Taylor’s question on algebraic descriptions of higher cohomology of Gelfand spectra
For every commutative complex Banach algebra , let denote its Gelfand spectrum. Taylor's question asks whether, and how, each higher integral cohomology group for can be described purely in terms of the underlying ring of , extending the known descriptions in the first three degrees. The even-degree case is claimed to be answered by canonical identifications with the corresponding étale Chow groups, while the odd-degree case and the full higher-degree question remain open.
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Progress summary
A new paper claims an answer for every even degree, while odd degrees and the full higher-degree question remain open.
Taylor’s question asks for algebraic descriptions of higher cohomology groups associated with Gelfand spectra, extending the classical low-degree theory.
September 9, 2026 even-degree result
On September 9, 2026, The Oka principle for étale Chow groups reported a canonical identification in every even cohomological degree for commutative complex Banach algebras. This is progress on Taylor’s question, but it does not address odd degrees or all higher degrees, and the claim is unverified.
Current status (as of September 2026): Even degrees are covered by a claimed but unverified result; odd degrees and the general higher-degree question remain open.
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