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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 30th 2010

split off Freyd-Mitchell embedding theorem from abelian category and added an expository survey reference.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 25th 2012

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.

1. 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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 5th 2019

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?

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 5th 2019
• (edited Oct 5th 2019)

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeOct 5th 2019

Performed the simple fix of the removed statement.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 5th 2019

Oh, I see, it was referring to the first half-sentence.

All right, thanks.