Bockstein homomorphism: Difference between revisions
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In [[mathematics]], the '''Bockstein homomorphism''' in [[homological algebra]] is a [[connecting homomorphism]] associated with a [[short exact sequence]] |
In [[mathematics]], the '''Bockstein homomorphism''' in [[homological algebra]] is a [[connecting homomorphism]] associated with a [[short exact sequence]] |
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:0 → P → Q → R → 0 |
:0 → ''P'' → ''Q'' → ''R'' → 0 |
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of [[abelian group]]s, when they are introduced as coefficients into a [[chain complex]] ''C'', and which appears in the [[homology]] groups as a homomorphism reducing degree by one, |
of [[abelian group]]s, when they are introduced as coefficients into a [[chain complex]] ''C'', and which appears in the [[homology]] groups as a homomorphism reducing degree by one, |
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:β: ''H''<sup>''i''</sup>(''C'', ''R'') → ''H''<sup>''i'' + 1</sup>(''C'', ''P''). |
:β: ''H''<sup>''i''</sup>(''C'', ''R'') → ''H''<sup>''i'' + 1</sup>(''C'', ''P''). |
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This is important as a [[cohomology operation]] (see [[Steenrod algebra]]). |
This is important as a source of [[cohomology operation]]s (see [[Steenrod algebra]]). For coefficients in a [[finite cyclic group]] of order ''n'' as ''R'', the mapping β can be combined with reduction modulo ''n''; and then iterated. |
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==Reference== |
==Reference== |
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Revision as of 11:16, 9 November 2005
In mathematics, the Bockstein homomorphism in homological algebra is a connecting homomorphism associated with a short exact sequence
- 0 → P → Q → R → 0
of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one,
- β: Hi(C, R) → Hi − 1(C, P).
To be more precise, C should be a complex of free abelian groups, or at least torsion free, and the homology is of the complexes formed by tensor product with C (some flat module condition should enter). The construction of β is by the usual argument (snake lemma).
A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have
- β: Hi(C, R) → Hi + 1(C, P).
This is important as a source of cohomology operations (see Steenrod algebra). For coefficients in a finite cyclic group of order n as R, the mapping β can be combined with reduction modulo n; and then iterated.
Reference
- Edwin Spanier, Algebraic Topology