Why global sections are not enough: the birth of cohomology
Guide 4 left us with coherent sheaves, Cartier divisors, and the line bundles that package twisting data on a scheme. The natural question — how many global sections does a line bundle O(D) have? — turns out to be the wrong question to ask in isolation. The functor that takes a sheaf F to its module of global sections F(X) is only LEFT exact: from a short exact sequence of sheaves 0 -> F' -> F -> F'' -> 0 you get an exact sequence of global sections 0 -> F'(X) -> F(X) -> F''(X), but the last map need not be onto. A section of F'' that is locally liftable to F may fail to lift globally. That failure is real geometric information, and sheaf cohomology is the bookkeeping device that records it.
Here is the smallest honest picture. On the circle S^1, cover it with two arcs U and V whose intersection is two small pieces. The constant sheaf R has plenty of local sections — a real number on each arc — yet the locally-constant functions on U and V need not agree on the overlap in a way that glues to a single global function with a prescribed jump. The obstruction to gluing is one real number, and that number is exactly H^1(S^1, R) = R. Cohomology in degree 0 counts what DOES glue (the global sections); cohomology in degree 1 and higher counts the obstructions to gluing, the ghosts of sections that exist locally but cannot be assembled globally.
Cech cohomology: a recipe you can run by hand
Fix an open cover U = {U_i} of X and a sheaf F. A Cech k-cochain assigns to every (k+1)-fold overlap U_{i_0...i_k} a section of F over it; the data on a triple overlap is what catches a failure to glue. The Cech differential d sends a cochain to its alternating restriction-difference on one-larger overlaps: for a 0-cochain (s_i), one i per open set, (d s)_{ij} = s_j - s_i on U_i intersect U_j. A 0-cochain is a cocycle (d s = 0) precisely when the s_i agree on overlaps — i.e. when they glue to a global section. So H^0 = ker d in degree 0 = global sections F(X), recovering the degree-0 story automatically.
In degree 1 the action lives. A 1-cochain is a family (t_ij) of sections on double overlaps; it is a cocycle if t_ij - t_ik + t_jk = 0 on triple overlaps (the cocycle condition you already met for transition functions of a line bundle in Guide 4). It is a coboundary if t_ij = s_j - s_i for some 0-cochain (s_i). Then H^1(X, F) = (cocycles)/(coboundaries): a 1-cocycle measures locally-consistent gluing data, and it is a coboundary exactly when that data is fake — already achievable by adjusting the local sections. A nonzero class in H^1 is genuine gluing data that no honest choice of local sections produces. This is precisely why line bundles up to isomorphism form a group H^1(X, O*): the transition cocycle modulo coboundaries IS the bundle.
Cech complex for a cover U = { U_i } of X, with a sheaf F:
C^0 ---d---> C^1 ---d---> C^2 ---d---> ...
(s_i) (t_ij) (a_ijk)
d on C^0 : (d s)_ij = s_j - s_i on U_i n U_j
d on C^1 : (d t)_ijk = t_jk - t_ik + t_ij on U_i n U_j n U_k
H^k(X, F) = ker(d : C^k -> C^{k+1}) / im(d : C^{k-1} -> C^k)
H^0 = global sections F(X) (things that glue)
H^1 = gluing obstructions (local-but-not-global sections)
Key vanishing fact (this is WHY affine schemes are the good building blocks):
X affine, F quasi-coherent ==> H^k(X, F) = 0 for all k > 0.Finiteness, vanishing, and the Euler characteristic
Three structural facts make this theory usable rather than merely defined. First, the affine vanishing theorem: on an affine scheme Spec R every coherent sheaf has zero higher cohomology, because a quasi-coherent sheaf there is just an R-module and modules have no gluing obstructions over a single ring. This is the precise sense in which affines are 'the good local model' promised back in Guide 2 — they are cohomologically trivial. Second, Grothendieck vanishing: on a Noetherian scheme of dimension n, H^k(X, F) = 0 for every k greater than n. So on a curve only H^0 and H^1 survive; on a surface only H^0, H^1, H^2. The cohomology lives in a finite window set by the dimension.
Third, and the one that makes Riemann-Roch tick: on a projective scheme over a field, every coherent sheaf F has finite-dimensional cohomology groups, all zero above the dimension. So we may form the Euler characteristic chi(F) = sum over i of (-1)^i dim H^i(X, F), a finite alternating sum of finite dimensions. The miracle of the Euler characteristic is its STABILITY: from a short exact sequence 0 -> F' -> F -> F'' -> 0 the long exact sequence in cohomology forces chi(F) = chi(F') + chi(F''). Individual cohomology dimensions jump around wildly as you deform a sheaf, but their alternating sum is rigid and additive. Riemann-Roch is, at heart, the statement that this rigid number has a clean geometric formula.
Serre duality: the pairing that closes the top
The Euler characteristic needs to know the top cohomology H^n, and that is where Serre duality earns its place. On a smooth projective variety X of dimension n with canonical sheaf omega_X (the top exterior power of the cotangent sheaf, the line bundle of top differential forms), Serre duality gives a perfect pairing H^i(X, F) x H^{n-i}(X, omega_X tensor F^dual) -> H^n(X, omega_X) = k for any locally free coherent sheaf F. In words: the i-th cohomology of F is the dual vector space of the (n-i)-th cohomology of the omega_X-twisted dual of F. Top cohomology becomes bottom cohomology read backwards. It is the algebraic-geometry sibling of Poincare duality and of the Hodge-star duality you met in the complex-geometry rung.
Concretely, on a smooth projective curve C of genus g, the canonical sheaf omega_C has a 1-dimensional space H^0(C, omega_C) when g = 1 and, in general, dim H^0(C, omega_C) = g — the holomorphic 1-forms are exactly the genus many. Serre duality on the curve reads H^1(C, L) = H^0(C, omega_C tensor L^dual)^dual for a line bundle L = O(D). That single identity converts the hard, unstable group H^1 into the dimension of a space of sections of another, often more tractable, line bundle. When deg L is large enough, omega_C tensor L^dual has negative degree, hence no sections, hence H^1(C, L) = 0 — this is exactly the vanishing the previous callout asked for, now derived rather than assumed.
Riemann-Roch: on a curve, then on a surface
Now the payoff. On a smooth projective curve C of genus g, for a divisor D with line bundle L = O(D), the Riemann-Roch theorem computes the Euler characteristic outright: chi(O(D)) = dim H^0 - dim H^1 = deg D + 1 - g. The right side knows only the DEGREE of D (how many points, with sign) and the genus g of C — pure topology and intersection numbers, no cohomology. Feeding in Serre duality, dim H^1(O(D)) = dim H^0(omega_C(-D)) = dim H^0(K - D) using the canonical divisor K with deg K = 2g - 2, gives the classical two-sided form: dim H^0(D) - dim H^0(K - D) = deg D + 1 - g. That is the Riemann-Roch theorem in the shape Riemann and Roch actually used.
- Read the genus from D = 0: O(0) is the trivial bundle, H^0 = constants (dimension 1) and by Serre duality H^1 = H^0(omega_C)^dual (dimension g). Riemann-Roch gives chi = 1 - g, and indeed 1 - g = deg 0 + 1 - g. The formula is forced to be self-consistent at the trivial bundle — a good sanity check before trusting it.
- Get effective sections for free: if deg D > 2g - 2, then deg(K - D) < 0 so H^0(K - D) = 0, and dim H^0(D) = deg D + 1 - g exactly. A line bundle of high enough degree has predictably many sections, no analysis required.
- Sharpen embeddings: on an elliptic curve (g = 1), deg D = 3 gives dim H^0 = 3, and those three sections embed the curve as a plane cubic in CP^2 — the Weierstrass equation reappears as a Riemann-Roch consequence. The count IS the geometry.
Lifting to a smooth projective surface S, the same skeleton grows a curvature-flavored correction term. For a divisor D with line bundle L = O(D), surface Riemann-Roch reads chi(O(D)) = (1/2) D.(D - K) + chi(O_S), where D.(D - K) is the intersection number of divisors on the surface (the algebraic shadow of the cup product), K is the canonical divisor, and chi(O_S) = 1 - q + p_g is a fixed invariant of S built from its irregularity q and geometric genus p_g. The structure is identical to the curve case — a quadratic intersection term plus a topological constant — and Serre duality on the surface, H^2(O(D)) = H^0(K - D)^dual, again converts the awkward top group into a space of sections you can hope to control.
Be honest about altitude here. Both forms are special cases of the Hirzebruch-Riemann-Roch theorem, chi(F) = integral over X of ch(F) . td(X), and ultimately of the Serre-Grothendieck-Riemann-Roch framework and the Atiyah-Singer index theorem — a Chern-character-times-Todd-class integral that we can state and motivate but absolutely cannot prove in a guide; a real proof is a course. The curve formula is the n = 1 instance and the surface formula the n = 2 instance of one grand statement. What this rung genuinely earns you is the machine that makes those statements MEAN something concrete: sheaves to carry the data, Cech cohomology to compute it, vanishing theorems to simplify it, Serre duality to close the top, and the Euler characteristic to make the whole alternating sum rigid. That is the working algebraic geometer's toolkit, and from here it powers everything from moduli problems to modern intersection theory.