unstable algebra. More precisely, we define a category of unstables algebra in the functor category already mentionned,and we prove this is equivalent to the category of unstable algebra localized by the morphisms whose kernel and cokernel are n-nilpotent. For n = 1, this gives an equivalence with the category of contravariant functors from finite dimensional vector spaces to profinite sets. We introduce a shift functor in this category, corresponding to Lannes T functor in the category of unstable algebra. We define a notion of centrality in the categories of functor that we study, such that the centre of a nil n-closed unstable algebra is the same as the centre of the associated functor. Using this, we get a refinement of the centre of an unstable algebra, making use of the nilpotent filtration. Finally, we use those results to study the classification of noetherian, nil-closed, connected, H*(V )-comodules in the category of unstable algebra, whose primitive elements are a fixed unstable algebra P.
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Submitted on : Tuesday, April 12, 2022 - 10:57:18 AM
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Ouriel Bloede. Filtration nilpotente, catégories de foncteurs et étude du centre d’une algèbre instable noethérienne sur l’algèbre de Steenrod. Algèbre commutative [math.AC]. Université d'Angers, 2021. Français. ⟨NNT : 2021ANGE0059⟩. ⟨tel-03638356⟩

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