Timeline for When an abelian category has enough flat objects?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2018 at 19:13 | comment | added | Tim Campion | @GeorgeC.Modoi Do you assert that if $QCoh(X)$ has enough projectives, then $X$ is affine? If so, I'd love to hear more about this -- in particular, it would answer one form of my question here. | |
| Feb 14, 2013 at 23:37 | comment | added | David White | @Aimin It seems to me that George has done a good job answering your second question. I doubt anyone has characterized abelian categories with enough flat objects. Has anyone even done this for having enough projectives? Enough injectives? Perhaps if you could tell us what application you have in mind we could figure out if there are enough flat objects in that particular setting. | |
| Apr 26, 2012 at 7:26 | comment | added | Aimin Xu | thanks a lot George C. Modoi. I find a new paper[Locally finitely presented categories with no flat objects ][1] [1]: arxiv.org/list/math.CT/recent which gives some examples | |
| Apr 21, 2012 at 14:29 | comment | added | George C. Modoi | Yes, the category of quasicoherent sheaves over a scheme has enough flat sheaves, but often (when the scheme is non-affine) no nonzero projective. | |
| Apr 20, 2012 at 11:50 | comment | added | Aimin Xu | Is there an abelian category which has enough flat objects but no projective objects? | |
| Apr 20, 2012 at 2:13 | comment | added | Aimin Xu | Can we define flat objects in any abelian category as follows: an object $M$ is flat if for every diagram $$ \xymatrix{ K \ar[r] & M \ar[d] & \\ A \ar[r] & B \ar[r] & 0} $$ with exact sequence $A \rightarrow B \rightarrow 0$ and $K$ finitely presented can be completed by a map $K\rightarrow A$ to be a commutative diagram | |
| Apr 19, 2012 at 22:47 | comment | added | George C. Modoi | In the case of a module category the existence of flat precovers (or with your terminology of enough flat objects) is sufficient for deriving the existence of flat covers. More generally the same is true for every precovering class which is closed under direct colimits. The same argument works for Grothendieck categories, giving a partial answer to the last question. For the rest, I think the paper by Rump whose link gave you already should offer a good answer. | |
| Apr 19, 2012 at 20:42 | comment | added | Joël | What is the definition of a flat object in an abelian category ? | |
| Apr 19, 2012 at 14:24 | history | asked | Aimin Xu | CC BY-SA 3.0 |