Open Access Dissertation
Date of Award
Daniel C. Isaksen
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.
Gheorghe, Bogdan, "The Motivic Cofiber Of Τ And Exotic Periodicities" (2017). Wayne State University Dissertations. 1804.