(perfect pairing)
A bilinear pairing is a map \(M \tensor_A N \to L\). By the tensor-hom adjunction this is the same data as a canonical map \[ M \to \Hom_A(N, L) . \] The pairing is perfect if this canonical map is an isomorphism. It is nondegenerate if the canonical maps \(M\to\Hom_A(N,L)\) and \(N\to\Hom_A(M,L)\) are injective.
Over a field, with \(M\) and \(N\) finite-dimensional and \(L\) the field, nondegenerate and perfect agree. Over \(\ZZ\), the pairing \(\ZZ\tensor_\ZZ\ZZ\to\ZZ\), \(a\tensor b\mapsto 2ab\), is nondegenerate and not perfect: the canonical map \(\ZZ\to\Hom_\ZZ(\ZZ,\ZZ)\cong\ZZ\) is multiplication by \(2\).
Let \(X\) be a nonsingular projective variety of dimension \(n\) over an algebraically closed field \(k\), and let \(\mcf\) be a locally free sheaf of finite rank. Serre duality asserts that the cup product \[ H^i(X;\mcf) \tensor H^{n-i}(X; \dualof{\mcf} \tensor \omega_X) \to H^n(X;\omega_X) \cong k \] is a perfect pairing of finite-dimensional \(k\)-vector spaces [@Har10a, Corollary III.7.7]; so \(h^i(\mcf) = h^{n-i}(\dualof{\mcf} \tensor \omega_X)\).
Dependencies
D-DEFTENSThe tensor product of modules, and extension of scalarsuses