Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)?

What can be said about the category of quasi-coherent sheaves?

  • 3
    $\begingroup$ What do you mean for "pure subobject" (pure quotient)? $\endgroup$ – Buschi Sergio Jul 26 '13 at 21:53
  • $\begingroup$ Let $\cal C$ be a category with tensor product (For example the category of modules over a ring, the category of representations of a quiver by modules, the category of sheaves of abelian groups,...). Let $$\varepsilon: 0\to A\to B\to C\to 0$$ be an exact sequence and $D$ be an arbitrary object. $\varepsilon$ is pure if The sequence $D\otimes \varepsilon$ is exact for each $D$. $\endgroup$ – Gholam Jul 27 '13 at 10:07
  • 1
    $\begingroup$ So, your abelian category is assumed to have a monoidal structure too. Do you want to assume any properties of the monoidal structure (e.g. how it interacts with colimits)? I think the question would be improved if you stated your assumptions. You can edit it; that's better than adding comments or leaving answers to your own question. $\endgroup$ – Tom Leinster Aug 13 '13 at 18:12
  • $\begingroup$ Actually by the category I mean very well known categories such as the category of modules over a ring, the category of sheaves of $\mathcal{O}_X$-modules, the category of quasi-coherent sheaves, the category of representations of a quiver by a modules and etc. All of these categories are Grothendieck categories. $\endgroup$ – Gholam Aug 13 '13 at 18:42
  • 5
    $\begingroup$ I suggest you edit your question to make your assumptions clear. At the moment it doesn't quite make sense, because "pure" doesn't make sense in the absence of a tensor product. As I said, that would be better than adding comments, and will make it more likely that people will answer your question. $\endgroup$ – Tom Leinster Aug 13 '13 at 19:08

I think the answer should be yes, because cokerenls are direct limits. Let $\mathcal{G}$ be a pure subsheaf of $\mathcal{H}$ and consider $$0\to \mathcal{G}\to \mathcal{H}\to \frac{\mathcal{H}}{\mathcal{G}}\to 0.$$ Since $\mathcal{G}$ and $\mathcal{H}$ are both in $\mathcal{F}$ and $\frac{\mathcal{H}}{\mathcal{G}}$ can be viewd as a direct limit, all terms in the above short exact sequence are in $\mathcal{F}$.

  • $\begingroup$ In general cokernels aren't directed colimit. $\endgroup$ – Buschi Sergio Aug 13 '13 at 19:13
  • $\begingroup$ Which conditions should be imposed to a grothendieck category to deduce that cokernels are direct limits? $\endgroup$ – Gholam Aug 13 '13 at 19:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.