Homological dimension may refer to the
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
of a ring. It may also refer to any other concept of dimension that is defined in terms of
homological algebra, which includes:
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Projective dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizat ...
of a module, based on projective resolutions
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Injective dimension
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of ...
of a module, based on injective resolutions
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Weak dimension
In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n' ...
of a module, or flat dimension, based on flat resolutions
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Weak global dimension
In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n ...
of a ring, based on the weak dimension of its modules
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Cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory.
Cohomologica ...
of a group
{{SIA, mathematics
Homological algebra