Bockstein homomorphism - Revision history

Vipul: 2 revisions

2008-05-11T19:32:10Z

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Vipul at 23:08, 13 December 2007

2007-12-13T23:08:35Z

← Older revision		Revision as of 23:08, 13 December 2007
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	<math>0 \to P \to Q \to R \to 0</math>		<math>0 \to P \to Q \to R \to 0</math>

	We can define a Bockstein homomorphism of type <math>(R,P)</math> and degree <math>1</math>. This is the connecting homomorphism in the associated long exact sequence of cohomology, for the cochain complex:		We can define a Bockstein homomorphism (denoted <math>\beta</math>) of type <math>(R,P)</math> and degree <math>1</math>. This is the connecting homomorphism in the associated long exact sequence of cohomology, for the cochain complex:

	<math>\to H^i(C;P) \to H^i(C;Q) \to H^i(C;R) \to H^{i+1}(C;P) \to \ldots</math>		<math>\to H^i(C;P) \to H^i(C;Q) \to H^i(C;R) \to H^{i+1}(C;P) \to \ldots</math>

			Thus, <math>\beta_i</math> is the map:

			<math>H^i(C;R) \to H^{i+1}(C;P)</math>

			The Bockstein homomorphism for a prime field of <math>p</math> elements is defined as the Bockstein homomorphism for the short exact sequence:

			<math>0 \to \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0</math>

Vipul at 23:03, 13 December 2007

2007-12-13T23:03:21Z

New page

{{stable cohomology operation}}

==Definition==

Given a short exact sequence of Abelian groups:

<math>0 \to P \to Q \to R \to 0</math>

We can define a Bockstein homomorphism of type <math>(R,P)</math> and degree <math>1</math>. This is the connecting homomorphism in the associated long exact sequence of cohomology, for the cochain complex:

<math>\to H^i(C;P) \to H^i(C;Q) \to H^i(C;R) \to H^{i+1}(C;P) \to \ldots</math>