Serre Duality - Origin and Generalisations

Origin and Generalisations

The origin of the theory lies in Serre's earlier work on several complex variables. In the generalisation of Alexander Grothendieck, Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot merely be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves, namely, the dualizing complex. Nevertheless, the statement of the theorem is recognisably Serre's.