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Short Exact Sequences: Splitting

  1. Mar 29, 2012 #1

    jgens

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    In Dummit and Foote, a short exact sequence of R-modules [itex]0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0[/itex] ([itex]\psi:A \rightarrow B[/itex] and [itex]\phi:B \rightarrow C[/itex]) is said to split if there is an R-module complement to [itex]\psi(A)[/itex] in [itex]B[/itex]. The authors are not really clear on what the phrase "an R-module complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some R-module [itex]C'[/itex] such that [itex]B[/itex] is the internal direct sum of [itex]\psi(A)[/itex] and [itex]C'[/itex]."

    In Dummit and Foote, a short exact sequence of groups [itex]1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1[/itex] ([itex]\psi:A \rightarrow B[/itex] and [itex]\phi:B \rightarrow C[/itex]) is said to split if there is a subgroup complement to [itex]\psi(A)[/itex] in [itex]B[/itex]. Again the authors are not really clear on what the phrase "a subgroup complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some group [itex]C'[/itex] such that [itex]B[/itex] is the internal semi-direct product of [itex]\psi(A)[/itex] and [itex]C'[/itex]."

    Are my interpretations correct?
     
    Last edited: Mar 29, 2012
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  3. Mar 29, 2012 #2

    micromass

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    The relevant definitions are on page 180 and page 383 (which is not so clear, but you get the idea).
     
  4. Mar 29, 2012 #3

    jgens

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    The definition on page 180 is clear and it appears that my interpretation was correct for groups. Since R-modules are an abelian category, I am pretty sure I got the right interpretation for modules too. Thanks!
     
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