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Access Type

WSU Access

Date of Award

January 2017

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Daniel C. Isaksen

Abstract

Consider the Tate twist τ ∈ H 0,1 (S 0,0 ) in the mod 2 cohomology of the motivic sphere.

After 2-completion, the motivic Adams spectral sequence realizes this element as a map

τ : S 0,−1 GGA S 0,0 . This thesis begins with the study of its cofiber, that we denote by Cτ.

We first show that this motivic 2-cell complex can be endowed with a unique E ∞ ring

structure. This promotes the known isomorphism π ∗,∗ Cτ ∼= Ext ∗,∗ BP ∗ BP (BP ∗ ,BP ∗ )

to an isomorphism of rings which also preserves higher products.

This structure allows us to consider its closed symmetric monoidal category of modules

( Cτ Mod,− ∧ Cτ −), which happens to live in the kernel of Betti realization. This category

has surprising applications, and moreover contains many interesting motivic spectra. In

particular, we construct exotic motivic fields K(w n ), detecting motivic w n -periodicity. This

theory of motivic w n -periodicity can be roughly seen as perpendicular to the v n -periodicity

story, detected by the motivic Morava K-theories K(n). Finally, we also explain why the category

Cτ Mod is so computable. The above isomor phism comes in a more structured version.

In work that is joint with Zhouli Xu and Guozhen Wang, we show that there is an equivalence

of ∞-categories D b ( MGL ∗,∗ MGL Comod ev ) ∼= GGGA Cτ Cell comp

between an algebraic derived category, and the subcategory Cτ Cell comp of cellular Cτ-

modules that are complete with respect to a version of the algebraic cobordism spectrum MGL.

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