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Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon \mathcal{A}\rightarrow\mathcal{A}/\mathcal{C}$ admits a right adjoint, the "section functor".) Then we can form the quotient category $\mathcal{A}/\mathcal{C}$.

Which properties inherits $\mathcal{A}/\mathcal{C}$ from $\mathcal{A}$? To be more precise:

  1. If $\mathcal{A}$ has enough injectives (resp. projectives), does $\mathcal{A}/\mathcal{C}$ too? If not, under which conditions?
  2. If $A\in \mathcal{A}$ is injective (resp. projective), is it $T(A)$, too? If not, under which conditions?
  3. If $A\in \mathcal{A}$ is a cogenerator, is it $T(A)$ too? If not, under which conditions?
  4. If $\mathcal{A}$ is complete, is it $\mathcal{A}/\mathcal{C}$ too? If not, under which conditions?

I know that:

  1. If $\mathcal{A}$ is cocomplete then so is $\mathcal{A}/\mathcal{C}$. ($T$ is a left adjoint)
  2. If $\{U_i\}$ is a set of generators then so is $\{T(U_i)\}$.
  3. If $\mathcal{A}$ is AB5 then so is $\mathcal{A}/\mathcal{C}$. ($T$ commutes with limits and one can prove that taking directed limits is exact.)
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This question continues on… . – archipelago Jun 2 '13 at 11:31

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