Access Type
Open Access Dissertation
Date of Award
January 2024
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
Andrew Salch
Abstract
The goal of this thesis is to calculate the cohomology of the extended height n Morava stabilizer group, with trivial coefficients, for all heights n and all primes p>>n. To do this, we construct a family of deformations – parametrized over an affine line and smooth away from a single point – of Ravenel's Lie algebra model for the Morava stabilizer group. The singular fiber of the resulting bundle of differential graded algebras is the Chevalley-Eilenberg DGA of Ravenel's Lie algebra model, while the smooth fibers have very understandable cohomology. From here, tools such as parallel transport, connections, and a derived version of the local invariant cycles theorem are used to compare the cohomology of the singular fiber with the cohomology of any smooth fiber. Among our conclusions is an isomorphism, for all heights n and all primes p>>n, between the cohomology ring of the extended height n Morava stabilizer group with trivial coefficients, and the cohomology of the unitary group of degree n with coefficients in the finite field with p elements. This isomorphism verifies a long-standing conjecture in stable homotopy theory.
Recommended Citation
Kang, Mohammad Behzad, "The Cohomology Of The Extended Morava Stabilizer Group, With Trivial Coefficients, At Large Primes" (2024). Wayne State University Dissertations. 4096.
https://digitalcommons.wayne.edu/oa_dissertations/4096