From rings to sheaves of modules
By guide 3 you can read a scheme X as a locally ringed space glued from affine pieces Spec R, each carrying its structure sheaf O_X of functions. The next step is the one that makes algebraic geometry feel like linear algebra over a varying base: instead of functions, study modules that live over O_X. A sheaf F of O_X-modules assigns to each open U an O_X(U)-module F(U), compatibly with restriction. On the affine chart Spec R, the cleanest such sheaves come from an honest R-module M by a recipe that exactly mirrors how O_X itself was built — localize at each prime.
Write the result as the sheaf M-twiddle on Spec R: its stalk at the prime p is the localization M_p, and its sections over a basic open D(f) are M[1/f]. When M = R you recover O_X. This quasi-coherent construction is the bridge: a quasi-coherent sheaf on a scheme is one that on every affine chart Spec R looks like M-twiddle for some module M. The slogan is honest and useful — quasi-coherent sheaves on Spec R are the SAME data as R-modules, an equivalence of categories. So all of commutative algebra becomes geometry, locally.
Coherent means finitely generated, with control
Quasi-coherent is generous; coherent is the disciplined subclass you actually compute with. Over a Noetherian scheme — the safe setting, where rings have no infinite ascending chains of ideals — a coherent sheaf is one that on each affine chart is M-twiddle for a FINITELY GENERATED module M. Finite generation is the analogue of finite-dimensionality: it is what makes ranks, dimensions, and eventually Euler characteristics finite numbers rather than wishful thinking.
Two examples anchor the idea. First, the structure sheaf O_X is coherent — finitely generated by the constant 1. Second, the ideal sheaf I_Z of a closed subscheme Z, whose sections over U are the functions vanishing on Z, is coherent and fits in a short exact sequence 0 -> I_Z -> O_X -> O_Z -> 0. Read that sequence: it says a closed subscheme is exactly the cokernel data of its vanishing ideal, the sheaf-level version of the coordinate ring quotient R/I you met for varieties. Coherent sheaves are closed under kernels, cokernels, and extensions, so this category is an abelian category — the right home for homological algebra.
Divisors: codimension-one geometry
Now switch from algebra to geometry for a moment. On a nice variety or scheme, a Weil divisor is a finite formal integer combination D = sum n_i Z_i of irreducible codimension-one closed subschemes — curves on a surface, points on a curve, hypersurfaces in P^n. The divisor of a rational function f records its zeros and poles with multiplicities: div(f) = sum (order of f along Z) times Z. Such divisors of functions are called principal, and two divisors are linearly equivalent when their difference is principal — D ~ D' iff D - D' = div(f) for some f. This is the exact graduate upgrade of the zeros-and-poles bookkeeping you saw on a Riemann surface, now legal on any reasonable scheme.
Weil divisors are intuitive but they need the scheme to be normal (so that 'order along Z' makes sense). The scheme-friendly cousin is the Cartier divisor: data given LOCALLY by a single rational function f_alpha on each open U_alpha, where on overlaps the ratio f_alpha / f_beta is an invertible regular function. A Cartier divisor is thus a recipe for cutting out a codimension-one locus by ONE equation locally, with the equations agreeing up to units. On a smooth scheme Weil and Cartier divisors coincide; on a singular one they can genuinely differ, and Cartier is the better-behaved notion because it is defined by the structure sheaf alone.
Line bundles, invertible sheaves & the Picard group
Here is the unification. The transition data of a Cartier divisor — the units f_alpha / f_beta on overlaps — is exactly the cocycle of a rank-one vector bundle, i.e. a line bundle. Algebraically, a line bundle is the same thing as an invertible sheaf: a coherent sheaf L that is locally free of rank one, meaning every point has a neighborhood where L looks like O_X itself. 'Invertible' is literal — there is a sheaf L^{-1} with L tensor L^{-1} = O_X, and tensor product makes these sheaves into a group.
That group is the Picard group Pic(X). The Picard group collects isomorphism classes of line bundles under tensor product, with O_X as identity and L^{-1} as inverse. The dictionary closes a loop: Cartier divisors modulo linear equivalence are isomorphic to Pic(X). A divisor D gives a line bundle O_X(D); D ~ D' precisely when O_X(D) and O_X(D') are isomorphic; and global sections of O_X(D) are exactly the rational functions f with div(f) + D effective (zeros and poles bounded by D). This last fact is the engine of the next guide — counting those sections is a cohomology problem.
Cartier divisors / linear equiv. <-> Pic(X) = { line bundles } / iso
D ~ D' (D - D' principal) <-> O_X(D) = O_X(D')
H^0(X, O_X(D)) = { f rational : div(f) + D >= 0 } u {0}
O_X(D) (x) O_X(D') = O_X(D + D'), O_X(0) = O_XA worked example: line bundles on projective space
Make it concrete on P^n, the example that organizes the whole subject. Every line bundle on P^n is O(d) = O(1)^{tensor d} for a unique integer d, so Pic(P^n) = Z. The bundle O(1) is the dual of the tautological sub-bundle; its global sections are the homogeneous LINEAR forms in the coordinates x_0, ..., x_n, an (n+1)-dimensional space. More generally H^0(P^n, O(d)) is the space of degree-d homogeneous polynomials when d >= 0, and it is ZERO when d < 0 — a negative line bundle has no global sections at all.
- Cover P^n by the standard charts U_i = { x_i != 0 }, each a copy of affine n-space — these are the affine pieces of guide 2.
- On the overlap U_i ∩ U_j, the transition function of O(d) is (x_i / x_j)^d, an invertible regular function — verify it is a cocycle so the local pieces glue to a genuine line bundle.
- Identify the associated divisor: O(d) = O_X(d H), where H is any hyperplane { linear form = 0 }, so the integer d literally counts hyperplanes.
- Read off sections: a global section of O(d) is a degree-d hypersurface (its zero locus), recovering 'a curve in P^2 of degree d is a section of O(d)' — the classical picture, now bundle-theoretic.
Notice what just happened: the geometry (hypersurfaces of degree d), the algebra (degree-d polynomials), and the bundle (O(d)) are three faces of one object, and the sheaf cohomology H^0 measures the algebra. Guide 5 picks up exactly here: the dimensions h^i = dim H^i(X, L) of the higher cohomology groups, their alternating sum the Euler characteristic, Serre duality pairing H^i with H^{n-i}, and the Riemann-Roch theorem turning all of it into an intersection-number count on a surface. Coherent sheaves, divisors, and line bundles are the nouns; cohomology is the verb that finally lets us count.