Bockstein homomorphism: Difference between revisions

Revision as of 23:08, 13 December 2007

This article defines a stable cohomology operation

Definition

Given a short exact sequence of Abelian groups:

$0 \to P \to Q \to R \to 0$

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

$\to H^{i} (C; P) \to H^{i} (C; Q) \to H^{i} (C; R) \to H^{i + 1} (C; P) \to \dots$

Thus, $β_{i}$ is the map:

$H^{i} (C; R) \to H^{i + 1} (C; P)$

The Bockstein homomorphism for a prime field of $p$ elements is defined as the Bockstein homomorphism for the short exact sequence:

$0 \to Z / p Z \to Z / p^{2} Z \to Z / p Z \to 0$

@@ Line 7: / Line 7: @@
 <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>
+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>