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cohomological dimension

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Cohomology

cohomology

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Special notions

Variants

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Theorems

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

An object in an (∞,1)-topos is said to have cohomological dimension n\leq n if all cohomology groups of degree k>nk \gt n vanish on that object.

Definition

Definition

For H\mathbf{H} an (∞,1)-topos and nn \in \mathbb{N} , an object XHX \in \mathbf{H} is said to have cohomological dimension n\leq n if for all Eilenberg-MacLane objects B kA\mathbf{B}^k A for k>nk \gt n the cohomology of XX with these coefficients vanishes:

H k(X,A):=π 0H(X,B A)*. H^k(X, A) := \pi_0 \mathbf{H}(X,\mathbf{B}^ A) \simeq * \,.

We say the (∞,1)-topos H\mathbf{H} itself has cohomological dimension n\leq n if its terminal object does.

This appears as HTT, def. 7.2.2.18.

Properties

Proposition

If 𝒳\mathcal{X} has homotopy dimension n\leq n then it also has cohomology dimension n\leq n.

The converse holds if 𝒳\mathcal{X} has finite homotopy dimension an n2n \geq 2.

This appears as HTT, cor. 7.2.2.30.

References

The general (,1)(\infty,1)-topos-theoretic notion is discussed in section 7.2.2 of

Revised on December 6, 2011 15:19:40 by Stephan Alexander Spahn (79.227.144.188)