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Edit:

Since I had some trouble making my previous question precise without diving into details about the origin of the homological objects I'm interested in, let me ask a more open-ended question:

Are there any known examples of the following setting:

A subclass $\mathfrak N$ of an Abelian category $\mathfrak C$ such that $\mathrm{Ext}^2(A,B)=0$ for all pairs of objects $A$, $B$ in $\mathfrak N$, but with infinitely many objects of $\mathfrak N$ having projective dimension greater than $1$?





Original question:

Given an Abelian category $\mathfrak C$ and a subclass $\mathfrak N\subset\mathfrak C$ of "nice" objects, I would like to prove $\mathrm{Ext}^2(A,B)=0$ for all $A,B\in\mathfrak N$.

This would be easy if I knew that all objects in $\mathfrak N$ had projective or injective dimension at most $1$. In this case, I would use that one of the variables is very nice (and also get a stronger result). Let's assume that objects in $\mathfrak N$ are, in general, not nice enough to ensure this.

Question: Are there any conditions for $\mathrm{Ext}^2(A,B)=0$ using that $A$ and $B$ are rather nice (without $A$ or $B$ being very nice)?

Edit: A more concrete description of the objects in the situation I am motivated by: Objects in $\mathfrak C$ are modules over a certain finite category, that is, a bunch of Abelian groups with a bunch of homomorphisms between them, which fulfill certain relations. These modules will often have infinite projective dimension. Objects in $\mathfrak N$ have several special properties: various sequences are exact, certain maps vanish, certain groups are free etc. Objects in $\mathfrak N$ have finite projective dimension.

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    @Rasmus: the two ways of understanding the phrase are equally inespecific :) Don't worry. If you think Theo will be able to deduce what you have in mind, then just wait for him.2011-09-18

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