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Encyclopedia > Ext functors

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 mathcal C= mathbf{Mod}(R) for the category of module over R, a ring. Let A be in mathcal C and set T(A)= mathrm{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 RnT). To this end, define

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

i.e., take a projective resolution

P(A) rightarrow A rightarrow 0,

compute

0 rightarrow mathrm{Hom}_{ mathcal C}(A,B) rightarrow mathrm{Hom}_{ mathcal C}(P(A),B),

and take the cohomology on the righthand side.


  Results from FactBites:
 
Ext functor - Wikipedia, the free encyclopedia (142 words)
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors.
Ext functors take their name from their relationship to extensions.
Given R-modules A and B, there is a bijective correspondence between equivalence classes of extensions of A by B and elements of
NationMaster - Encyclopedia: Derived functor (1358 words)
The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H Derived functors and the long exact sequences are "natural" in several technical senses.
Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
  More results at FactBites »

 

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