K-theory, reality, and duality

Abstract

Abstract. We show that the real K-theory spectrum KO is Anderson selfdual using the method previously employed in the second author's calculation of the Anderson dual of T mf . Indeed the current work can be considered as a lower chromatic version of that calculation. Emphasis is given to an algebrogeometric interpretation of this result in spectrally derived algebraic geometry. We finish by applying the result to a calculation of 2-primary Gross-Hopkins duality at height 1, and obtain an independent calculation of the group of exotic elements of the K(1)-local Picard group.

Equivariant Anderson Duality and Mackey Functor Duality

“…In this section, we study the relationship between non-equivariant Anderson duality and our Z/2-equivariant version via the forgetful functor (−) e and fixed points (−) Z/2 . As observed in [HS14], fixed points commutes with Anderson duality only for strongly complete Z/2-spectra. We will make this statement precise in Proposition 3.38.…”

“…Anderson duality was introduced by Anderson in [And], and used in [Kai71] and more recently in [Sto11,HS14]. The goal of this section is to understand equivariant Anderson duality in terms of Mackey functors.…”

“…We start by a definition of Anderson duality, following [Kai71,HS14]. The first step is to define Brown-Comenetz duality, as in [BC76] Proposition 3.7.…”

“…Results in this direction are also used, in the non-equivariant setting, in the proof of the principal result of [HS14].…”

Gorenstein duality for real spectra

“…The C 2 -spectra KR (see Heard and Stojanoska [14]) and Tmf 1 .3/ [27] are also C 2 -equivariantly Anderson self-dual, at least if we allow suspensions by representation spheres.…”