About higher Ext in R-Mod - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:40:41Z http://mathoverflow.net/feeds/question/15016 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15016/about-higher-ext-in-r-mod About higher Ext in R-Mod some guy on the street 2010-02-11T18:29:21Z 2010-02-11T18:44:02Z <p>So, in $R-Mod$, we have the rather short sequence</p> <ul> <li><p>$\mathrm{Ext}^0(A,B)\cong Hom_R(A,B)$</p></li> <li><p>$\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv$, equivalence classes of "good" factorizations of $0\in Hom_R(A,B)\cong\mathrm{Ext}^0(A,B)$, with the Baer sum.</p></li> </ul> <p>Question:</p> <ul> <li>$\mathrm{Ext}^{2+n}(A,B) \cong\ ???$</li> </ul> <p>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 <em>what sort</em> of extension? Or is that already completely wrong?</p> http://mathoverflow.net/questions/15016/about-higher-ext-in-r-mod/15017#15017 Answer by Jack Schmidt for About higher Ext in R-Mod Jack Schmidt 2010-02-11T18:37:14Z 2010-02-11T18:37:14Z <p>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. </p> http://mathoverflow.net/questions/15016/about-higher-ext-in-r-mod/15018#15018 Answer by Tim Porter for About higher Ext in R-Mod Tim Porter 2010-02-11T18:41:51Z 2010-02-11T18:41:51Z <p>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.</p> <p>It is also in <a href="http://en.wikipedia.org/wiki/Ext_functor" rel="nofollow">http://en.wikipedia.org/wiki/Ext_functor</a></p>