nLab
Ext

Idea

In the context of homological algebra the right derived functor of the hom-functor is called the $\mathrm{Ext}$-functor . It derives its name from the fact that the derived hom $\mathrm{Ext}(A,K)$ between abelian groups classifies group extensions of $A$ by $K$.

(This is really a special case of the general discussion at cohomology and group cohomology.)

Together with the Tor-functor it is one of the central objects of interest in homological algebra.

Details

Given an abelian category $A$ we may consider the hom-functor ${\mathrm{Hom}}_A:A^{\mathrm{op}}\times A\to \mathrm{Ab}$ either as a functor in first or in second variable, and compute the corresponding right derived functors. If they exist, the classical right derived functors of either functor agree and also agree with the homology of the mixed double complex obtained by taking simultaneously an injective resolution of the first contravariant argument and projective resolution of the second covariant argument. The last construction is called the balanced $\mathrm{Ext}$.

Alternatively, one can consider the derived category $D(A)$ and define

$${\mathrm{Ext}}^i(M,N):={\mathrm{Hom}}_{D(A)}(M,N[i])$$Ext^i(M,N) := Hom_{D(A)}(M,N[i])

or define ${\mathrm{Ext}}^i$-groups as groups of extensions of length $i$.

References

• H. Cartan, S. Eilenberg, Homological algebra, Princeton Univ. Press 1956.

• M. Kashiwara and P. Schapira, Categories and Sheaves, Springer (2000)

• S. I . Gelfand, Yu. I. Manin, Methods of homological algebra

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, CUP 1994