The Ring Spectrum of Stable Power Operations

Specialeforsvar ved Julie Zangenberg Rasmussen

Titel: The Ring Spectrum of Stable Power Operations

Abstract: For a commutative ring spectrum E, we recall the description of the associative ring spectrum Pow(E) of stable power operations due to Lurie and Glasman-Lawson, which naturally acts on the underlying spectrum of any commutative E-algebra in a way compatible with stable operations [Lur07, Lec.24], [GL20]. The homotopy of this ring describes the stable power operations on the generalized cohomology theory E(􀀀) associated to E. It further ts into a string of associative ring spectra E ! Pow(E) ! End(E), which describes the restriction of stable power operations to stable operations.
We show that Pow(HFp) is isomorphic to a natural completion of the big Steenrod algebra, which shows that it captures the structure of the Steenrod operations and all sums hereof, hence it describes all the stable power operations on ordinary mod-p cohomology. We further consider the case of Morava E-theory E(Fpn; 􀀀), where 􀀀 is the Honda formal group law of height n. This will be done by constructing an associative ring spectrum dPow(E), which is the K(n)-local version of Pow(E), where K(n) denotes the Morava K-theory at height n. We end this thesis by sketching the calculation of dPow(E) at
height 1 by using results of Rezk [Rez09].

Vejleder:  Piotr Pstra˛gowski
Censor:   Iver Mølgaard Ottosen, Aalborg Universitet