# Can not tell colimits from limits

Let $$X$$ be a concentrated scheme and $$F$$ a quasi-coherent sheaf of modules. The following are equivalent: (a) The functor $$\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$$ preserves direct limits. In other words, $$F$$ is a finitely presented object of $$Qco(X)$$. (b) The functor $$\mathit{Hom}(F, −) : Qco(X)\rightarrow Mod(X)$$ preserves direct limits.

The question is: is there a typo? My understanding is that the covariant $$\mathrm{Hom}$$ (not the internal one) preserves limits for any category. Should it be "colimits" instead of "limits" in the point (a)?

• The following will also be helpful to you: Filtered colimits are called "direct limits" and "inductive limits", and cofiltered limits are called "inverse limits" and "projective limits", i.e. the limits of diagrams whose opposite categories are filtered. Unfortunately this terminology was defined before products and coproducts were known to be kinds of limit and colimit, so the terminology is backwards. – Robert Furber Apr 28 at 14:30

EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)

No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$$^{op}$$"; thankfully, in general there isn't too much confusion about what one means in a given situation.

The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.

No there is no typo : indeed the covariant hom always preserves limits, but when $$F$$ is "small enough" (here : finitely presented) it also preserves direct limits.

Here's a sketch of proof where I replace sheaves by mere modules : let $$F$$ be an $$R$$-module; then $$F$$ is finitely presented iff $$\hom(F,-)$$ preserves direct limits.

Indeed if you have a directed system $$(M_i)_{i\in I}$$ and a map $$F\to \varinjlim M_i$$, the generators land in some $$M_{i_0}$$ (there's finitely many of them) and every relation is satisfied in some $$M_{j_0}, j_0\geq i_0$$ because the relations are satisfied in the colimit and there's only finitely many of them too.

For the converse you may want to take the directed system of finitely generated submodules of $$F$$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)

This works for modules and actually all algebraic structures so that "$$\hom(F,-)$$ preserves direct limits" is a good generalization of "$$F$$ is finitely presented"

• I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right? – user74900 Apr 28 at 11:58
• @AknazarKazhymurat yes, I'll add a word about that ! – Max Apr 28 at 11:59
• @AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP ! – Max Apr 28 at 12:00