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Monthly Archives: April 2014

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).