5 Projective space
References: [ GW1 ] Section (3.5).
Jan. 13, 2026
a family \((U_i)_{i\in I}\) of schemes,
for all \(i,j\in I\), an open subscheme \(U_{ij} \subseteq U_i\),
for all \(i,j \in I\), an isomorphism \(\varphi _{ji}\colon U_{ij}\to U_{ji}\),
for all \(i\), \(U_{ii} = U_i\) (and \(\varphi _{ii} = \operatorname{id}_{U_i}\)),
for all \(i\), \(j\), \(k\),
\[ \varphi _{ij}\circ \varphi _{jk} = \varphi _{ik}\quad \text{on}\ U_{kj}\cap U_{ki} \](part of this condition is that \(\varphi _{jk}(U_{kj}\cap U_{ki}) \subseteq U_{ji}\)).
Note that the “cocycle condition” (b) implies in particular that \(\varphi _{ii} = \operatorname{id}_{U_i}\) (i.e., we could omit this condition from (a)), and that the isomorphisms \(\varphi _{ij}\) and \(\varphi _{ji}\) are inverse to each other.
Let \(((U_i)_i, (U_{ij})_{i,j}, (\varphi _{ij})_{i,j})\) be a gluing datum. There exists a scheme \(X\) together with open immersions \(\psi _i\colon U_i\to X\), \(i\in I\), such that \(X = \bigcap _i \psi (U_i)\), and for all \(i\), \(j\), the restriction \(\psi _{i|U_{ij}}\) is an isomorphism \(U_{ij}\xrightarrow {\cong } \psi _i(U_i)\cap \psi _j(U_j)\), and the composition \(\psi _{i|U_{ij}}^{-1}\circ \psi _{j|U_{ji}}\) equals the isomorphism \(\varphi _{ij}\).
The scheme \(X\) together with the \(\psi _i\) is uniquely determined up to unique isomorphism.
If all the \(U_i\), \(U_{i,j}\) are \(S\)-schemes and the \(\varphi _{ij}\) are morphisms of \(S\)-schemes for some scheme \(S\), then \(X\) carries a (unique) \(S\)-scheme structure so that all the \(\psi _{i}\) are morphisms of \(S\)-schemes.
Construct the topological space \(X\) as in Problem 28, and the structure sheaf \({\mathscr O}_X\) using Proposition 3.10, applied to the basis of the topology of \(X\) consisting of all those open subsets that are contained in (at least) one of the \(\psi _i(U_i)\). See [ GW1 ] Proposition 3.10 for a few more details.
The uniqueness and the final statement follow from gluing of morphisms (Proposition 4.12).
(Disjoint union) For any family \((U_i)_{i\in I}\) of schemes we may set \(U_{ij} = \emptyset \) for all \(i\), \(j\). This gives a gluing datum (with the uniquely determined \(\varphi _{ij}\)), and the scheme obtained by gluing is called the disjoint union of the \(U_i\), and denoted by \(\bigsqcup _{i\in I} U_i\).
(Gluing of two schemes) Let \(U_1\), \(U_2\) be schemes, and let \(U_{12} \subseteq U_1\), \(U_{21}\subseteq U_2\) be open subschemes with an isomorphism \(U_{12}\cong U_{21}\). There is a unique way to extend this to a gluing datum. The cocycle condition is automatically satisfied.
(Affine line with doubled origin) Let \(k\) be a field, \(U_1 = \mathbb {A}^1_k\), \(U_2 = \mathbb {A}^1_k\), \(U_{12} = U_1 \setminus \left\{ 0 \right\} \), \(U_{21} = U_2 \setminus \left\{ 0 \right\} \), and \(U_{12}\cong U_{21}\) the identity. This defines a gluing datum, and the resulting scheme is called the “affine line with doubled origin”. One can show that this (somewhat “pathological”, because we applied the gluing in a “stupid” way (in the above situation the isomorphism \(\varphi _{ij}\) extends to an isomorphism \(U_1\cong U_2\)); more precisely, it is a non-separated scheme, see below) scheme is not affine.
(Projective line) Let \(k\) be a field, \(U_1 = \mathbb {A}^1_k\), \(U_2 = \mathbb {A}^1_k\), \(U_{12} = U_1 \setminus \left\{ 0 \right\} \), \(U_{21} = U_2 \setminus \left\{ 0 \right\} \), and \(U_{12}\cong U_{21}\) the morphism \(x\mapsto x^{-1}\) (i.e., on affine coordinate rings, it is given by \(k[T, T^{-1}]\to k[T, T^{-1}]\), \(T\mapsto T^{-1}\)). This defines a gluing datum, and the resulting scheme is called the “projective line over \(k\)”, cf. also the next section.
References: [ GW1 ] Sections (3.6), (3.7); for a slightly different approach see [ Ha ] II.2 and/or [ GW1 ] Chapter 13.
Let \(R\) be a ring.
Jan. 14, 2026
Consider the following gluing datum which mimicks the covering of \(\mathbb {P}^n(k)\) (from the introduction) by the open subsets \(\left\{ (x_0: \cdots : x_n);\ x_i\ne 0 \right\} \), \(i=0, \dots , n\). (Note that each of these has a natural bijection with \(k^n\) by \((x_0 : \cdots : x_n)\mapsto \left( \frac{x_0}{x_i}, \dots , \widehat{\frac{x_i}{x_i}}, \dots , \frac{x_n}{x_i} \right)\).)Let
\(U_i = \operatorname{Spec}R \left[ \frac{X_0}{X_i}, \dots , \widehat{\frac{X_i}{X_i}}, \dots , \frac{X_n}{X_i} \right]\) (where we view all these rings as subrings of \(R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\)),
\(U_{ij} = D \left( \frac{X_j}{X_i} \right) \subseteq U_i\),
\(\varphi _{ij}\colon U_{ji}\to U_{ij}\) is the identity (note that the localizations \(\Gamma (U_i, {\mathscr O}_{U_i})_{\frac{X_j}{X_i}}\) and \(\Gamma (U_j, {\mathscr O}_{U_j})_{\frac{X_i}{X_j}}\) are equal as subrings of \(R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\)).
Because the maps \(\varphi _{ij}\) are defined as the identity of subrings of the ring \(R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\), it is immediate that the cocycle condition is satisfied.
Note that for every \(i\), the ring \( R \left[ \frac{X_0}{X_i}, \dots , \widehat{\frac{X_i}{X_i}}, \dots , \frac{X_n}{X_i} \right]\) is isomorphic to a polynomial ring over \(R\) in \(n\) variables, i.e., \(U_i\cong \mathbb {A}^n_R\).
The open subschemes \(U_i\cong \mathbb {A}^n_R\) are also called the standard charts of \(\mathbb {P}^n_R\).
See Problem sheet 11.
Otherwise the structure morphism \(\mathbb {P}^n_R\to \operatorname{Spec}(R)\) would be an isomorphism because of Proposition 5.5 and Theorem 4.5. But (the underlying continuous map of) this morphism clearly is not injective (e.g., because even the composition \(\mathbb {A}^n_R\to \mathbb {P}^n_R\to \operatorname{Spec}(R)\) with one of the standard charts is not injective).
We have corresponding identifications for the standard charts, and they are compatible with the gluing datum.
Recall how we defined the Zariski topology on \(\mathbb {P}^n(k)\) in the first chapter, by defining closed subsets \(V_+(I)\) for ideals \(I \subset k[X_\bullet ]\) generated by homogeneous polynomials. We can transfer this definition to the context of schemes by once more using gluing of schemes. For a homogeneous polynomial \(f(X_0, \dots , X_n)\) we write \(\Phi _i(f) = f \left( \frac{X_0}{X_i}, \dots , \frac{X_n}{X_i} \right)\in R \left[ \frac{X_0}{X_i}, \dots , \frac{X_n}{X_i} \right] \subseteq R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\) for its dehomogenization with respect to \(X_i\).
Let \(R\) be a ring and let \(I \subseteq R[X_0, \dots , X_n]\) be a homogeneous ideal, i.e., an ideal generated by homogeneous polynomials. For each \(i\), let \(V_i\) be the closed subscheme of \(U_i = \operatorname{Spec}R[ \frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}]\) defined by the polynomials \(\Phi _i(f)\), where \(f\) ranges over all homogeneous polynomials in \(I\), i.e.,
We extend this to a gluing datum by setting
One checks that the isomorphisms \(\varphi _{ij}\colon U_{ji}\to U_{ij}\) above induce isomorphisms \(\varphi _{ij}\colon V_{ji}\to V_{ij}\). In this way we obtain a gluing datum.
Topologically, \(V_+(i)\) is a closed subset of \(\mathbb {P}^n_R\) (since we can check this by intersecting with each \(U_i\)). The \(k\)-valued points of \(V_+(I)\) coincide with the set \(V_+(I)\) defined in the first chapter (as a subset of \(\mathbb {P}^n_k\), cf. Proposition 5.7).
As a special case, \(D_+(X_i) = U_i\). For a homogeneous polynomial \(f\) we have
and under the identification \(D_+(X_i)=U_i\), this set is identified with \(D(\Phi _i(f))\) (as follows from the definition and the construction of \(V_+(f)\)).
It follows that the sets \(D_+(f)\) for varying homogeneous \(f\) form a basis of the topology of \(\mathbb {P}^n_R\).
Jan. 20, 2026
Here in the description of the map on \(k\)-valued points we fix a representative \(x_\bullet = (x_0, \dots , x_n)\in k^{n+1}\). Note that while the value \(f_j(x_0, \dots , x_n)\) of an individual \(f_j\) is not well-defined, the points \((f_0(x_\bullet ): \cdots : f_m(x_\bullet ))\) is well-defined, because of our assumption that all \(f_j\) are homogeneous of the same degree.
One can show moreover that the morphism \(f\) is uniquely characterized by the behavior on \(k\)-valued points whenever \(R\) is a reduced ring (i.e., has no non-trivial nilpotent elements).
We use \(R[X_0, \dots , X_n]\) as “the polynomial ring of \(\mathbb {P}^n_R\)”, and \(R[Y_0, \dots , Y_m]\) as “the polynomial ring of \(\mathbb {P}^m_R\)”, in order to distinguish between the two sides. So, for instance, \(D_+(X_i) \subset \mathbb {P}^n_R\) and \(D_+(Y_j) \subset \mathbb {P}^m_R\) are the standard charts.
Now we first construct a morphism
which on \(k\)-valued points induces the desired morphism.
To do so,
(Using the same cover, one can show that \(D_+(f_j)\) is affine and compute its coordinate ring, and use this to specify the morphism \(f_j\) directly, but we proceed slightly differently.)
We have \(D_+(X_i f_j) = \operatorname{Spec}(R \left[ \frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}, \frac{X_i^d}{f_j} \right])\), because \(\frac{f_j}{X_i^d}\) is the dehomogenization of \(f_j\) with respect to \(X_i\) (denoted by \(\Phi _i(f_j)\) above), and \(D_+(X_i f_j)\) is the principal open inside \(D_+(X_i)\cong \mathbb {A}^n_R\) defined by this element. We then have a morphism \(D_+(X_i f_j) \to D_+(Y_j)\) attached to the ring homomorphism
Note that
is really an element of the left hand side.
It is then easy to check that these morphisms by gluing of morphisms give rise to a morphism \(D_+(f_j) \to D_+(Y_j)\) with the desired behavior on \(k\)-valued points.
Finally, one uses gluing of morphisms again, now applied to the compositions \(D_+(f_j) \to D_+(Y_j) \subset \mathbb {P}^m_R\) in order to obtain a morphism