(Ext)
There are two definitions of the \(\Ext\) functors, which are seen to agree by a spectral sequence argument. Fixing an \(A\)-module \(M\), define \(\Ext^i_A(M, N)\) to be the right derived functors of the left exact functor \(\Hom_A(M, \wait)\); this gives the usual long exact sequence in the second variable.
Alternatively, fixing \(N\), take the right derived functors of the contravariant left exact functor \(\Hom_A(\wait, N)\) to be \(\Ext^i_A(\wait, N)\); this gives a long exact sequence in the first variable.
\(\Ext^0_A(M,N) = \Hom_A(M,N)\), and \(\Ext^i_A(M,N) = 0\) for \(i>0\) whenever \(M\) is projective or \(N\) is injective. Since the two definitions agree, \(\Ext\) can be computed from a projective resolution of \(M\) or from an injective resolution of \(N\): over \(\ZZ\), the resolution \(0 \to \ZZ \mapsvia{n} \ZZ \to \ZZ/n \to 0\) gives \(\Ext^1_\ZZ(\ZZ/n, N) = N/nN\).
\(\Ext^1(M,N)\) classifies extensions \(0 \to N \to E \to M \to 0\) up to equivalence, with the split extension as the zero class. For \(X\) projective, Cohen–Macaulay, and equidimensional of dimension \(n\) over an algebraically closed field \(k\), with dualizing sheaf \(\omega_X\), and for \(\mcf\) coherent, Serre duality gives \(\Ext^i(\mcf, \omega_X)\cong H^{n-i}(X,\mcf)^\vee\) [@Har10a, Theorem III.7.6].
Dependencies
D-DEFDERIVDerived functorsuses