International audienceThe aim of this paper is to study sub-algebras of the Z/2-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor F 2) which come from quotient Hopf algebroids of the Z/2-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical A(n) and E(n) and show that the Steenrod algebra is free as a module over these

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory, we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.

Abstract. We show that the Z/2-equivariant n th integral Morava Ktheory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of Z/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO(Z/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.Conventions: In this paper, F denotes the field with two elements. When considering the Steenrod algebra and the chromatic tower, the prime number is assumed to be p = 2. The category of abelian groups is denoted Ab. For E an object in the category of spectra (resp. Z/2-equivariant spectra), we denote E * (resp. E ⋆ ) the cohomology theory represented by E, and E * (resp. E ⋆ ) the homology theory represented by E. The homotopy of E is denoted E * (resp. E ⋆ ). Equivariant cohomology theories are graded over the orthogonal representation ring, thus ⋆ is an orthogonal representation of Z/2. We denote 1 the trivial one dimensional representation, and α the sign representation.

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

Using a form of descent in the stable category of A(2)-modules, we show that there are no exotic elements in the stable Picard group of A(2), i.e. that the stable Picard group of A( 2) is free on 2 generators. AcknowledgmentsAuthors would like to thank Bob Bruner for some fruitful conversations.Convention. Through out this paper, F will denote the field with two elements. Every algebraic structure is implicitly over the base field F, and tensor products are taken over F. The Hopf algebras under consideration in this paper are connected, cocommutative finite dimensional graded Hopf algebras, unless explicitly specified otherwise.

No abstract

In this paper, we produce a cellular motivic spectrum of motivic modular forms over R and C, answering positively to a conjecture of Dan Isaksen. This spectrum is constructed to have the appropriate cohomology, as a module over the relevant motivic Steenrod algebra. We first produce a C 2 -equivariant version of this spectrum, and then use a machinery to construct a motivic spectrum from an equivariant one. We believe that this machinery will be of independent interest.

Let A be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over A. The starting point is the observation that the stable category of A-modules can be reconstructed, as an ∞-category, as the totalization of a cosimplicial ∞-category whose layers are related to the stable categories of modules over the quasi-elementary sub-Hopf-algebras of A. This leads to a spectral sequence computing the Picard group which, in some cases, is completely understood. This also leads to a spectral sequence answering a lifting problem in the category of A-modules. We then show how to apply this machinery to compute Picard groups and solve lifting problems in the case of A(1)-modules, where A(1) is the subalgebra of the Steenrod algebra generated by the two first Steenrod squares.2010 Mathematics Subject Classification. 55S10,55P42,19L41. Key words and phrases. Stable category of modules, Steenrod algebra, Homotopical descent. The author is indebted to Akhil Mathew for suggesting such an approach to the study of Picard groups of Hopf algebras.