# Cohomological dimension

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The cohomological dimension $( \mathop{\rm dim} _ {G} X )$ of a topological space $X$ relative to the group of coefficients $G$ is the maximum integer $p$ for which there exists closed subsets $A$ of $X$ such that the cohomology groups $H ^ {p} ( X , A ; G )$ are non-zero. The homological dimension $h \mathop{\rm dim} _ {G} X$ is similarly defined (cf. Homological dimension of a space). Finite Lebesgue dimension (covering dimension) is the same as $\mathop{\rm dim} _ {G}$( or $h \mathop{\rm dim} _ {G}$) if $G$ is the subgroup of the integers (or real numbers modulo 1). In Euclidean space $X \subset \mathbf R ^ {n}$ the equation $\mathop{\rm dim} _ {G} X = p$ is equivalent to the property that $X$ is locally linked by $( n - p - 1 )$- dimensional cycles (with coefficients in $G$). For paracompact spaces $X$, the inequality $\mathop{\rm dim} _ {G} X \leq p$ is equivalent to the existence of soft resolutions (cf. Soft sheaf and Resolution) for $G$ of length $p$. Since soft sheaves are acyclic, in this way a connection is established with the general definition of dimension in homological algebra; for example, the injective (or projective) dimension of a module is $\leq p$ if it has an injective (or projective) resolution of length $p$; the global dimension of a ring is the maximum of the injective (or projective) dimensions of the modules over the ring and is the analogue of the Lebesgue dimension of $X$.

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How to Cite This Entry:
Cohomological dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomological_dimension&oldid=46386
This article was adapted from an original article by E.G. Sklyarenko, I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article