Posted by: louisyangliu | April 6, 2008

Exactness & Splitting

In abelian category, if B is splittable, B=A\oplus C, then clearly we have a short exact sequence, 0\rightarrow A\rightarrow B\rightarrow C\rightarrow0. But exactness dosen’t necessarily imply splitting, generalized from Hatcher’s example, we have a short exact sequence

0\rightarrow\mathbb{Z}_{p^{m}}\rightarrow\mathbb{Z}_{p^{m+k}}\oplus\mathbb{Z}_{p^{n-k}}\rightarrow\mathbb{Z}_{p^{n}}\rightarrow0,     (1)

where m, n, and k are positive integers, k<n, and p is a prime. And we define the homomorphism

\varphi:\mathbb{Z}_{p^{m}}\rightarrow \mathbb{Z}_{p^{m+k}}\oplus \mathbb{Z}_{p^{n-k}}      (2)

by \varphi(1)=(p^{k},1), and the homomorphism

\psi:\mathbb{Z}_{p^{m+k}}\oplus \mathbb{Z}_{p^{n-k}}\rightarrow \mathbb{Z}_{p^{n}}

by \psi(p^{k},1)=0 and \psi(1,0)=1. However, \mathbb{Z}_{p^{m+k}}\oplus \mathbb{Z}_{p^{n-k}} is not split as the direct sum of \mathbb{Z}_{p^{m}} and \mathbb{Z}_{p^{n}}.

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