Building global schemes from affine pieces — the projective line P¹ and beyond
Just as a smooth manifold is built by gluing together open subsets of ℝn, a scheme is built by gluing together affine schemes along common open subsets. The simplest non-affine example is the projective line ℙ1: take two copies of the affine line 𝔸1 = Spec(k[t]) and glue them by identifying the open subset where t ≠ 0 in the first copy with the open subset where s ≠ 0 in the second, via the transition function t = 1/s.
This gluing construction produces genuinely new objects. The projective line is compact (proper over k) and has no non-constant global regular functions — properties that no affine scheme can have. The general Proj construction builds projective schemes from graded rings, giving a systematic way to construct ℙn, projective varieties, and weighted projective spaces.
To glue two affine schemes U1 = Spec(R1) and U2 = Spec(R2), we specify open subsets U12 ⊆ U1 and U21 ⊆ U2together with an isomorphism φ: U12 ≅ U21. The resulting scheme X has both U1 and U2 as open subschemes, and the structure sheaf is determined by the sheaves on U1and U2 plus the gluing isomorphism on the overlap.
For ℙ1, the two affine patches are Spec(k[t]) and Spec(k[s]), glued along Spec(k[t, t−1]) ≅ Spec(k[s, s−1]) via t ↔ 1/s. A global regular function on ℙ1 would be a polynomial in both t and s = 1/t — hence constant. This is the algebraic version of the fact that every holomorphic function on the Riemann sphere is constant.
Given a graded ring S = S0 ⊕ S1 ⊕ S2 ⊕ …, the scheme Proj(S) is constructed usinghomogeneous prime ideals that do not contain the irrelevant ideal S+. For S = k[x0, …, xn], this gives ℙn. The standard affine cover of ℙnconsists of D+(xi) ≅ Spec(S(xi)), where S(xi) is the degree-zero part of the localization at xi.
The Proj construction is the scheme-theoretic version of projectivization. By varying the graded ring, it produces projective varieties (quotients of polynomial rings), weighted projective spaces (non-standard gradings), and blowups (Rees algebras). It is the main machine for constructing compact schemes in algebraic geometry.