An Expansion of the Continuity Property

TL;DR

This paper extends the continuity property of Čech-type cohomology to systems of presheaves over inverse-limit spaces. It introduces the $+$ construction to lift presheaves on a basis to presheaves on the space, and analyzes when this yields a sheaf, along with the sheafification $\hat{\Gamma}=\Gamma^{++}$ and stalk-level interpretations. Čech cohomology is reformulated for presheaves on bases, with a limiting process that preserves cohomology via a continuity theorem for surjective inverse systems of compact Hausdorff spaces, culminating in an isomorphism $\varinjlim \check{H}^{n}(X_i,\Gamma_i) \cong \check{H}^{n}(X,\hat{\Gamma})$. The main application concerns profinite groups: for $G=\varprojlim G_i$ and discrete $G$-module $A$, one has $H^{k}(G,A) \cong \varinjlim H^{k}(G_i,A_i)$, which is linked to classifying spaces through $BG_i$ and Čech cohomology with locally constant coefficient systems, yielding a coherent bridge between algebraic and topological cohomology in inverse-limit contexts.

Abstract

One of the advantages of working with Alexander-Spanier-Čech type cohomology theory is the continuity property: For inverse systems of sufficiently well-behaved spaces, the result of taking the cohomology of their limit is a direct limit of their cohomologies. However, Čech cohomology natively works with presheaves of modules rather than modules themselves. We define the notion of a system of presheaves for an inverse system of topological spaces, and show that, under the same circumstances as the ordinary continuity property, a suitable limit of the system provides the Čech cohomology of the inverse limit of the spaces. We then show one application of this result in comparing the cohomology of an inverse limit of finite groups to that of the inverse limit of their classifying spaces.

Theorems & Definitions (23)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Lemma 1.4
• proof
• Lemma 1.5
• proof