User Jeremy Rickard

80 votes

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$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

answered Jan 1, 2016 at 17:07

58 votes

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Are there any nontrivial abelian categories with only finitely many objects?

answered Nov 10, 2012 at 14:33

49 votes

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Example of an unnatural isomorphism

answered Aug 14, 2013 at 8:25

46 votes

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Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

answered Jan 11, 2018 at 14:45

46 votes

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Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

answered Oct 23, 2020 at 19:48

35 votes

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Does the symmetric group on an infinite set have a minimal generating set?

answered Jul 10, 2014 at 16:44

30 votes

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Are there only finitely many associative algebras of fixed dimension?

answered Jun 17, 2014 at 9:54

30 votes

If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

answered Mar 4, 2023 at 19:18

28 votes

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Cocomplete but not complete abelian category

answered Oct 15, 2014 at 11:31

27 votes

A question about representations of finite groups

answered Nov 5, 2013 at 14:22

25 votes

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Is this ring isomorphic to a quotient of a group algebra?

answered Oct 13 at 17:54

24 votes

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Combinatorics Problem: $\sum _{k=0}^{s-1} \binom{n}{k}=\sum _{k=1}^s 2^{k-1} \binom{n-k}{s-k}$

answered Jun 28, 2014 at 19:28

24 votes

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Non-isomorphic graphs with bijective graph homomorphisms in both directions between them

answered Sep 4, 2017 at 14:34

23 votes

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Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

answered Feb 9, 2016 at 10:01

22 votes

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Example of an abelian category with enough projectives and injectives which are not dual

answered Feb 9, 2018 at 14:54

21 votes

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Grothendieck group of derived category

answered Feb 15, 2017 at 10:28

21 votes

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Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?

answered Aug 2, 2013 at 9:54

20 votes

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Is the homotopy category of an abelian model category abelian?

answered Jan 3, 2018 at 13:54

20 votes

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Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

answered Mar 26, 2023 at 18:21

20 votes

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Classification of subgroups of finitely generated abelian groups

answered Jul 19, 2020 at 11:13

20 votes

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Recovering an abelian category from the Ext of its simple objects

answered May 1, 2021 at 8:41

18 votes

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Monoids of endomorphisms of nonisomorphic groups

answered Nov 12, 2019 at 13:21

18 votes

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Are all vector-space valued functors on sets free?

answered Jul 14, 2014 at 16:33

17 votes

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A group whose automorphism group is cyclic

answered Feb 24, 2015 at 12:44

17 votes

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Tilting Objects in BGG Categories $\mathcal{O}$

answered Dec 2, 2018 at 14:02

17 votes

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Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

answered Feb 5, 2017 at 11:22

16 votes

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When the restriction of a derived functor to a subcategory is the derived functor of the restriction

answered Jul 15, 2015 at 19:04

16 votes

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Intersection of all normalizers

answered Apr 18, 2013 at 9:56

16 votes

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In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?

answered Dec 16, 2023 at 21:07

15 votes

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Which spaces have the (weak) homotopy type of compact Hausdorff spaces?

answered Dec 1, 2013 at 11:48

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396 Answers

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there any nontrivial abelian categories with only finitely many objects?

Example of an unnatural isomorphism

Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

Does the symmetric group on an infinite set have a minimal generating set?

Are there only finitely many associative algebras of fixed dimension?

If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

Cocomplete but not complete abelian category

A question about representations of finite groups

Is this ring isomorphic to a quotient of a group algebra?

Combinatorics Problem: $\sum _{k=0}^{s-1} \binom{n}{k}=\sum _{k=1}^s 2^{k-1} \binom{n-k}{s-k}$

Non-isomorphic graphs with bijective graph homomorphisms in both directions between them

Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Example of an abelian category with enough projectives and injectives which are not dual

Grothendieck group of derived category

Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?

Is the homotopy category of an abelian model category abelian?

Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

Classification of subgroups of finitely generated abelian groups

Recovering an abelian category from the Ext of its simple objects

Monoids of endomorphisms of nonisomorphic groups

Are all vector-space valued functors on sets free?

A group whose automorphism group is cyclic

Tilting Objects in BGG Categories $\mathcal{O}$

Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Intersection of all normalizers

In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?

Which spaces have the (weak) homotopy type of compact Hausdorff spaces?