Ext functor

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.

Related Topics:
Mathematics - Homological algebra - Derived functor - Hom functor - Algebraic topology

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More precisely, write mathcal C=mathbf{Mod}(R) for the category of module over R, a ring. Let A be in mathcal C and set T(A)=operatorname{Hom}_{mathcal C}(A,B), for fixed B in mathcal C. (This is a left exact functor (contravariant) so we want its right derived functors R^nT). To this end, define

Related Topics:
Category - Module - Ring - Left exact functor - Contravariant - Derived functor

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:operatorname{Ext}_R^n(A,B)=(R^nT)(A),

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i.e., take a projective resolution

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:P(A) ightarrow A ightarrow 0,

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compute

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:0 ightarrowoperatorname{Hom}_{mathcal C}(A,B) ightarrowoperatorname{Hom}_{mathcal C}(P(A),B),

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and take the cohomology on the righthand side.

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