Gluing Let {Xi: i ∈ I} be a family of locally ringed spaces

1. Gluing

Let {X_i: i ∈ I} be a family of locally ringed spaces. Assume that for each i, j ∈ I, there exist an open subset Uij of Xi and an open subset Uji of Xj so that for i, j, k ∈ I,

(1) ϕ_ij : U_ij → U_ji is an isomorphism of locally ringed spaces, (2) ϕ⁻¹_ij (U_ji∩ U_jk) = U_ij ∩ U_ik,

(3) ϕ_ik= ϕ_jk◦ ϕ_ij.

Here we set Uii= Xi and ϕii= idXi. A collection {(I, Xi, Uij, ϕij)} of such data is called a gluing data.

Proposition 1.1. Given a gluing data of locally ringed spaces, there is a locally ringed space X together with a family of open subspaces {U_i} and a family of isomorphisms of locally ringed spaces {ϕi : Xi → U_i} such that

(1) ϕ(U_ij) = U_i∩ U_j and (2) ϕij = ϕ⁻¹_j |_Ui∩Uj◦ ϕ_i|_Uij.

Assume that we prove Proposition 1.1. We immediately obtain a scheme X when X_i are all schemes given in a gluing data:

Corollary 1.1. In Lemma 1.1 above, assume that X_i are all schemes. Then X is also a scheme.

Proof. Let x be a point in X. Then x is contained in some U_i. Since U_i is a scheme, there is an open neighborhood V of x in Ui so that V is an affine scheme. Since x is arbitrary, we prove that every point has an open neighborhood such that the open neighborhood is

also an affine scheme. By definition X is a scheme. 

Let`

iXi be the union of all {Xi : i ∈ I}. On `

iXi, we define an equivalent relation ∼ as follows. Let x_i ∈ X_i and x_j ∈ X_j. We say x_i ∼ x_j if and only if x_i ∈ U_ij and x_j ∈ U_ji and ϕij(xi) = xj.

Lemma 1.1. ∼ is an equivalence relation.

Proof. Let x_i, x_j and x_k be elements of X_i, X_j and X_k respectively. It is obvious that x_i ∼ x_i since ϕ_ii is the identity map on X_i and if x_i ∼ x_j, then x_j ∼ x_i by the fact that ϕij : Uij → U_ji is an isomorphism. We only need to prove the relation is transitive.

Let x_i ∼ x_j and x_j ∼ x_k for some i, j, k. We want to show that x_i ∼ x_k. Since x_i ∼ x_j, xi ∈ U_ij and xj ∈ U_ji with ϕij(xi) = xj. Since xj ∼ x_k, xj ∈ U_jk and xk ∈ U_kj with ϕ_jk(x_j) = x_k. We find x_j ∈ U_ji∩ U_jk. Hence

x_i = ϕ⁻¹_ij (x_j) ∈ ϕ⁻¹_ij (U_ji∩U_jk) = U_ij∩U_ik and x_k = ϕ_jk(x_j) ∈ ϕ_jk(U_ji∩U_jk) = U_ki∩U_kj Hence xi ∈ U_ik and x_k∈ U_ki. Moreover,

ϕ_ik(x_i) = ϕ_jk(ϕ_ij(x_i)) = ϕ_jk(x_j) = x_k.

We conclude x_i∼ x_k. 

The quotient space `

iXi/ ∼ is denoted by X. Let ϕi be the composition of the maps X_i → `

iX_i → X, where the first map X_i → `

iX_i is the inclusion map and the second map`

iXi → X is the quotient map. Denote U_i = ϕi(Xi). Then ϕi : Xi→ U_i is a bijection by the definition of ϕ_i. We can also verify that the conditions (1) and (2) in Proposition 1.1 hold.

Lemma 1.2. Let T be the family of subsets of X consisting of U so that ϕ⁻¹_i (V ) is open in Xi for all i ∈ I. The family forms a topology on X.

1

2

Proof. Since ϕ⁻¹_i (∅) = ∅ for all i, ∅ ∈ T . Since ϕ⁻¹_i (X) = Xi is open in Xi, X ∈ T . If {Vα} is a family of elements of T , then ϕ⁻¹_i (Vα) is open in Xifor all α. ByS

αϕ⁻¹_i (Vα) = ϕ⁻¹_i (S

αVα) and the union of open sets is open, ϕ⁻¹_i (S

αVα) is open in Xi for all i. Hence S

αVα ∈ T . If V₁, V2 ∈ T , ϕ⁻¹_i (Vα) is open in Xi for α = 1, 2. Since ϕ⁻¹_i (V1 ∩ V₂) = ϕ⁻¹_i (V1) ∩ ϕ⁻¹_i (V2), and the finite intersection of open sets is open, ϕ⁻¹_i (V1∩ V₂) is open in

X_i for all i. This shows V₁∩ V₂ ∈ T . 

Since ϕ⁻¹_i (Uj) = Uij is open in Xi for all i, Uj is open in X. It also follows from the definition that ϕi : Xi → U_i is continuous for all i. We also have {Ui} forms an open covering of X.

To show ϕi is a homeomorphism, we show ϕi is an open mapping. Let W be an open subset of X_i. Then W ∩ U_ij is open in X_i. Claim ϕ_i(W ) is open in X. Since ϕ⁻¹_j (ϕ_i(W )) = ϕij(W ∩ Uij) and ϕij is a homeomorphism for all i, j, ϕ⁻¹_j (ϕi(W )) is open in Xj for all j.

By definition, ϕi(W ) is open in X and hence is open in Ui.

Now, we only need to construct a sheaf of ring O_X on X satisfying the requirement. Let O_Ui = ϕi,∗O_Xi. We obtain an isomorphism ϕi : Xi → U_i of locally ringed spaces. By the isomorphism of locally ringed spaces ϕ_ij, we have an isomorphism of sheaf of rings

O_Ui|_Ui∩U_j → O_Uj|_Ui∩U_j

satisfying a cocycle condition from {ϕ_ij}. We obtain a sheaf O_X of rings so that O_X|_Ui ∼= O_Ui for each i compatible with the gluing maps above. Then (X, OX) is a ringed space. If x ∈ X, then x ∈ Ui for some i. Then O_X,x ∼= OUi,x is a local ring.