Definition 12.10.1 (02MO)—The Stacks project

Definition 12.10.1. Let $\mathcal{A}$ be an abelian category.

A Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence ¹

\[ A \to B \to C \]

with $A, C \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, then also $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
A weak Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence

\[ A_0 \to A_1 \to A_2 \to A_3 \to A_4 \]

with $A_0, A_1, A_3, A_4$ in $\mathcal{C}$, then also $A_2$ in $\mathcal{C}$.

[1] By Definition 12.5.7 this means $\mathop{\mathrm{Im}}(A \to B) = \mathop{\mathrm{Ker}}(B \to C)$.

Comments (6)

Comment #3951 by Shane on January 25, 2019 at 01:53

Is it possible to put a warning in (1) that the sequence is not a short exact sequence? That trips me up every time I look at this.

Comment #3957 by Johan on January 25, 2019 at 11:48

Maybe your comment here can serve as a warning. The trouble with putting warnings is where does one stop? In every lemma or definition there is some absolutely essential thing that should not be overlooked (and in well written lemmas all the assumptions are necessary). If we try to emphasize certain of these conditions over others, then when are we going to stop?

Comment #6735 by Alejandro González Nevado on November 20, 2021 at 03:37

The comment of Mohammed in the general Section 02MN refers to this definition. The condition between $A,B,C$ under the arrows is an "iff" condition not an "if". At least as long as I saw in other literature. Example: look at the definition in https://ncatlab.org/nlab/show/Serre+subcategory

Comment #6738 by Johan on November 20, 2021 at 15:32

@#6735 OK, I cannot speak about the correctness of the site you linked to, but I can say that we definitively do not want to say "if and only if" in the definition as given here. For example the category consisting just of the zero object is (or should be) a Serre subcategory but doesn't satisfy the if and only if condition. Note, note, note: exactness of $A \to B \to C$ means only that the image of $A \to B$ is equal to the kernel of $B \to C$ and does not imply that $A \to B$ is injective and does not imply that $B \to C$ is surjective.

Comment #6740 by Johan on November 20, 2021 at 15:34

Postscript: I guess I should have listend to Shane!

Comment #6924 by Johan on January 19, 2022 at 14:12

OK, I have added a bit of text on what an exact sequence really is! Then I have added a precise reference to Serre's paper (which uses the exact same defintiion) and I have added a footnote to the definition of exact sequences. Here are the changes.

There are also:

20 comment(s) on Section 12.10: Serre subcategories

Comments (6)

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