Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be abelian categories and $F: \mathcal{A} \to \mathcal{C}$ and $G: \mathcal{B} \to \mathcal{C}$ be functors between them. It is then possible to define the pullback category $\mathcal{D} := \mathcal{A}\times_\mathcal{C} \mathcal{B}$, which is abelian again. One way to define $\mathcal{D}$ concretely is the following:
- The objects of $\mathcal{D}$ consist of pairs $(A,B)$, where $A$ is an object of $\mathcal{A}$ and $\mathcal{B}$ is an object of $B$, together with an isomorphism $f: FA \to GB$.
- Morphisms between $(A,B,f)$ and $(A', B', f')$ consist of two morphisms $A\to A'$ and $B\to B'$ such that the corresponding square diagram in $\mathcal{C}$ commutes.
Such situations sometimes occur in algebraic geometry, where the relevant abelian categories are categories of quasi-coherent modules.
By definition, we get an exact sequence \begin{eqnarray*}0\to Hom_{\mathcal{D}}((A,B,f), (A',B',f')) \to Hom_{\mathcal{A}}(A,A') \oplus Hom_{\mathcal{B}}(B,B') \to Hom_{\mathcal{C}}(FA,GB')\end{eqnarray*}
Now suppose that all occuring abelian categories have, say, enough injectives. Then my question is the following:
Can we define boundary morphisms \begin{eqnarray*}Ext^i_{\mathcal{C}}(FA, > GB') \to Ext^{i+1}_{\mathcal{D}}((A, > B, f), (A',B',f'))\end{eqnarray*} such that the emerging sequence of Ext groups is long exact?
Probably one needs at least some additional assumption for this to hold, such as exactness of $F$ and $G$ or the existence of some adjoints.
This is an attempt to control the cohomological dimension of $\mathcal{D}$ in terms of the cohomological dimension of $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ - other suggestions to this enterprise are also welcome.
