Serre duality
/ sair /
On a compact orientable manifold, Poincare duality says the cohomology in degree i is dual to the cohomology in the complementary degree. Serre duality is the algebraic-geometry counterpart for the cohomology of coherent sheaves: it pairs the high-degree cohomology of a sheaf with the low-degree cohomology of a complementary, 'twisted-dual' sheaf, with the twist supplied by a single canonical object, the canonical sheaf. It is the symmetry that turns hard-to-compute high cohomology into easy-to-compute global sections, and it is the engine that makes Riemann-Roch usable.
Precisely, let X be a smooth projective variety of dimension n over a field k, with canonical sheaf omega_X (the top exterior power of the cotangent sheaf, the sheaf of top-degree differential forms). Serre duality asserts a perfect pairing, for any coherent locally free sheaf (vector bundle) F and each i, between H^i(X, F) and H^{n-i}(X, F^dual tensor omega_X), realized via the cup product into the top cohomology H^n(X, omega_X), which is canonically isomorphic to k (the trace map). In particular the dimensions match: h^i(X, F) = h^{n-i}(X, F^dual tensor omega_X). For a line bundle L on a smooth projective curve (n = 1) this reads h^1(L) = h^0(omega_X tensor L^{-1}) = h^0(K - D) in divisor notation, which is exactly the term that converts the Riemann-Roch theorem from an inequality-with-error into a clean equality: h^0(D) - h^0(K - D) = deg D + 1 - g, where g is the genus and K the canonical divisor. So Serre duality computes the otherwise-mysterious h^1.
Serre duality is foundational: it is what makes the cohomology of curves and surfaces tractable, underlies the Riemann-Roch theorem on curves and surfaces and the meaning of the arithmetic genus, and generalizes (Grothendieck-Serre duality) to singular and relative settings via a dualizing complex. Honest cautions. First, the clean statement above needs hypotheses: smoothness and properness (projective) over a field; for SINGULAR or non-Cohen-Macaulay schemes the canonical sheaf must be replaced by a dualizing complex, and the simple sheaf-level pairing becomes a derived-category statement. Second, omega_X is the canonical sheaf of THIS theory, an algebraic object (top differentials); do not confuse this duality with topological Poincare duality (which is over Z and sees torsion) or with the Atiyah-Singer index theorem (an analytic-index statement) — Serre duality is a statement about coherent-sheaf cohomology and the canonical bundle, full stop. Third, the pairing is between F and its twisted dual, not F with itself; the canonical twist by omega_X is essential and is exactly where the geometry of the variety enters.
On a smooth projective curve X of genus g, take D a divisor and L = O(D). Serre duality gives h^1(O(D)) = h^0(omega_X(-D)) = h^0(K - D). Plug into Riemann-Roch h^0(D) - h^1(D) = deg D + 1 - g to get the usable form h^0(D) - h^0(K - D) = deg D + 1 - g. For D = 0 this yields h^0(K) = g: the genus is exactly the dimension of the space of global holomorphic differentials.
Serre duality turns h^1(D) into h^0(K - D), making Riemann-Roch on curves an exact equality.
The clean omega_X pairing needs smoothness and properness over a field; on singular schemes omega_X becomes a dualizing complex and the statement is derived-categorical. And this is coherent-sheaf duality with the canonical bundle — not topological Poincare duality and not the Atiyah-Singer index theorem.