Every ∞-commutative monoid has a universal (n−t)-fold semiadditive character that blue-shifts height, recovers the transchromatic character on Morava E-theory, and computes L_Q(S^A_{K(n)}) via GL_{n−t}(Z_p)-fixed points.
Ramzi, Separability in homotopical algebra, Preprint arXiv:2305.17236 (2023)
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abstract
We study the notion of \emph{separable algebras} in the context of symmetric monoidal stable $\infty$-categories. In the first part of this paper, we compare this context to that of tensor-triangulated categories and show that separable algebras and their modules in a symmetric monoidal stable $\infty$-category are, in large parts, controlled by the (tensor-triangulated) homotopy category. We also study a variant of this notion, which we call ind-separability. Among other things, this provides a partially new proof of the Goerss--Hopkins--Miller theorem about the uniqueness of $\mathbb E_\infty$-structures on Morava $E$-theory. We later initiate a study of separable algebras \textit{\`a la} Auslander-Goldman by relating them to Azumaya algebras, and prove in some restrictive cases that centers of separable algebras are separable. Finally, we study the Hochschild homology of separable algebras and prove some descent results in topological Hochschild homology.
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Authors develop an ∞-categorical approach to Hopfological algebra that refines Khovanov-Qi foundations and generalizes to arbitrary rigidly-compactly generated symmetric monoidal stable ∞-categories.
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
Separable extensions of noetherian rings and finite étale morphisms of noetherian schemes give rise to separable extensions of singularity categories.
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Higher Semiadditive Character Theory
Every ∞-commutative monoid has a universal (n−t)-fold semiadditive character that blue-shifts height, recovers the transchromatic character on Morava E-theory, and computes L_Q(S^A_{K(n)}) via GL_{n−t}(Z_p)-fixed points.
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Hopfological algebra, revisited
Authors develop an ∞-categorical approach to Hopfological algebra that refines Khovanov-Qi foundations and generalizes to arbitrary rigidly-compactly generated symmetric monoidal stable ∞-categories.
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The Galois theory of $G$-spectra and the Burnside ring
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
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The singularity category of a separable extension
Separable extensions of noetherian rings and finite étale morphisms of noetherian schemes give rise to separable extensions of singularity categories.