Ext functor

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In mathematics, the Ext functors of homological algebra are derived functors of Hom 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)=\operatorname{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>\operatorname{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\operatorname{Hom}_{\mathcal C}(A,B)\rightarrow\operatorname{Hom}_{\mathcal C}(P(A),B),</math>

and take the cohomology on the righthand side.Template:Topology-stub Template:Cattheory-stub