Brown–Gersten descent

Builds on Algebraic K-theory Noetherian topological space Krull dimension Spectrum Presheaf Homotopy limit Sheaf cohomology Spectral sequence

Statement

Let $X$ be a Noetherian topological space of finite Krull dimension $d$ . Let $E$ be a presheaf of spectra on the open subsets of $X$ such that:

$E(\emptyset)$ is contractible.
For all open sets $U, V \subseteq X$ the square formed by $E(U \cup V) \to E(U)$ , $E(U \cup V) \to E(V)$ , $E(U) \to E(U \cap V)$ and $E(V) \to E(U \cap V)$ is homotopy cartesian.

Let $\mathbb{H}(X; E)$ be the hypercohomology spectrum, the homotopy limit over the Zariski site of a fibrant replacement of $E$ . Then $E$ satisfies Zariski descent: the map $E(X) \to \mathbb{H}(X; E)$ is a weak equivalence. There is a strongly convergent spectral sequence $E_2^{p,q} = H^p(X; \pi_{-q} E) \Rightarrow \pi_{-p-q} E(X).$ Here $\pi_{-q} E$ is the sheaf associated with $U \mapsto \pi_{-q} E(U)$ , and $H^p$ is sheaf cohomology. The $E_2$ terms vanish for $p > d$ .

Remarks

The idea: Brown and Gersten compare $E$ with a fibrant replacement in a model structure on presheaves on $X$ . On a Noetherian space every open set is quasi-compact, so condition 2 propagates by induction to all finite open covers, and finite dimension gives the convergence of the spectral sequence.

Example: for a regular separated Noetherian scheme $X$ of finite Krull dimension, Quillen's K-theory $U \mapsto K(U)$ satisfies condition 2 by Quillen's localization theorem. So there is a spectral sequence $H^p(X; \mathcal{K}_{q}) \Rightarrow K_{q-p}(X)$ , with $\mathcal{K}_q$ the Zariski sheaf of K-groups. Combined with the Gersten resolution, it links K-groups with Chow groups as in Bloch's formula.

Condition 2 is not automatic: connective K-theory of singular schemes can fail it. Thomason and Trobaugh showed that Bass's nonconnective K-theory satisfies it on every quasi-compact quasi-separated scheme, which gives Zariski descent there.

Going deeper

For a student

Brown and Gersten proved this to treat algebraic K-theory as a generalized sheaf cohomology. For K-theory of regular schemes it gives the descent spectral sequence from Zariski cohomology of K-sheaves to K-groups. It is the Zariski analogue of later results on Nisnevich descent, where condition 2 is replaced by a condition on elementary distinguished squares.

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Tag	`N3GQA1TZ9Z`
URI	`m2://N3GQA1TZ9Z`
Kind	statement, theorem
Writing	W0 stub
MSC	19E08, 18F20, 55P42
Status	This copy of the site has no build data. Run `m2 status N3GQA1TZ9Z`.
About	Algebraic K-theory, Spectrum
Uses	Algebraic K-theory, Noetherian topological space, Krull dimension, Spectrum, Presheaf, Homotopy limit, Sheaf cohomology, Spectral sequence
Changed	3 November 2026

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