Ext functor
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In mathematics, the Ext functors of homological algebra are derived functors of <math>\mathrm{Hom}<math> functors. They were first used in algebraic topology, but are common in many areas of mathematics.
More precisely, write <math>\mathcal C=\mathbf{Mod}(R)<math> for the category of module over <math>R<math>, a ring. Let <math>A<math> be in <math>\mathcal C<math> and set <math>T(A)=\mathrm{Hom}_{\mathcal C}(A,B)<math>, for fixed <math>B<math> in <math>\mathcal C<math>. (This is a left exact functor (contravariant) so we want its right derived functors <math>R^nT<math>). To this end, define
- <math>\mathrm{Ext}_R^n(A,B)=(R^nT)(A),<math>
i.e., take a projective resolution
- <math>P(A)\rightarrow A\rightarrow 0,<math>
compute
- <math>0\rightarrow\mathrm{Hom}_{\mathcal C}(A,B)\rightarrow\mathrm{Hom}_{\mathcal C}(P(A),B),<math>
and take the cohomology on the righthand side.Template:Math-stub