# Cohomological dimension

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The cohomological dimension of a topological space relative to the group of coefficients is the maximum integer for which there exists closed subsets of such that the cohomology groups are non-zero. The homological dimension is similarly defined (cf. Homological dimension of a space). Finite Lebesgue dimension (covering dimension) is the same as (or ) if is the subgroup of the integers (or real numbers modulo 1). In Euclidean space the equation is equivalent to the property that is locally linked by -dimensional cycles (with coefficients in ). For paracompact spaces , the inequality is equivalent to the existence of soft resolutions (cf. Soft sheaf and Resolution) for of length . 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 if it has an injective (or projective) resolution of length ; 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 .