Hyperdescent and étale K-theory

Abstract

We study the étale sheafification of algebraic K-theory, called étale K-theory. Our main results show that étale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that étale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on étale sites.

arXiv comment89 pages, v3: various corrections and edits

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Cited by in corpus (13)

• Higher Semiadditive Algebraic K-Theory and Redshift
• Some recent advances in topological Hochschild homology
• Purity in chromatically localized algebraic $K$-theory
• Algebraic Cobordism and Étale Cohomology
• The mod $p$ Riemann-Hilbert correspondence and the perfect site
• Descent and cyclotomic redshift for chromatically localized algebraic K-theory
• On Lurie's theorem and applications
• The Galois action on symplectic $K$-theory
• Compact sheaves on a locally compact space
• Syntomic cycle classes and prismatic Poincaré duality
• Thomason filtration via $T(1)$-local $\mathrm{TC}$
• The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory
• Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory