4 Schemes

(4.1) Ringed spaces, locally ringed spaces

Recall that we have attached to every ring $R$ the topological space $X= \operatorname{Spec}(R)$ and the structure sheaf ${\mathscr O}_X$. From these data we can recover the ring $R$ as $R={\mathscr O}_X(X)$, so in some sense we have reached our goal to attach to $R$ a “geometric object” without losing any information. This leads to the following definition.

Definition 4.1

A ringed space is a pair $(X, {\mathscr O}_X)$ where $X$ is a topological space and ${\mathscr O}_X$ is a sheaf of rings on $X$.

Often one writes $X$ instead of $(X, {\mathscr O}_X)$. The sheaf ${\mathscr O}_X$ is usually called the structure sheaf of $X$. As we will see soon, however, this is not yet the “good” notion of geometric object that we should use in order to talk about spectra of rings; the problem lies in the notion of morphism. In order to understand this, we first define the natural notion of morphisms of ringed spaces. To define a morphism $(X, {\mathscr O}_X)\to (Y, {\mathscr O}_Y)$ of ringed spaces, it is natural to start with a continuous map $f\colon X\to Y$. But in addition, the structure sheaves should also be related. For this, recall that we think of the ring ${\mathscr O}_X(U)$ as the ring of functions defined on $U$. Then it is natural to expect that for every function defined on $V \subseteq Y$, by “composition with $f$” we obtain a function on $f^{-1}(V)$. We need to put this in quotes because there is no reason why the elements of the ring ${\mathscr O}_Y(V)$ should really be function on $V$, so “composition” does not really make sense. However, we can still ask that the morphism $(X, {\mathscr O}_X)\to (Y,{\mathscr O}_Y)$ comes with maps ${\mathscr O}_Y(V)\to {\mathscr O}_X(f^{-1}(V))$ for every open $V \subseteq Y$. Clearly, these should be compatible with the restriction maps of these sheaves. Putting everything together, we arrive at the following definition.

Definition 4.2

A morphism $f\colon (X, {\mathscr O}_X)\to (Y, {\mathscr O}_Y)$ of ringed spaces is a pair $(f, f^\flat )$ where $f\colon X\to Y$ is a continuous map and $f^\flat \colon {\mathscr O}_Y\to f_*{\mathscr O}_X$ is a morphism of sheaves of rings.

There are obvious notions of identity morphisms and composition of morphisms, and we obtain the category ${\rm (Ringed Spaces)}$ of ringed spaces.

Dec. 10, 2025

By adjunction, the morphism $f^\flat $ corresponds to a morphism $f^\sharp \colon f^{-1}{\mathscr O}_Y\to {\mathscr O}_X$ of sheaves of rings on $X$. In particular, for every $x\in X$, we obtain a ring homomorphism $f^\sharp _x\colon {\mathscr O}_{Y, f(x)} = (f^{-1}{\mathscr O}_Y)_x\to {\mathscr O}_{X, x}$ between the stalks. One checks that this ring homomorphism may also be constructed explicitly by using the universal property of the colimit ${\mathscr O}_{Y, f(x)}$ for the maps ${\mathscr O}_Y(V)\to {\mathscr O}_{X, x}$ (for $V \subseteq Y$ open, $f(x)\in V$) obtained as the composition

\[ {\mathscr O}_Y(V)\to (f_*{\mathscr O}_X)(V) = {\mathscr O}_X(f^{-1}(V))\to {\mathscr O}_{X, x} \]

where the first map is obtained from $f^\flat $, and the final map is the natural map into the stalk, using that $f(x)\in V$ implies $x\in f^{-1}(V)$.

With this definition of the category of ringed spaces, we obtain a functor

\[ {\rm (Rings)}^{\rm opp}\longrightarrow {\rm (Ringed Spaces)}. \]

In fact, to every ring $R$ we may attach its spectrum $(\operatorname{Spec}(R), {\mathscr O}_{\operatorname{Spec}(R)})$ with the structure sheaf. If $\varphi \colon R\to S$ is a ring homomorphism, we have the continuous map ${}^a\varphi \colon \operatorname{Spec}(S)\to \operatorname{Spec}(R)$ between the spectra. Furthermore, for every principal open $D(s) \subseteq \operatorname{Spec}(R)$, $s\in R$, we have $({}^a\varphi )^{-1}(D(s)) = D(\varphi (s))$, and thus the natural homomorphism $R_s\to S_{\varphi (s)}$ is a ring homomorphism ${\mathscr O}_{\operatorname{Spec}(R)}(D(s))\to {\mathscr O}_{\operatorname{Spec}(S)}(({}^a\varphi )^{-1}(D(s)))$. Since the $D(s)$ form a basis of the topology, these define the desired sheaf homomorphism ${\mathscr O}_{\operatorname{Spec}(R)}\to {}^a \varphi _*{\mathscr O}_{\operatorname{Spec}(S)}$. (Cf. Proposition 3.10.)

From now on we usually write $\operatorname{Spec}(R)$ for the ringed space $(\operatorname{Spec}(R), {\mathscr O}_{\operatorname{Spec}(R)})$.

However, it turns out that this functor is not fully faithful. Therefore, passing from rings to ringed spaces remains problematic, because there are morphisms between spectra as ringed spaces that we “do not want to allow”.

In fact, morphisms coming from ring homomorphisms have a special property that we can see by inspecting the stalks.

Proposition 4.3

Let $R$ be a ring, $X=\operatorname{Spec}(R)$ (as a ringed space). For every $x\in X$, the stalk ${\mathscr O}_{X, x}$ is isomorphic to the localization $R_{x}$ of $R$ with respect to the prime ideal $x$. In particular, the stalk is a local ring (i.e., it has a unique maximal ideal).
Let $\varphi \colon R \to S$ be a ring homomorphism, let $X=\operatorname{Spec}(S)$, $Y=\operatorname{Spec}(R)$ and let $f\colon X\to Y$ be the morphism of ringed spaces attached to $\varphi $ as above. Then for every $x\in X$ the ring homomorphism ${\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X, x}$ is a local homomorphism of local rings, i.e., it maps the maximal ideal of ${\mathscr O}_{Y, f(x)}$ into the maximal ideal of ${\mathscr O}_{X, x}$.

Proof

We have already seen the first point. For the second one, note that $\varphi $ induces a commutative diagram

$\begin{tikzcd} \Oscr_Y(Y)\ar[r]\ar[d] & \Oscr_X(X)\ar[d]\\ \Oscr_{Y, f(x)}\ar[r] & \Oscr_{X, x} \end{tikzcd}$

which we may rewrite as

$\begin{tikzcd} R\ar[r, "\varphi"]\ar[d] & S\ar[d]\\ R_{f(x)}\ar[r] & S_x. \end{tikzcd}$

This means that the homomorphism $R_{f(x)}\to S_x$ is simply the natural homomorphism between localizations induced by $\varphi $, i.e., $\frac{a}{s}\mapsto \frac{\varphi (a)}{\varphi (s)}$. In particular, it maps the maximal ideal $f(x) R_{f(x)} = \varphi ^{-1}(x) R_{\varphi ^{-1}(x)}$ into the maximal ideal $x S_x$.

Definition 4.4

A locally ringed space is a ringed space $(X, {\mathscr O}_X)$ such that for every $x\in X$ the stalk ${\mathscr O}_{X,x}$ is a local ring (also called the local ring of $X$).
Let $X$, $Y$ be locally ringed spaces. A morphism $X\to Y$ of locally ringed spaces is a morphism $X\to Y$ of ringed spaces such that for every $x\in X$ the induced ring homomorphism $f^\sharp _x\colon {\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X,x}$ is a local homomorphism.

The above discussion shows that $\operatorname{Spec}$ actually is a contravariant functor from the category of rings to the category of locally ringed spaces.

Theorem 4.5

The contravariant functor $R\mapsto (\operatorname{Spec}(R), {\mathscr O}_{\operatorname{Spec}(R)})$ from the category of rings to the category of locally ringed spaces is fully faithful.

Proof

Let $R$, $S$ be rings and let $X=\operatorname{Spec}(S)$, $Y=\operatorname{Spec}(R)$ (considered as locally ringed spaces). We have natural maps

\[ \operatorname{Spec}\colon \operatorname{Hom}(R, S)\to \operatorname{Hom}(X, Y),\quad \Gamma \colon \operatorname{Hom}(X, Y)\to \operatorname{Hom}(R, S), \]

and we want to show that they are inverse to each other. It follows directly from the construction of $\operatorname{Spec}(\varphi )$ that $\varphi \mapsto \operatorname{Spec}(\varphi ) \mapsto \Gamma (\operatorname{Spec}(\varphi ))$ is the identity morphism.

Now consider the composition $f\mapsto \Gamma (f)\mapsto \operatorname{Spec}(\Gamma (f))$. We write $f = (f, f^\flat )$ and $\operatorname{Spec}(\Gamma (f)) = (g, g^\flat )$. For every $x\in X$, we have the commutative diagram

$\begin{tikzcd} \Oscr_Y(Y) \ar[r]\ar[d] & (f_*\Oscr_X)(Y)\ar[d] & =\Oscr_X(X)\\ \Oscr_{Y, f(x)}\ar[r] & \Oscr_{X,x}. \end{tikzcd}$

In terms of the rings $R$, $S$ this diagram may be written as

$\begin{tikzcd} R \ar[r, "\Gamma(f)"]\ar[d] & S\ar[d]\\ R_{f(x)}\ar[r] & S_x. \end{tikzcd}$

Now $f$ is a morphism of locally ringed spaces, so the ring homomorphism in the lower row of this diagram is local. This implies that the preimage of the maximal ideal of $S_x$ is the maximal ideal of $R_{f(x)}$. It follows that $f(x) = \Gamma (f)^{-1}(x)$. In other words, as continuous maps we have $f = g$. It then follows from the above diagram that the sheaf morphisms $f^\flat $ and $g^\flat $ induce the same maps between the stalks $R_{f(x)}$ and $S_x$ for all $x$. Therefore they coincide by Proposition 3.28.

(4.2) Schemes

References: [ GW1 ] Sections (2.9) – (2.12), Chapter 3; [ Ha ] Chapter II.2.

Dec. 16, 2025

In view of the previous discussion, we arrive at the following definition.

Definition 4.6

An affine scheme is a locally ringed space that is isomorphic to a locally ringed space of the form $(\operatorname{Spec}(R), {\mathscr O}_{\operatorname{Spec}(R)})$ for a ring $R$.

Rephrasing Theorem 4.5, we can say that the functor $\operatorname{Spec}$ is a contravariant equivalence between the category of rings and the category of affine schemes. Therefore, we have succeeded in attaching to each ring a “geometric object” (namely a locally ringed space), and this construction preserves the information of the ring, and also of ring homomorphisms.

From this point, it is easy to extend the definition to include many more geometric objects that are interesting from the point of algebraic geometry, and accessible to the methods of commutative algebra. We start with the following simple remark.

Remark 4.7

Let $X=(X, {\mathscr O}_X)$ be a locally ringed space, and let $U \subseteq X$ be an open subset. Then $(U, {\mathscr O}_{X|U})$ is a locally ringed space (which we often just denote by $U$). The natural morphism $U\to X$ of locally ringed spaces is called an open immersion.

As an example, let $X=\operatorname{Spec}(R)$ be an affine scheme, and let $s\in R$. Then $D(s)$ (in the sense of the above construction) is a locally ringed space, and as a locally ringed space is isomorphic to $\operatorname{Spec}(R_s)$, i.e., it is again an affine scheme.

We can now give Grothendieck’s definition of a scheme. (In the beginning, what we call a scheme was called a prescheme (e.g. in the first edition of [ Mu ] and in [Diedonné, Grothendieck: Éléments de Géométrie Algébrique]), but the terminology has changed later.)

Definition 4.8

A scheme is a locally ringed space $X$ such that there exists an open cover $X = \bigcup _i U_i$, such that for every $i$, $(U_i, {\mathscr O}_{X|U_i})$ is an affine scheme. A morphism of schemes is a morphism of the underlying locally ringed spaces.

Proposition/Definition 4.9

Let $X$ be a scheme, and let $U \subseteq X$ be an open subset, seen as a locally ringed space as above. Then $U$ is a scheme. We call $U$ an open subscheme of $X$.

Proof

We need to show that we can cover $U$ by affine schemes. To do so, cover $X = \bigcup _i U_i$ by affine schemes. Then $U\cap U_i$ is open in $U_i$, and hence can be covered by principal open subsets of $U_i$. But every principal open of an affine scheme is itself an affine scheme, as remarked above.

Example 4.10

(affine space) Let $R$ be a ring. Then we call $\mathbb {A}^n_R := \operatorname{Spec}(R[T_1, \dots , T_n])$ affine space of relative dimension $n$ over $R$. The inclusion $R\to R[T_{\bullet }]$ gives us a morphism $\mathbb {A}^n_R\to \operatorname{Spec}(R)$ of affine schemes.
Let $k$ be a field. By $0$ we denote the “origin” in $\mathbb {A}^n_k$, i.e., the closed point corresponding to the maximal ideal $(T_1, \dots , T_n)$.

For $n \ge 2$, the open subscheme $U = \mathbb {A}^n_k \setminus \{ 0\} $ is not affine (cf. Problem Sheet 9). In particular, open subschemes of affine schemes need not be affine.
(spectrum of a domain) Let $R$ be a domain. Then $\eta = (0)$ is the unique minimal prime ideal, and hence the generic point of $\operatorname{Spec}(R)$. The stalk ${\mathscr O}_{\operatorname{Spec}(R), \eta }$ is the field of fractions $K:=\operatorname{Frac}(R)$. For every non-empty open $U \subseteq X :=\operatorname{Spec}(R)$, the natural map ${\mathscr O}_X(U)\to {\mathscr O}_{\operatorname{Spec}(R), \eta } = K$ is injective, thus all the rings ${\mathscr O}_X(U)$ ($U\ne \emptyset $) and hence also all the stalks ${\mathscr O}_{X, x}$ may be considered as subrings of $K$ in a natural way. Furthermore, for every non-empty open $U$ we have

\[ {\mathscr O}_X(U) = \bigcap _{x\in U} {\mathscr O}_{X,x} \]

as subrings of $K$.
Dec. 17, 2025
(closed subschemes of affine schemes) Let $R$ be a ring and ${\mathfrak a}\subseteq R$ an ideal. We have seen that the continuous map $i\colon \operatorname{Spec}(R/{\mathfrak a})\to \operatorname{Spec}(R)$ induced by the canonical projection $R\to R/{\mathfrak a}$ is a homeomorphism onto its image $V({\mathfrak a})$. From now on we write $V({\mathfrak a})$ for the affine scheme obtained in this way, i.e., the topological space is $V({\mathfrak a})$, and the structure sheaf is $i_*{\mathscr O}_{\operatorname{Spec}(R/{\mathfrak a})}$. Then $V({\mathfrak a})\cong \operatorname{Spec}(R/{\mathfrak a})$ as schemes.
Let $k$ be a field. For every $n\ge 1$, the scheme $\operatorname{Spec}(k[T]/(T^n))$ has the form of the previous example. Each of these schemes topologically is just one point, however, they are pairwise non-isomorphic. We think of the scheme structure of $\operatorname{Spec}(k[T]/(T^n))$ for $n {\gt} 1$ as giving us an “infinitesimal neighborhood” of this point, which is larger when $n$ is large, because the image of a polynomial in the ring $k[T]/(T^n)$ gives us not only the information about the value of the polynomial at $T=0$, but also the first, …, $(n-1)$-th derivative of the polynomial at $0$.
(schematic intersection of closed subscheme of affine scheme) Let $R$ be a ring, ${\mathfrak a}, {\mathfrak b}\subseteq R$ ideals. We define the schematic intersection

\[ V({\mathfrak a})\cap V({\mathfrak b}) = V({\mathfrak a}+{\mathfrak b}) \]

as a “closed subscheme” of $\operatorname{Spec}(R)$ in the sense of Part (5). The scheme structure allows us to see, in addition to the set (and topological space) $V({\mathfrak a})\cap V({\mathfrak b})$, also the “type of intersection”. For instance, if $k$ is an algebraically closed field and $R =k[X, Y]$ the polynomial ring in two variables, then for ${\mathfrak a}= (Y)$ and ${\mathfrak b}= (Y-f(X))$, $f\in k[X]$, the scheme $V({\mathfrak a})\cap V({\mathfrak b})$ sees the zeros of $f$ with their multiplicities.

However, there are also schemes (e.g., “projective space” as constructed below) that are not isomorphic to an open subscheme of an affine scheme. Understanding how to construct these is one of the main next steps.

(4.3) Morphisms into affine schemes

Jan. 6, 2026

References: [ GW1 ] Section (3.3).

We have shown that morphisms between affine schemes correspond to ring homomorphisms between the rings of global sections of their structure sheaves. This result generalizes as follows. Recall the notation $\Gamma (U, {\mathscr F}) = {\mathscr F}(U)$ (for a presheaf ${\mathscr F}$).

Theorem 4.11

Let $X$ be a locally ringed space, $R$ a ring and $Y = \operatorname{Spec}(R)$. The map

\begin{align*} \operatorname{Hom}_{\rm locRS}(X, Y) & \longrightarrow \operatorname{Hom}_{\rm Ring}(R, \Gamma (X, {\mathscr O}_X)), \\ f & \mapsto (R = \Gamma (Y, {\mathscr O}_Y)\xrightarrow {f^\flat (Y)}\Gamma (f^{-1}(Y), f_*{\mathscr O}_X) = \Gamma (X, {\mathscr O}_X)), \end{align*}

is a bijection.

Proof

It is not too hard to prove the theorem along the same lines as Theorem 4.5 (see [ GW1 ] Proposition 3.4).

We give a simpler proof in the case that $X$ is a scheme in order to illustrate the principle of gluing of morphisms.

Proposition 4.12

(Gluing of morphisms) Let $X$, $Y$ be schemes (or: topological spaces, (locally) ringed spaces), and let $X = \bigcup _i U_i$ be a cover by open subsets. Let $f_{i}\colon U_i\to Y$ be morphisms such that $f_{i|U_i\cap U_j} = f_{j|U_i\cap U_j}$ for all $i$, $j$. Then there exists a unique morphism $f\colon X\to Y$ such that $f_{|U_i} = f_{i}$ for all $i$.

Proof

On sets, define $f(x) = f_i(x)$ where $i$ is chosen such that $x\in U_i$. This is independent of $i$ and thus defines a map $f\colon X\to Y$ of sets such that $f_{|U_i}=f_i$ for all $i$. Continuity can be checked locally on $X$, hence $f$ is continuous, because all the $f_i$ are.

It remains to define a sheaf homomorphism ${\mathscr O}_Y\to f_*{\mathscr O}_X$. For $V \subseteq Y$ open, combining the data of the $f_i$ gives us a homomorphism

\[ \Gamma (V, {\mathscr O}_Y) \longrightarrow \prod _i \Gamma (f^{-1}(V)\cap U_i, {\mathscr O}_X). \]

Applying the sheaf axioms for the sheaf ${\mathscr O}_X$ and the cover $f^{-1}(V) = \bigcup _i (f^{-1}(V)\cap U_i)$ one checks that this homomorphism factors through $\Gamma (f^{-1}(V), {\mathscr O}_X)$. This defines the desired homomorphism ${\mathscr O}_Y\to f_*{\mathscr O}_X$.

Proof of Theorem 4.11

Let $X=\bigcup _i U_i$ be a cover by affine open subschemes. We also chose covers $U_i\cap U_j = \bigcup U_{ijk}$ by affine open subschemes. Consider the diagram

$\begin{tikzcd} \Hom(X, Y) \ar[r]\ar[d] & \prod\limits_i \Hom(U_i, Y)\ar[r, yshift=-1mm]\ar[r, yshift=1mm]\ar[d] & \prod\limits_{i,j,k} \Hom(U_{ijk}, Y)\ar[d]\\ \Hom(R, \Gamma(X, \Oscr_X) \ar[r] & \prod\limits_i \Hom(R, \Gamma(U_i, \Oscr_X)\ar[r, yshift=-1mm]\ar[r, yshift=1mm] & \prod\limits_{i,j,k} \Hom(R, \Gamma(U_{ijk}, \Oscr_X)) \end{tikzcd}$

where the vertical maps are as in the statement of the theorem, the top horizontal morphisms are given by restriction of morphisms to open subspaces, and the bottom horizontal row is given by restriction of sections (this is basically $\operatorname{Hom}(R, -)$ of the “sheaf sequence” for ${\mathscr O}_X$).

The top row is exact by “gluing of morphisms”, the bottom row is exact because ${\mathscr O}_X$ is a sheaf and the $\operatorname{Hom}$ functor is left exact. By Theorem 4.5, $\operatorname{Hom}(U_i, Y) = \operatorname{Hom}(R, \Gamma (U_i, {\mathscr O}_X))$, the vertical morphisms in the middle and on the right hand side are isomorphisms. It follows that the vertical morphism on the left is an isomorphism as well. This is what we had to show.

Example 4.13

Let $X$ be a scheme. There is a unique morphism $X\to \operatorname{Spec}(\mathbb {Z})$ of schemes.
Let $k$ be a field (or any ring), and let $X$ be a scheme. A morphism $X\to \operatorname{Spec}(k)$ is the same as a $k$-algebra structure on $\Gamma (X, {\mathscr O}_X)$ (which is by definition a ring homomorphism $k\to \Gamma (X, {\mathscr O}_X)$).

Often, a $k$-algebra structure (for some field or ring $k$) on a scheme $X$ is given (e.g., on $\operatorname{Spec}(k[T_\bullet ]/{\mathfrak a})$, one of our examples of interesting schemes), and in this case, similarly as for $k$-algebras, it is usually useful to consider only morphisms which are compatible with this structure. This leads to the following notion of “relative schemes”, or “schemes over a fixed base scheme $S$”.

Definition 4.14

Let $S$ be a scheme. The

category (Sch/S$)$ of $S$-schemes is defined as follows. The objects are morphisms $X\to S$ of schemes. The morphisms between $S$-schemes $X\to S$ and $Y\to S$ are scheme morphisms $X\to Y$ such that the triangle

$\begin{tikzcd} X\ar[rr]\ar[dr] & & Y\ar[dl]\\ & S \end{tikzcd}$

is commutative. Composition of morphisms is the usual composition of morphisms of schemes.

None

If $f\colon X\to S$ is an $S$-scheme, we call $f$ its structure morphism. Usually we omit it from the notation and simply speak of $X$ as an $S$-scheme. The $\operatorname{Hom}$-sets in the category of $S$-schemes are usually denoted as $\operatorname{Hom}_S(X, Y)$. If $S=\operatorname{Spec}(R)$ is affine, we also speak of $R$-schemes instead.

The scheme $S$ is also called the base scheme.

By 4.13 (1), we have $\text{(Sch)}=$(Sch/S$)$.

Remark 4.15

Let $R$ be a ring. If $X$ is an $R$-scheme, then the rings $\Gamma (U, {\mathscr O}_X)$ fur $U \subseteq X$ open, and the local rings ${\mathscr O}_{X, x}$ for $x\in X$ as well as their residue class fields $\kappa (x) = {\mathscr O}_{X, x}/{\mathfrak m}_x$ all are equipped with the structure of an $R$-algebra, since the $R$-algebra $\Gamma (X, {\mathscr O}_X)$ naturally maps to all of them, and all the natural homomorphisms between these rings are homomorphisms of $R$-algebras.

Jan. 7, 2026

Remark 4.16

Theorem 4.11 has the following variant for $S$-schemes. If $S =\operatorname{Spec}(R)$ and $X = \operatorname{Spec}(A)$ are affine schemes, where $A$ is an $R$-algebra, so that $X$ is an $S$-scheme, then for every $S$-scheme $T$ the natural map

\[ \operatorname{Hom}_S(T, X) \longrightarrow \operatorname{Hom}_R(A, \Gamma (T, {\mathscr O}_T)) \]

is bijective. Here on the left we have morphisms of $S$-schemes and on the right we have homomorphisms of $R$-algebras.

(None)

(4.4) Morphisms from spectrum of a field into scheme $X$

References: [ GW1 ] Section (3.4).

Let $X$ be a scheme. To study $X$, it is often essential to understand the sets of morphisms $\operatorname{Hom}(T, X)$ for varying schemes $T$. Even though these sets are often hard to understand, their importance justifies introducing a shorter symbol for them.

Definition 4.17

Let $X$ be a scheme. For every scheme $T$ we write

\[ X(T) := \operatorname{Hom}(T, X) \]

and call this the set of $T$-valued points of $X$.
If $X$, $T$ are $S$-schemes for some scheme $S$, then we usually (by abuse of notation) write

\[ X(T) = \operatorname{Hom}_S(T, X) \]

for the set of morphisms $T\to X$ of $S$-schemes and call this the set of $T$-valued points of the $S$-scheme $X$. (If it is necessary to distinguish between the two sets, we could also write $X_S(T)$.)
If $T=\operatorname{Spec}(R)$ is affine, we also write $X(R) = X(\operatorname{Spec}(R))$ and speak of $R$-valued points.

Example 4.18

Let $R$ be a ring.

For the affine space $\mathbb {A}^n_R$ over $R$ we have

\[ \mathbb {A}^n_R(T) = \operatorname{Hom}_{\operatorname{Spec}(R)}(T, \mathbb {A}^n_R) = \operatorname{Hom}_R(R[T_1, \dots , T_n], \Gamma (T, {\mathscr O}_T)) = \Gamma (T, {\mathscr O}_T)^n. \]

Note that here it is important that we take only morphisms of $R$-schemes on the right (and correspondingly, of $R$-algebras on the left).
More generally, if $X = V({\mathfrak a})$ for an ideal ${\mathfrak a}\subseteq R[T_1, \dots , T_n]$, then for every $R$-algebra $A$ we can identify

\[ X(A) = \left\{ (x_i)_i\in A^n;\ \forall f\in {\mathfrak a}: f(x_1, \dots , x_n)=0 \right\} , \]

where we again take morphisms of $R$-schemes on the left hand side. (And similarly for $X(T)$ where $T$ is any $R$-scheme, not necessarily affine.)

Remark 4.19

There is another, more formal but also very useful justification why the $T$-valued points of a scheme are important. We will take this up again later in more detail, and hence at this point give only a brief sketch.

The argument applies to any (locally small) category ${\mathcal C}$ (for us, it would be the category of schemes, or the category of $S$-schemes for some scheme $S$). To each object $X$ of ${\mathcal C}$ we may attach the functor

\[ h_X\colon {\mathcal C}^{\rm opp}\to {\rm (Sets)},\quad X\mapsto \operatorname{Hom}_{{\mathcal C}}(T, X), \]

(on morphisms, $h_X$ is given by composition).

In this way, we obtain a functor ${\mathcal C}\to \text{Fun}({\mathcal C}^{\rm opp}, {\rm (Sets)})$ from ${\mathcal C}$ to the category of (contravariant) functors from ${\mathcal C}$ to the category of sets (with morphisms of functors as morphisms). The Yoneda lemma states that this functor is fully faithful. In particular, given objects $X,Y$ of ${\mathcal C}$, we have $X\cong Y$, if and only if the functors $h_X$ and $h_Y$ are isomorphic. (Proving this result is not difficult, you should try it! Or see, e.g., [ GW1 ] Section (4.2).)

Note however that it is (almost always) a very special property of a functor ${\mathcal C}^{\rm opp}\to {\rm (Sets)}$ to be of the form $h_X$ for some $X$; for most functors this is not true.

For schemes, with our terminology of $T$-valued points, this means that a scheme $X$ is determined by the collection of all of its $T$-valued points together with the maps $X(T)\to X(T')$ induced by scheme morphisms $T\to T'$.

While in general it is difficult to make the set $X(T)$ explicit, we can give a useful description at least in the case where $T$ is the spectrum of a field, see Proposition 4.21 below. To prepare for this, we start with the following remarks.

Let $X$ be a scheme and let $x\in X$.

Definition 4.20

Let $X$ be a scheme and let $x\in X$. We call the stalk ${\mathscr O}_{X, x}$ the local ring of $X$ (or ${\mathscr O}_X$) at $x$ and usually denote its (unique) maximal ideal by ${\mathfrak m}_x$. The quotient $\kappa (x) := {\mathscr O}_{X, x}/{\mathfrak m}_x$ is called the residue class field of $X$ at $x$.

By definition of the notion of morphism of scheme (i.e., of morphism of locally ringed space), the residue class field behaves functorially in the following sense. For a morphism $f\colon X\to Y$ of schemes and a point $x\in X$, we have the ring homomorphism $f^\sharp _x\colon {\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X,x}$, a local homomorphism between local rings. This induces a ring homomorphism $\kappa (f(x))\to \kappa (x)$ between the residue class fields, which in this section we will denote by $\overline{f^\sharp _x}$.

If $U \subseteq X$ is an affine open neighborhood of $x$ (i.e., $U \subseteq X$ is open and the scheme $U$ is affine), say $U=\operatorname{Spec}(A)$, then $x$ corresponds to a prime ideal ${\mathfrak p}_x \subset A$ and we obtain a ring homomorphism

\[ {\mathscr O}_{X, x}\cong {\mathscr O}_{U,x} = A_{{\mathfrak p}_x} \longleftarrow A. \]

Passing to the spectra, we obtain a morphism

\[ j_x\colon \operatorname{Spec}({\mathscr O}_{X,x})\to U \to X, \]

where the morphism $U\to X$ is the inclusion of the open subscheme $U$. This morphism is independent of the choice of $U$. (It is easy to see that the morphism is not changed if $U$ is replaced by a principal open subscheme of $U$. From this, one may deduce the independence of $U$.) It maps the closed point of $\operatorname{Spec}({\mathscr O}_{X,y})$ to $x$ (because the inverse image of the maximal ideal of $A_{{\mathfrak p}_x}$ in $A$ is ${\mathfrak p}_x$).

Furthermore, the projection ${\mathscr O}_{X, x}\to \kappa (x)$ induces a morphism $\operatorname{Spec}(\kappa (x))\to \operatorname{Spec}({\mathscr O}_{X,x})$ and composing this with $j_x$ we obtain a morphism

\[ i_{x}\colon \operatorname{Spec}(\kappa (x))\to X \]

which maps the unique point of $\operatorname{Spec}(\kappa (x))$ to $x$.

Proposition 4.21

Let $X$ be a scheme, and let $K$ be a field. The maps

$\begin{tikzcd} X(K)\ar[r, yshift=1.2mm] & \left\{ (x, \alpha);\ x\in X,\ \alpha\colon \kappa(x)\to K\right\} \ar[l, yshift=-1.2mm] \\[-.7cm] f\ar[r, mapsto] & ({\rm Im}(f), \overline{f^\sharp})\\[-.7cm] i_x\circ \Spec(\alpha)& \ar[l, mapsto] (x,\alpha) \end{tikzcd}$

are inverse to each other and in particular are bijective.

Here by abuse of notation we denote by ${\rm Im}(f)$ the unique point in the image of $f$.

Proof

It follows from the above discussion that the given maps are inverse to each other.

Similarly, if $k$ is a field, we have the following version for $k$-schemes.

Proposition 4.22

Let $k$ be a field, $X$ a $k$-scheme, and let $K$ be an extension field of $k$ (so that $\operatorname{Spec}(K)$ also is a $k$-scheme). The maps of Proposition 4.21 restrict to bijections

$\begin{tikzcd} X(K)\ar[r, yshift=1.2mm] & \left\{ (x, \alpha);\ x\in X,\ \alpha\colon \kappa(x)\to K\ \text{$k$-homom.}\right\}. \ar[l, yshift=-1.2mm] \end{tikzcd}$

Here $X(K)$ denotes the set $\operatorname{Hom}_k(\operatorname{Spec}(K), X)$ of $K$-valued points of $X$ as a $k$-scheme.

Algebraic Geometry 1, Winter term 2025/26.
Lecture notes.

Content

4 Schemes