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For a coalgebra C over a field K, the category C-Comod of left C-comodules is an example of locally finitely presented Grothendieck category. For this type of categories a notion of flat object exists due to Stenström this paper

It is known that every module has a flat cover by this paper

We also know that each locally finitely presented Grothendieck category has enough flat objects by the paper

More generally, When an abelian category has enough flat objects? Let $A$ be an abelian category with enough flat objects, does each object in $A$ has a flat cover?

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  • $\begingroup$ What is the definition of a flat object in an abelian category ? $\endgroup$
    – Joël
    Commented Apr 19, 2012 at 20:42
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    $\begingroup$ 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. $\endgroup$
    – George C. Modoi
    Commented Apr 19, 2012 at 22:47
  • $\begingroup$ 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 $\endgroup$
    – Aimin Xu
    Commented Apr 20, 2012 at 2:13
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    $\begingroup$ Yes, the category of quasicoherent sheaves over a scheme has enough flat sheaves, but often (when the scheme is non-affine) no nonzero projective. $\endgroup$
    – George C. Modoi
    Commented Apr 21, 2012 at 14:29
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    $\begingroup$ 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 $\endgroup$
    – Aimin Xu
    Commented Apr 26, 2012 at 7:26

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