 The higher Ext functors measure the non-exactness of the Hom-functor.The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring "R" with residue field "k" is regular if and only if
vanishes for all large enough "n".Moreover, the dimensions of these Ext-groups, known as Betti numbers, grow polynomially in "n" if and only if "R" is a local complete intersection ring.