Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 RR-modules is an abelian category but lacks these properties.” This is not true, see e.g. https://stacks.math.columbia.edu/tag/0AZ5

    Anonymous

    diff, v8, current

    • 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 RR is not Noetherian, then for a non-finitely generated ideal II of RR, the quotient map RR/IR \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.

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeSep 14th 2022
    • (edited Sep 14th 2022)

    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.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    added pointer to:

    diff, v13, current