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On Serre Duality

Serre Duality is the statement, for X a smooth projective (integral) variety and \mathcal{E} a locally-free sheaf on X,

H^i(X,\mathcal{E}) \cong H^{n-i}(X, \mathcal{E}^* \otimes \omega_X)^*,

where \omega_X is the canonical bundle and n = \dim X. This isomorphism is almost canonical: it depends on the choice of an isomorphism

t: H^n(X,\omega_X) \to k,

called a trace map. I’m going to sketch out my understanding of what’s going on with this duality statement and how it comes up (non-rigorously).