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I am reading Dummit and Foote, Chapter 10, Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms.
In the proof of the first part of the theorem (see image below) D&F make the following statement:
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Conversely, if F is in the image of [itex] \psi'[/itex] then [itex] F = \psi' (F') [/itex] for some [itex] F' \in Hom_R (D, L) [/itex] and so [itex] \phi ( F (d) )) = \phi ( \psi ( F' (d))) [/itex] for any [itex]d \in D[/itex]. ...
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My problem is that surely [itex] F = \psi' (F') [/itex] implies that [itex] \phi ( F (d) )) = \phi ( \psi' ( F' (d))) [/itex] and NOT [itex] \phi ( F (d) )) = \phi ( \psi ( F' (d))) [/itex]?
Hoping someone can help>
Theorem 28 and the first part of the proof read as follows:
Peter
I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms.
In the proof of the first part of the theorem (see image below) D&F make the following statement:
-------------------------------------------------------------------------------
Conversely, if F is in the image of [itex] \psi'[/itex] then [itex] F = \psi' (F') [/itex] for some [itex] F' \in Hom_R (D, L) [/itex] and so [itex] \phi ( F (d) )) = \phi ( \psi ( F' (d))) [/itex] for any [itex]d \in D[/itex]. ...
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My problem is that surely [itex] F = \psi' (F') [/itex] implies that [itex] \phi ( F (d) )) = \phi ( \psi' ( F' (d))) [/itex] and NOT [itex] \phi ( F (d) )) = \phi ( \psi ( F' (d))) [/itex]?
Hoping someone can help>
Theorem 28 and the first part of the proof read as follows:
Peter
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