Equivariant Anderson Duality and Mackey Functor Duality

Abstract

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.

Gorenstein duality for real spectra

“…Our main theorem also recovers the main result of [28] about the Anderson self-duality of integral real Morava K-theory.…”

“…For example, we can follow [18] and Hu [17] and define BPRhni D BPR=.v nC1 ; v nC2 ; : : : / and ER.n/ D BPRhniOEv These spectra are still strongly even, as we will show. Apart from the extensive literature on K-theory with reality (eg Atiyah [4], Dugger [8] and Bruner and Greenlees [7]), real spectra have been studied by Hu and Kriz, in a series of papers by Kitchloo and Wilson (see eg [21] for one of the latest instalments), by Banerjee [5], by Ricka [28] and by Lorman [24]. A crucial point is that a morphism between strongly even C 2 -spectra is an equivalence if it is an equivalence of underlying spectra [27,Lemma 3.4].…”

The C2–spectrum Tmf1(3) and its invertible modules

“…We prove the theorem by an application of the slice spectral sequence. There has been similar work by Ricka [Ric14] on Anderson duality of integral versions of Morava K-theory; our results have been obtained independently.…”

“…By smashing X with representation spheres, we see that it even refines to an RO(G)graded sequence. Equivariant Anderson duality in the case G = C 2 has been explored in some detail in [Ric14].…”

Equivariant cohomology for cyclic groups of square-free order

“…to Bredon cohomology using Anderson duality [2]. Equivariant Anderson duality was studied for the group C 2 in [22], and formulated for general finite groups in [14]. We briefly recall the results here.…”