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Results tagged with abelian-categories
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user 82179
2
votes
The category of abelian group objects
I'm not sure why this hasn't been mentioned yet, but the category of abelian group objects in smooth proper geometrically integral schemes is not abelian, already if $S = \operatorname{Spec} k$. Indee …
- 28.7k
2
votes
Accepted
Is every filtration on an abelian category strict?
This is not true.
Example. Let $\mathscr A = \mathbf{Ab}$ (you may restrict to finitely generated abelian groups if you like), and consider the functorial two-step filtration $F^0 \supseteq F^1 \supse …
- 28.7k
19
votes
Accepted
Abelian category with enough injectives but not functorially
Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions.
I will show that the category $\mathbf{Ab}^{\operatornam …
- 28.7k
17
votes
Accepted
When is the category of finitely presented modules abelian?
Wojowu's idea is right:
Lemma. Let $R$ be a ring, let $\mathbf{Mod}_R$ be the category of (left) $R$-modules, and let $\mathbf{Mod}_R^{\text{fp}}$ be the subcategory of finitely presented modules. The …
- 28.7k
6
votes
Accepted
Do idempotents in an abelian category constitute a lattice?
With minor modifications, the example of Jeremy Rickard also shows that there is no meet in general.
Example. Let $\mathscr C = \mathrm{Ab}$, and let $X = (\mathbf Z/4)^3$. Consider the subgroups $A = …
- 28.7k