# Ext & Complexes

I have heard that given two sheaves $A$ and $B$ on a variety, one can identify elements of $Ext^d(A,B)$ with complexes of sheaves $$0\to B \to C_1 \to \cdots \to C_d \to A \to 0.$$

My questions are,

How do I see that this is true?

and

If I have obtained an element of $Ext^n$ by some other method, can I explicitly construct the $C_j$ sheaves and the differentials?

I am sure this is well-known, so I'm marking it also as "reference-request".

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Are you familiar with how to do this in the setting without sheaves? For example $Ext$ of $R$-modules and how to get extensions from cocycles? – Sean Tilson Dec 5 '10 at 18:34
I've never gone through the general case, but I found that working out the special case of exact sequences $0 \to S \to E \to Q \to 0$ of vector bundles (i.e. of $Ext^1(S,Q)$) gives a pretty good idea of what's going on (it also makes you never want to check the details in the general case). – Gunnar Þór Magnússon Dec 5 '10 at 18:43
@Gunnar, to get a feeling for the general case, you need to do at least $\mathrm{Ext}^2$. – Mariano Suárez-Alvarez Dec 9 '10 at 1:51

In Vista 3.4.6, Weibel says "... the set of equivalence classes ... (if this is indeed a set)", and then claims that $Ext^1(A,B)$ is an abelian group. How can $Ext^1$ not be a set, and still be an abelian group? – James Davidoff Dec 5 '10 at 19:13