Homological dimension

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Homological dimension may refer to the global dimension of a ring. It may also refer to any other concept of dimension that is defined in terms of homological algebra, which includes:

In ring theory and homological algebra, the global dimension of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.

Homological algebra area of mathematics

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

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 TorR
n(M,N) is nonzero for some left R-module N, and the weak dimension of a left R-module is defined similarly. The weak dimension was introduced by Cartan and Eilenberg (1956, p.122). The weak dimension is sometimes called the flat dimension as it is the shortest length of a resolution of the module by flat modules. The weak dimension of a module is at most equal to its projective 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.

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