Ext functor

(Redirected from Ext group)

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 $R n T$ ). 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.

Categories: Homological algebra

Last updated: 08-26-2005 02:17:59

Ext functor - Your Art History Reference Guide!