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. the name "ext" comes from the connection between the functors and extensions in abelian categories. == definition and computation == let r be a ring and let modr be the category of modules over r. let b be in modr and set t(b) = homr(a,b), for fixed a in modr. this is a left exact functor and thus has right derived functors rnt. the ext functor is defined by :\operatorname{ext}_r^n(a,b)=(r^nt)(b). this can be calculated by taking any injective resolution :0 \rightarrow b \rightarrow i^0 \rightarrow i^1 \rightarrow \dots, and computing :0 \rightarrow \operatorname{hom}_r(a,i^0) \rightarrow \operatorname{hom}_r(a,i^1) \rightarrow \dots. then (rnt)(b) is the homology of this complex. note that homr(a,b) is excluded from the complex. an alternative definition is given using the functor g(a)=homr(a,b). for a fixed module b, this is a contravariant left exact functor, and thus we also have right derived functors rng, and can define :\operatorname{ext}_r^n(a,b)=(r^ng)(a). this can be calculated by choosing any projective resolution :\dots \rightarrow p^1 \rightarrow p^0 \rightarrow a \rightarrow 0, and proceeding dually by computing :0\rightarrow\operatorname{hom}_r(p^0,b)\rightarrow \operatorname{hom}_r(p^1,b) \rightarrow \dots. then (rng)(a) is the homology of this complex. again note that homr(a,b) is excluded. these two constructions turn out to yield isomorphic results, and so both may be used to calculate the ext functor. == ext and extensions == ===equivalence of extensions=== ext functors derive their name from the relationship to extensions of modules. given r-modules a and b, an extension of a by b is a short exact sequence of r-modules :0\rightarrow b\rightarrow e\rightarrow a\rightarrow0. two extensions :0\rightarrow b\rightarrow e\rightarrow a\rightarrow0 :0\rightarrow b\rightarrow e^\prime\rightarrow a\rightarrow0 are said to be equivalent (as extensions of a by b) if there is a commutative diagram image:equivalenceofextensions.png. note that the five lemma implies that the middle arrow is an isomorphism. an extension of a by b is called split if it is equivalent to the trivial extension :0\rightarrow b\rightarrow a\oplus b\rightarrow a\rightarrow0. there is a bijective correspondence between equivalence classes of extensions :0\rightarrow b\rightarrow e\rightarrow a\rightarrow 0 of a by b and elements of :\operatorname{ext}_r^1(a,b). ===the baer sum o
Binary_operations Homological_algebra