Timeline for Exact subcategory with trivial Grothendieck group: what are the consequences and examples
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2020 at 14:31 | answer | added | Ollie | timeline score: 6 | |
| Mar 29, 2020 at 0:34 | vote | accept | GSM | ||
| Mar 5, 2020 at 13:33 | comment | added | Jeremy Rickard | @MartinBrandenburg I'm using Quillen's definition en.wikipedia.org/wiki/Exact_category, which I think is what the OP meant. The ambient abelian category is the category of left exact functors from the exact category to abelian groups. | |
| Mar 5, 2020 at 13:23 | comment | added | Martin Brandenburg | Maybe we use two different definitions of an exact category. But where does that ambient abelian category come from? | |
| Mar 5, 2020 at 13:10 | comment | added | Jeremy Rickard | @MartinBrandenburg Every essentially small exact category is an extension closed full subcategory of an abelian category, which may be assumed to be essentially small (take the smallest extension closed abelian subcategory containing the original exact category), and so Freyd-Mitchell can be applied to that abelian category. | |
| Mar 5, 2020 at 12:59 | comment | added | Martin Brandenburg | @JeremyRickard It's usually only stated for abelian categories - why does exactness suffice here? | |
| Mar 5, 2020 at 12:50 | answer | added | Johannes Hahn | timeline score: 6 | |
| Mar 5, 2020 at 10:58 | comment | added | Jeremy Rickard | Doesn't it follow from the Freyd-Mitchell embedding theorem that every essentially small exact category is equivalent to a full exact subcategory of a module category? | |
| Mar 5, 2020 at 10:40 | comment | added | GSM | @YCor $R$ is a ring with 1, that is all what we assume | |
| Mar 5, 2020 at 10:33 | comment | added | YCor | could you say what is $R$? probably a ring, but hard to guess if you assume it to be commutative | |
| Mar 5, 2020 at 10:32 | history | edited | YCor | CC BY-SA 4.0 |
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| Mar 5, 2020 at 10:28 | history | asked | GSM | CC BY-SA 4.0 |