Serre Duality

High Quality Content by WIKIPEDIA articles! In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn i. If the variety is defined over the complex numbers, this yields different information from Poincaré duality, which relates Hi to H2n i, considering V as a real manifold of dimension 2n. The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hi vanish for i 1, but H1 does enter implicitly. In fact, the basic relation of the theorem involves l(D) and l(K D), where D is a divisor and K is a divisor of the canonical class. After Serre we recognise l(K D) as the dimension of H1(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H0(D) and H1(KD ), and we are reading off dimensions (notation: K is the canonical line bundle, D is the dual line bundle, and juxtaposition is the tensor product of line bundles).... Mehr