Drew Heard

K -theory, reality, and duality

We present a new proof of Andersons result that the real K -theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complex K -theory spectrum KU is C 2 -equivariantly equivalent to Σ 4 KU , where C 2 acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of the K (1)-local Picard group.

Local duality for structured ring spectra

Abstract We use the abstract framework constructed in our earlier paper [8] to study local duality for Noetherian E ∞ -ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.

Picard groups of higher real K-theory spectra at height p - 1

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real

Local duality in algebra and topology

K

Algebraic chromatic homotopy theory for BP∗BP-comodules: ALGEBRAIC CHROMATIC HOMOTOPY THEORY FOR BP∗BP-COMODULES

-theory spectra of Hopkins and Miller at height

The algebraic chromatic splitting conjecture for Noetherian ring spectra

n=p-1

The chromatic splitting conjecture for Noetherian commutative ring spectra

, for

On equivariant and motivic slices.

p

Derived completion for comodules.

an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra

Picard groups and duality for Real Morava

E_n^{hG}