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split off Freyd-Mitchell embedding theorem from abelian category and added an expository survey reference.
I have touched the formatting of Freyd-Mitchell embedding theorem a little: moved all the cited references to the References-section. And added Weibel’s book.
With a little luck I find occasion to write out the full proof in the entry soon.
Removed “But for instance the category of finitely generated $R$-modules is an abelian category but lacks these properties.” This is not true, see e.g. https://stacks.math.columbia.edu/tag/0AZ5
Anonymous
Maybe I am misreading what just happened.
But an infinite sum or product of non-trivial finitely generated modules is not itself finitely generated anymore. That’s an example of the sentence which you removed.
How is the StacksProject page you point to related to any of this?
I think Anonymous is referring to item 5 on the stacks page, and has a point: if $R$ is not Noetherian, then for a non-finitely generated ideal $I$ of $R$, the quotient map $R \to R/I$ between finitely generated modules does not have a finitely generated kernel, hence the category of f.g. modules is not abelian.
Performed the simple fix of the removed statement.
Oh, I see, it was referring to the first half-sentence.
All right, thanks.
What is the formal link between the Freyd-Mitchell embedding theorem and the Gabriel-Popescu theorem? Is there a general theorem that specialises to both, or does one follow from the other? (This MathOverflow answer perhaps seems relevant.)
added pointer to:
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