# Non-trivial conditions for $\mathrm{Ext}^2(A,B)=0$?

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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If you described more concretely your situation, it would be easier to come up with something useful—«an abelian category with a class of objects» is a rather generic situation – Mariano Suárez-Alvarez Sep 18 '11 at 7:04
Dear Mariano, I added a description to the question. – Rasmus Sep 18 '11 at 9:03
Does it help in this specific situation to use dimension-shifting to make the question "when is $\operatorname{Ext}^1 = 0$? – m_t_ Sep 18 '11 at 11:21
@mt_: I don't see how this would help. – Rasmus Sep 18 '11 at 11:39
The added details are not exactly details. You basically wrote «the objects have some a bunch of properties» without explaining what the properties were (of every class of objects in every abelian category you can say that «various sequences are exact, certain maps vanish, certain groups are free etc.») Similarly, Freyd's embedding theorem implies that every abelian category is equivalent to one consisting of modules over a finit category. – Mariano Suárez-Alvarez Sep 18 '11 at 16:13