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I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 30 and its proof (D&F, pages 389-390)
I need some help in order to fully understand the proof of $$(3) \Longrightarrow (4)$$.
Proposition 30 reads as follows:
View attachment 2507
The relevant part of the proof i.e. $$(3) \Longrightarrow (4) $$ is as follows:
View attachment 2508
https://www.physicsforums.com/attachments/2509
In the exact sequence $$ 0 \longrightarrow ker \ \phi \longrightarrow \mathcal{F} \stackrel{\phi}{\longrightarrow} P \longrightarrow 0 $$ , the mapping between $$ ker \ \phi \text{ and } \mathcal{F} $$ is the inclusion map and the map $$ \phi $$ would be the unique map identity on the (finite) generators for P ...
BUT ... how do we know that P is finitely generated ...
Indeed, I am assuming that a free module is one with a finite basis ... is this the case?
Can someone please clarify the above issues?
Peter
I am studying Proposition 30 and its proof (D&F, pages 389-390)
I need some help in order to fully understand the proof of $$(3) \Longrightarrow (4)$$.
Proposition 30 reads as follows:
View attachment 2507
The relevant part of the proof i.e. $$(3) \Longrightarrow (4) $$ is as follows:
View attachment 2508
https://www.physicsforums.com/attachments/2509
In the exact sequence $$ 0 \longrightarrow ker \ \phi \longrightarrow \mathcal{F} \stackrel{\phi}{\longrightarrow} P \longrightarrow 0 $$ , the mapping between $$ ker \ \phi \text{ and } \mathcal{F} $$ is the inclusion map and the map $$ \phi $$ would be the unique map identity on the (finite) generators for P ...
BUT ... how do we know that P is finitely generated ...
Indeed, I am assuming that a free module is one with a finite basis ... is this the case?
Can someone please clarify the above issues?
Peter
Last edited: