So, in $R-Mod$, we have the rather short sequence


* $\mathrm{Ext}^0(A,B)\cong Hom\_R(A,B) $
* $\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of "good" factorizations of $0\in Hom\_R(A,B)\cong\mathrm{Ext}^0(A,B)$, with the Baer sum.


Question:


* $\mathrm{Ext}^{2+n}(A,B) \cong\ ??? $


While I suppose one could pose a conjugate question in algebraic topology/geometry, where the answer might look "simpler", I'm asking for a more directly algebraic/diagramatic understanding of the higher $\mathrm{Ext}$ functors. For instance, I'd expect $\mathrm{Ext}^2(A,B)$ to involve diagrams extending the split exact sequence $A\rightarrow A\oplus B\rightarrow B$, but precisely *what sort* of extension? Or is that already completely wrong?



About higher Ext in R-Mod
They correspond to longer exact sequences under an equivalence relation due to Yoneda. See chapter III.3 (p. 82ff) of MacLane's Homology (or briefly on the wikipedia page for the Ext functor). There are also many online sources for "higher extension modules and yoneda", but MacLane's presentation is clear and describes the Baer addition very nicely. Yoneda also describes a product from Ext^n x Ext^m to Ext^(n+m) that can turn certain Ext's into rings. This is popular to do with Ext^\*(k,k) where k is the trivial module for a k-algebra. 


