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started projective module
(will need to move some material around with projective object. Also, I am splitting off now projective resolution from resolution )
I have been expanding the Properties-section at projective module. Made fully explicit in detailed proofs what was previously just alluded to as “clearly”: that every direct summand of a free module is projective, assuming the axiom of choice. This is meant to be expository and serve newbies.
added a section Definition - F-resolutions of an object.
Will now add a corresponding Properties-section…
(Sorry, this should go to the thread on projective resolution…)
Added to projective module the many ways to characterize finitely generated projective modules.
For completeness, I have added to the beginning of the entry the elementary proof of the first claim, here.
projective module, Prop. 5 number 4:
There exist elements $x_1,\ldots,x_n \in P$ and linear forms $\vartheta_1,\ldots,\vartheta_n \in Hom(P,R)$ such that $x = \sum_i \vartheta_i(x) x_i$ for all $x \in P$.
In my memory, these data were called a normal basis of the projective module ?
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