- #1
Math Amateur
Gold Member
MHB
- 3,998
- 48
I am reading Dummit and Foote, CH 10 Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
As they introduce split sequences, D&F write the following:View attachment 2462
I am concerned at the following statement:
"In this case the module \(\displaystyle B \) contains a sub-module \(\displaystyle C'\) isomorphic to \(\displaystyle C\) (namely \(\displaystyle C' = 0 \oplus C\)) as well as the submodule A, and this submodule complement to A "splits" B into a direct sum ... ... "
Maybe I am being pedantic or I am just confused but it does not seem to me that B contains A as a sub-module or that C' is complement to A (regarding A as an abelian group).
It seems to me that if \(\displaystyle B = A \oplus B \) then certainly B contains both \(\displaystyle A' = A \oplus 0 \) (and NOT A) and \(\displaystyle C' = 0 \oplus C\).
To check that the submodule C' is complement to A' we note that
(1) \(\displaystyle B = A' + C' \)
since
\(\displaystyle A' + C' = \{ (a,0) + (0,c) = (a,c) | a \in A, c \in C \} \)
and we also note that
(2) \(\displaystyle A' \cap C' = (0,0) \)
Given (1) and (2), the submodule \(\displaystyle C'\) is complement to the submodule \(\displaystyle A'\)
Can someone please indicate whether my analysis is correct, and also comment of the D&F text? Should D&F be talking about \(\displaystyle A'\) being a submodule and specifying \(\displaystyle C'\) as complement to \(\displaystyle A'\) (not \(\displaystyle A\))?
Peter
As they introduce split sequences, D&F write the following:View attachment 2462
I am concerned at the following statement:
"In this case the module \(\displaystyle B \) contains a sub-module \(\displaystyle C'\) isomorphic to \(\displaystyle C\) (namely \(\displaystyle C' = 0 \oplus C\)) as well as the submodule A, and this submodule complement to A "splits" B into a direct sum ... ... "
Maybe I am being pedantic or I am just confused but it does not seem to me that B contains A as a sub-module or that C' is complement to A (regarding A as an abelian group).
It seems to me that if \(\displaystyle B = A \oplus B \) then certainly B contains both \(\displaystyle A' = A \oplus 0 \) (and NOT A) and \(\displaystyle C' = 0 \oplus C\).
To check that the submodule C' is complement to A' we note that
(1) \(\displaystyle B = A' + C' \)
since
\(\displaystyle A' + C' = \{ (a,0) + (0,c) = (a,c) | a \in A, c \in C \} \)
and we also note that
(2) \(\displaystyle A' \cap C' = (0,0) \)
Given (1) and (2), the submodule \(\displaystyle C'\) is complement to the submodule \(\displaystyle A'\)
Can someone please indicate whether my analysis is correct, and also comment of the D&F text? Should D&F be talking about \(\displaystyle A'\) being a submodule and specifying \(\displaystyle C'\) as complement to \(\displaystyle A'\) (not \(\displaystyle A\))?
Peter
Last edited: