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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?

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up vote 11 down vote accepted

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.

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Another reference is section IV.9 of Hilton & Stammbach, A course in homological algebra. –  Charles Rezk Feb 12 '10 at 0:19
    
Perhaps it is worth noting that there is a description of cohomology classes in $H^n(G,M)$ for $G$ a group and $M$ a $G$-module, that uses crossed n-fold extensions. I do not know if the nice detailed higher structure of such cohomology groups of groups has been looked at using this description. –  Tim Porter Sep 25 '10 at 7:25
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I think what you are asking is for the standard description in terms of n-term exact sequences starting with A and ending with B (or the other way around). One usual reference is MacLane Chapter III section 5, p.82 in my copy. It is known as the Yoneda description.

It is also in http://en.wikipedia.org/wiki/Ext_functor

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