Standard complex
Technique for constructing resolutions in homological algebra From Wikipedia, the free encyclopedia
In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by Samuel Eilenberg and Saunders Mac Lane (1953) and Henri Cartan and Eilenberg (1956, IX.6) and has since been generalized in many ways.
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The name "bar complex" comes from the fact that Eilenberg & Mac Lane (1953) used a vertical bar | as a shortened form of the tensor product in their notation for the complex.
Definition
Summarize
Perspective
If A is an associative algebra over a field K, the standard complex is
with the differential given by
If A is a unital K-algebra, the standard complex is exact. Moreover, is a free A-bimodule resolution of the A-bimodule A.
Normalized standard complex
The normalized (or reduced) standard complex replaces with .
Monads
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See also
References
- Cartan, Henri; Eilenberg, Samuel (1956), Homological algebra, Princeton Mathematical Series, vol. 19, Princeton University Press, ISBN 978-0-691-04991-5, MR 0077480
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: ISBN / Date incompatibility (help) - Eilenberg, Samuel; Mac Lane, Saunders (1953), "On the groups of . I", Annals of Mathematics, Second Series, 58: 55–106, doi:10.2307/1969820, ISSN 0003-486X, JSTOR 1969820, MR 0056295
- Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math.AG/0506603.
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