##  [Serre Duality](/serre-duality) 

  ##  [Serre Duality](https://algebra.quantumdictionary.io/serre-duality-0) 

  

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- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A duality theorem for proper smooth n-dimensional algebraic or complex-analytic spaces giving a natural perfect pairing between the cohomology of a coherent sheaf and the cohomology of its dual twisted by the canonical (dualizing) sheaf, typically expressed as H^i(X,F) ≅ Hom(H^{n-i}(X, ω_X ⊗ F^∨), k)^* or an isomorphism H^i(X,F)^* ≅ H^{n-i}(X, ω_X ⊗ F^∨).