The Galois group of a stable homotopy theory

By A. Mathew. 2014

164 citations recorded by the source corpus.

To a "stable homotopy theory" (a presentable, symmetric monoidal stable $\infty$-category), we naturally associate a category of finite \'etale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic $\mathbf{E}_\infty$-algebra of topological modular forms is trivial and that the Galois group of $K(n)$-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the stable module category of a finite group. A fundamental idea throughout is the purely categorical notion of a "descendable" algebra object and an associated analog of faithfully flat descent in this context.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5297c89e47ffb44c718477e3ed9c232fd3df774f

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In Geometry and Topology

In Mathematical Physics

In Nuclear and High Energy Physics

Citations in the corpus

Prisms and prismatic cohomology (2019, methodology): “so assume Jis ﬁnite. But then Proposition 7.10 and the construction of R′ show that indeed Sperfd = R′, as desired. We used the following lemma, which relies on Mathew’s notion of descendability from [Mat16]; see also [BS17, §11.2] for a quick review. PRISMS AND PRISMATIC COHOMOLOGY 55 Lemma 8.6 (St”
Projectivity of the Witt vector affine Grassmannian (2015, methodology): “used in the rest of the paper, we extend Theorem 1.2 to the derived category of quasi-coherent sheaves; here we use the language of ∞-categories, and rely on a notion recently introduced by Mathew in [Mat14]. This section also contains some other results of independent interest: h-descent for Witt v”
Affineness and chromatic homotopy theory (2013, background): “Moreover, one can develop [42] a version of Grothendieck’s étale fundamental group formalism in this setting.”
Nilpotence and descent in equivariant stable homotopy theory (2015, methodology): “We refer to [51] for preliminaries on Galois theory in a symmetric monoidal, stable ∞-category.”