- #1
Chris L T521
Gold Member
MHB
- 915
- 0
Here's this week's problem.
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Problem: Let $U$, $V$, and $W$ be three left $K$-vector spaces, and $\psi$, $\phi$ linear maps, fitting into a short exact sequence:
\[ 0\rightarrow U \xrightarrow{\psi} V \xrightarrow{\phi} W \rightarrow 0.\]
Define
\[S = \{\sigma \in \text{Hom}_K(W,V) : \phi \circ \sigma = \text{Id}_W\}.\]
(An element of S is called a splitting of the short exact sequence). Prove that there exists a bijection from $\text{Hom}_K(W,U)$ to $S$.
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Problem: Let $U$, $V$, and $W$ be three left $K$-vector spaces, and $\psi$, $\phi$ linear maps, fitting into a short exact sequence:
\[ 0\rightarrow U \xrightarrow{\psi} V \xrightarrow{\phi} W \rightarrow 0.\]
Define
\[S = \{\sigma \in \text{Hom}_K(W,V) : \phi \circ \sigma = \text{Id}_W\}.\]
(An element of S is called a splitting of the short exact sequence). Prove that there exists a bijection from $\text{Hom}_K(W,U)$ to $S$.
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