Chapter I — The Language of Schemes

§1. Affine Schemes

1.1. The prime spectrum of a ring

(1.1.1) Notation. Let $A$ be a (commutative) ring and $M$ an $A$-module. In this chapter and the following ones we shall constantly use the notation:

• $\mathrm{Spec}(A)$ = the set of prime ideals of $A$, also called the prime spectrum of $A$; for $x\in X=\mathrm{Spec}(A)$, it is often convenient to write ${\mathfrak{j}}_x$ in place of $x$. For $\mathrm{Spec}(A)$ to be empty, it is necessary and sufficient that $A=0$.

• $A_x=A_{{\mathfrak{j}}_x}=S^{-1}A$, where $S=A\setminus {\mathfrak{j}}_x$ — the (local) ring of fractions.

• ${\mathfrak{m}}_x={\mathfrak{j}}_x{A}_{{\mathfrak{j}}_x}$ = maximal ideal of $A_x$.

• $\kappa (x)=A_x/{\mathfrak{m}}_x$ = residue field of $A_x$, canonically isomorphic to the field of fractions of $A/{\mathfrak{j}}_x$, with which we identify it.

• $f(x)$ = class of $f$ mod ${\mathfrak{j}}_x$ in $A/{\mathfrak{j}}_x\subset \kappa (x)$, for $f\in A$ and $x\in X$; we also call $f(x)$ the value of $f$ at the point $x\in \mathrm{Spec}(A)$. The relations $f(x)=0$ and $f\in {\mathfrak{j}}_x$ are equivalent.

• $M_x=M{\otimes }_A{A}_x$ = module of fractions with denominators in $A\setminus {\mathfrak{j}}_x$.

• $r(E)$ = radical of the ideal of $A$ generated by $E$, for $E\subset A$.

• $V(E)=\{x\in X:E\subset {\mathfrak{j}}_x\}$ (equivalently, $\{x\in X:f(x)=0\text{ for all }f\in E\}$), for $E\subset A$. Thus

  $$r(E)=\bigcap _{x\in V(E)}{\mathfrak{j}}_x.\text{\hspace{1em}(1.1.1.1)}$$

• $V(f)=V(\{f\})$ for $f\in A$.

• $D(f)=X\setminus V(f)=\{x\in X:f(x)\ne 0\}$.

Proposition (1.1.2).

(i) $V(0)=X$, $V(1)=\varnothing$. (ii) $E\subset E^{\prime }$ implies $V(E)\supset V(E^{\prime })$. (iii) For every family $(E_{\lambda })$ of subsets, $V({\bigcup }_{\lambda }{E}_{\lambda })=V({\sum }_{\lambda }{E}_{\lambda })={\bigcap }_{\lambda }>V(E_{\lambda })$. (iv) $V(E{E}^{\prime })=V(E)\cup V(E^{\prime })$. (v) $V(E)=V(r(E))$.

Proof. (i), (ii), (iii) are trivial; (v) follows from (ii) and (1.1.1.1). For (iv), $V(E{E}^{\prime })\supset V(E)\cup V(E^{\prime })$ is clear; conversely, if $x\notin V(E)$ and $x\notin V(E^{\prime })$, pick $f\in E$, $f^{\prime }\in E^{\prime }$ with $f(x),f^{\prime }(x)\ne 0$ in $\kappa (x)$; then $f(x)f^{\prime }(x)\ne 0$, so $x\notin V(E{E}^{\prime })$.

Proposition (1.1.2) shows the sets $V(E)$ are the closed sets of a topology on $X$, called the spectral topology¹; unless otherwise stated, $X=\mathrm{Spec}(A)$ is always equipped with this topology.

(1.1.3) For a subset $Y\subset X$, write $\mathfrak{j}(Y)=\{f\in A:f(y)=0\text{ for all }y\in Y\}={\bigcap }_{y\in Y}{\mathfrak{j}}_y$. Clearly $Y\subset Y^{\prime }⟹\mathfrak{j}(Y)\supset \mathfrak{j}(Y^{\prime })$, and

$$\mathfrak{j}\left(\bigcup _{\lambda }{Y}_{\lambda }\right)=\bigcap _{\lambda }\mathfrak{j}(Y_{\lambda })\text{\hspace{1em}(1.1.3.1)}$$

$$\mathfrak{j}(\{x\})={\mathfrak{j}}_x.\text{\hspace{1em}(1.1.3.2)}$$

Proposition (1.1.4).

(i) For every $E\subset A$, $\mathfrak{j}(V(E))=r(E)$. (ii) For every $Y\subset X$, $V(\mathfrak{j}(Y))=>\bar{Y}$, the closure of $Y$.

Corollary (1.1.5). The closed subsets of $X=\mathrm{Spec}(A)$ and the ideals of $A$ equal to their radicals correspond bijectively under the inclusion-reversing maps $Y\mapsto \mathfrak{j}(Y)$ and $\mathfrak{a}\mapsto V(\mathfrak{a})$. To $Y_1\cup Y_2$ corresponds $\mathfrak{j}(Y_1)\cap \mathfrak{j}(Y_2)$; to ${\bigcap }_{\lambda }{Y}_{\lambda }$ corresponds the radical of ${\sum }_{\lambda }\mathfrak{j}(Y_{\lambda })$.

Corollary (1.1.6). If $A$ is Noetherian, $X=\mathrm{Spec}(A)$ is a Noetherian space.

The converse fails: any non-Noetherian integral ring with a unique nonzero prime ideal (e.g. a non-discrete rank-1 valuation ring) gives a counterexample. For an example whose spectrum is not Noetherian, take $A=\mathcal{C}(Y)$, the ring of continuous real-valued functions on an infinite compact space $Y$; then $Y$ (as a set) corresponds to the maximal ideals of $A$, and the topology of $Y$ is induced from the spectral topology of $X=\mathrm{Spec}(A)$.

Corollary (1.1.7). For $x\in X$, $\bar{x}=\{y\in X:{\mathfrak{j}}_x\subset {\mathfrak{j}}_y\}$. $\{x\}$ is closed iff ${\mathfrak{j}}_x$ is maximal.

Corollary (1.1.8). $X=\mathrm{Spec}(A)$ is a Kolmogorov space.

(1.1.9) For $f,g\in A$, by (1.1.2)(iv),

$$D(fg)=D(f)\cap D(g).\text{\hspace{1em}(1.1.9.1)}$$

Also $D(f)=D(g)$ iff $r(f)=r(g)$, iff the minimal primes containing $(f)$ and $(g)$ agree. In particular $D(f)=D(g)$ whenever $f=ug$ with $u$ invertible.

Proposition (1.1.10).

(i) The $D(f)$ ($f\in A$) form a basis of the topology of $X$. (ii) For every $f\in A$, $D(f)$ is quasi-compact. In particular, $X=D(1)$ is quasi-compact.

Proof. (i) follows from $U=X\setminus V(E)={\bigcup }_{f\in E}D(f)$. For (ii): if $D(f)\subset {\bigcup }_{\lambda \in L}D(f_{\lambda })$, let $\mathfrak{a}\subset A$ be the ideal generated by the $f_{\lambda }$; then $V(f)\supset V(\mathfrak{a})$, so $r(f)\subset r(\mathfrak{a})$, hence $f^n\in \mathfrak{a}$ for some $n$. Then $f^n$ lies in the ideal $\mathfrak{b}$ generated by a finite subfamily $(f_{\lambda }{)}_{\lambda \in J}$, so $D(f)=D(f^n)\subset {\bigcup }_{\lambda \in J}D(f_{\lambda })$.

Proposition (1.1.11). For every ideal $\mathfrak{a}\subset A$, $\mathrm{Spec}(A/\mathfrak{a})$ is canonically identified with the closed subspace $V(\mathfrak{a})$ of $\mathrm{Spec}(A)$.

Recall (0.1.1.1) that the nilradical $\mathfrak{N}=r(0)$ is the intersection of all prime ideals of $A$.

Corollary (1.1.12). $\mathrm{Spec}(A)$ and $\mathrm{Spec}(A/\mathfrak{N})$ are canonically homeomorphic.

Proposition (1.1.13). $X=\mathrm{Spec}(A)$ is irreducible (0.2.1.1) if and only if $A/\mathfrak{N}$ is integral (equivalently, $\mathfrak{N}$ is prime).

Corollary (1.1.14).

(i) In the bijection of (1.1.5), the irreducible closed subsets of $X$ correspond to the prime ideals of $A$; the irreducible components correspond to minimal primes. (ii) $x\mapsto \bar{x}$ is a bijection between $X$ and the set of closed irreducible subsets — that is, every closed irreducible subset of $X$ admits a unique generic point.

Proposition (1.1.15). If $\mathfrak{J}\subset A$ contains the radical of $A$, then the only neighborhood of $V(\mathfrak{J})$ in $X=\mathrm{Spec}(A)$ is $X$ itself.

Proof. Every maximal ideal belongs to $V(\mathfrak{J})$. Every ideal $\mathfrak{a}\ne A$ is in some maximal ideal, so $V(\mathfrak{a})\cap V(\mathfrak{J})\ne \varnothing$.

1.2. Functorial properties of prime spectra

(1.2.1) For a ring homomorphism $\varphi :A^{\prime }\to A$ and $x={\mathfrak{j}}_x\in \mathrm{Spec}(A)$, $A^{\prime }/{\varphi }^{-1}({\mathfrak{j}}_x)$ injects into $A/{\mathfrak{j}}_x$, so ${\varphi }^{-1}({\mathfrak{j}}_x)$ is prime. Write ${}^a\varphi (x)={\varphi }^{-1}({\mathfrak{j}}_x)$, defining

$${}^a\varphi :X=\mathrm{Spec}(A)\to X^{\prime }=\mathrm{Spec}(A^{\prime })$$

(also written $\mathrm{Spec}(\varphi )$) — the map associated to $\varphi$. Let ${\varphi }^x$ be the injective homomorphism $A^{\prime }/{\varphi }^{-1}({\mathfrak{j}}_x)\to A/{\mathfrak{j}}_x$ induced by $\varphi$, and also its extension to fields $\kappa ({}^a\varphi (x))\to \kappa (x)$. For $f^{\prime }\in A^{\prime }$,

$${\varphi }^x\left(f^{\prime }({}^a\varphi (x))\right)=(\varphi (f^{\prime }))(x)\text{for }x\in X.\text{\hspace{1em}(1.2.1.1)}$$

Proposition (1.2.2).

(i) For every $E^{\prime }\subset A^{\prime }$, ${}^a{\varphi }^{-1}(V(E^{\prime }))=V(\varphi (E^{\prime }))$; in particular for $f^{\prime }\in A^{\prime }$, ${}^a{\varphi }^{-1}(D(f^{\prime }))=D(\varphi (f^{\prime }))$. (1.2.2.1, 1.2.2.2) (ii) For every ideal $\mathfrak{a}\subset A$, ${}^a\varphi \overline{V(\mathfrak{a})}=V({\varphi }^{-1}(\mathfrak{a}))$. (1.2.2.3)

Corollary (1.2.3). ${}^a\varphi$ is continuous.

If $A^{\prime \prime }$ is a third ring and ${\varphi }^{\prime }:A^{\prime \prime }\to A^{\prime }$ a homomorphism, ${}^a(\varphi \circ {\varphi }^{\prime })={}^a{\varphi }^{\prime }\circ {}^a\varphi$. So $\mathrm{Spec}(A)$ is a contravariant functor in $A$ from rings to topological spaces.

Corollary (1.2.4). If every $f\in A$ is of the form $h\varphi (f^{\prime })$ with $h$ invertible (e.g., if $\varphi$ is surjective), then ${}^a\varphi$ is a homeomorphism from $X$ onto ${}^a\varphi (X)$.

(1.2.5) When $\varphi :A\to A/\mathfrak{a}$ is the canonical quotient, ${}^a\varphi$ is the canonical injection of $V(\mathfrak{a})\cong \mathrm{Spec}(A/\mathfrak{a})$ into $X=\mathrm{Spec}(A)$.

Corollary (1.2.6). If $S\subset A$ is multiplicative, $\mathrm{Spec}(S^{-1}A)$ is canonically identified (with topology) with $\{x\in \mathrm{Spec}(A):{\mathfrak{j}}_x\cap S=\varnothing \}$.

Corollary (1.2.7). ${}^a\varphi (X)$ is dense in $X^{\prime }$ iff every element of $\mathrm{Ker}\varphi$ is nilpotent.

1.3. Sheaf associated to a module

(1.3.1) Let $A$ be a commutative ring, $M$ an $A$-module, $f\in A$, $S_f=f^n:n\ge 0$. Write $A_f=S_f^{-1}A$, $M_f=S_f^{-1}M$. The saturated multiplicative subset $S_f^{\prime }$ of divisors of elements of $S_f$ gives $A_f=S_f^{\prime -1}A$, $M_f=S_f^{\prime -1}M$ (0.1.4.3).

Lemma (1.3.2). The following are equivalent: (a) $g\in S_f^{\prime }$; (b) $S_g^{\prime }\subset S_f^{\prime }$; (c) $f\in r(g)$; (d) $r(f)\subset r(g)$; (e) $V(g)\subset V(f)$; (f) $D(f)\subset D(g)$.

(1.3.3) If $D(f)\supset D(g)$, there is a canonical functorial homomorphism ${\rho }_{g,f}:M_f\to M_g$, satisfying ${\rho }_{h,g}\circ {\rho }_{g,f}={\rho }_{h,f}$ (1.3.3.1) when $D(f)\supset D(g)\supset D(h)$.

For fixed $x\in X$, the $S_f^{\prime }$ (with $f\in A\setminus {\mathfrak{j}}_x$) form a directed family with union $A\setminus {\mathfrak{j}}_x$, so $M_x=\underrightarrow{\lim }{M}_f$ is the inductive limit along $({\rho }_{g,f})$. Write ${\rho }_x^f:M_f\to M_x$ for the canonical map ($f\in A\setminus {\mathfrak{j}}_x$, i.e. $x\in D(f)$).

Definition (1.3.4). The structure sheaf of $X=\mathrm{Spec}(A)$, written $A$ or ${\mathcal{O}}_X$, and the sheaf associated to $M$, written $M$, are the sheaf of rings (resp. $\tilde{A}$-module) associated with the presheaf $D(f)\mapsto A_f$ (resp. $D(f)\mapsto M_f$) on the basis $\mathfrak{B}=\{D(f):f\in A\}$ (using (1.1.10), (0.3.2.1), and (0.3.5.6)).

By (0.3.2.4), the stalk $A_x$ (resp. $M_x$) is identified with $A_x$ (resp. $M_x$). Write ${\theta }_f:A_f\to \Gamma (D(f),A)$ (resp. ${\theta }_f:M_f\to \Gamma (D(f),M)$) for the canonical map; for $x\in D(f)$ and $\xi \in M_f$, $({\theta }_f(\xi ){)}_x={\rho }_x^f(\xi )$.

Proposition (1.3.5). $M$ is an exact covariant functor from $A$-modules to $A$-modules.

Proof. A homomorphism $u:M\to N$ yields $u_f:M_f\to N_f$ compatible with restrictions, hence $u:M\to N$; stalkwise $u_x=u_x$, so functoriality and exactness follow from (0.1.3.2). Plainly $\mathrm{Supp}(\tilde{M})=\mathrm{Supp}(M)$.

Proposition (1.3.6). For every $f\in A$, $D(f)\subset X$ is canonically identified with $\mathrm{Spec}(A_f)$, and $M_f$ is canonically identified with $M|D(f)$.

Theorem (1.3.7). For every $A$-module $M$ and $f\in A$, the map ${\theta }_f:M_f\to \Gamma (D(f),M)$ is bijective. In particular, $M\cong \Gamma (X,M)$ via ${\theta }_1$.

Proof sketch. Injectivity: if ${\theta }_f(\xi )=0$ and $\xi \in M_f$, then for every prime $\mathfrak{p}\subset A_f$, some $h\notin \mathfrak{p}$ kills $\xi$; the annihilator is contained in no prime, hence equals $A_f$, so $\xi =0$. Surjectivity (case $f=1$): given a section $s\in \Gamma (X,\tilde{M})$, by (1.1.10) cover $X$ by finitely many $D(f_i)$ with $s|D(f_i)={\theta }_{f_i}({\xi }_i)$, ${\xi }_i\in M_{f_i}$. After multiplying through by powers, write ${\xi }_i=z_i/f_i^n$ with all $n$ equal and compatibility relations $f_j^n{z}_i=f_i^n{z}_j$. Since $(f_i^n)$ generate $A$, pick $g_i\in A$ with ${\sum }_i{g}_i{f}_i^n=1$; set $z={\sum }_i{g}_i{z}_i$. Then $f_i^{n}z=z_i$, so ${\theta }_1(z)|D(f_i)=s|D(f_i)$. The general case $f\ne 1$ follows by passing to $D(f)\cong \mathrm{Spec}(A_f)$.

Corollary (1.3.8). $u\mapsto u$ is a bijection ${\mathrm{Hom}}_A(M,N)\stackrel{\sim }{\to }{\mathrm{Hom}}_A(M,N)$. In particular $M=0⟺\tilde{M}=0$.

Corollary (1.3.9).

(i) For $u:M\to N$, the sheaves associated with $\mathrm{Ker}u$, $\mathrm{Im}u$, $\mathrm{Coker}u$ are $\mathrm{Ker}u$, $\mathrm{Im}u$, $\mathrm{Coker}u$. In particular, $u$ is injective (resp. surjective, bijective) iff $u$ is. (ii) The functor $M\mapsto \tilde{M}$ commutes with inductive limits and direct sums.

So sheaves isomorphic to associated sheaves of $A$-modules form an abelian category.

(1.3.10) A submodule $N\subset M$ gives an injective $N\to M$; we identify $N$ with an $A$-submodule of $\tilde{M}$. For submodules $N,P\subset M$,

$$\widetilde{N+P}=N+P,\text{\hspace{1em}(1.3.10.1)}$$

$$\widetilde{N\cap P}=N\cap P.\text{\hspace{1em}(1.3.10.2)}$$

So $N=P⟹N=P$.

Corollary (1.3.11). $\Gamma$ is exact on the category of sheaves isomorphic to associated sheaves of $A$-modules.

Corollary (1.3.12). Let $M$, $N$ be $A$-modules.

(i) $\widetilde{M{\otimes }_{A}N}\cong M{\otimes }_{A}N$ canonically. (ii) If $M$ admits a finite presentation, $\widetilde{{\mathrm{Hom}}_A(M,N)}\cong \mathcal{H}o{m}_A(M,N)$ canonically.

(1.3.13) An $A$-algebra structure on $B$ is the same as $A$-module structure plus $A$-homomorphism $\varphi :B{\otimes }_{A}B\to B$ and unit $e\in B$ satisfying associativity, commutativity, and unit axioms. By (1.3.12), $\varphi :B{\otimes }_{A}B\to B$ makes $B$ an $A$-algebra. Similarly, $B$-module structure on $N$ corresponds to $A$-module homomorphism $\psi :B{\otimes }_{A}N\to N$; $N$ is thus a $\tilde{B}$-module.

For $B$-modules $M$, $N$, $M{\otimes }_{B}N\cong \widetilde{M{\otimes }_{B}N}$, and $\mathcal{H}o{m}_B(M,N)\cong \widetilde{{\mathrm{Hom}}_B(M,N)}$ when $M$ has a finite presentation. For $\mathfrak{J}\subset B$, $\widetilde{\mathfrak{J}N}=\mathfrak{J}\cdot N$. Graded $A$-algebras carry over to graded $\tilde{A}$-algebras.

1.4. Quasi-coherent sheaves over a prime spectrum

Theorem (1.4.1). Let $X=\mathrm{Spec}(A)$, $V\subset X$ a quasi-compact open subset, and $\mathcal{F}$ an $({\mathcal{O}}_X|V)$-module. The following are equivalent:

(a) $\mathcal{F}\cong M|V$ for some $A$-module $M$. (b) $V$ is covered by finitely many $V_i=D(f_i)>\subset V$ with $\mathcal{F}|V_i\cong M_i$ for some $A_{f_i}$-module $M_i$. (c) $\mathcal{F}$ is quasi-coherent (0.5.1.3). (d) The following hold: (d1) For every $D(f)\subset V$ and $s\in \Gamma (D(f),>\mathcal{F})$, $f^{n}s$ extends to a section over $V$ for some $n\ge 0$. (d2) For every $D(f)\subset V$ and $t\in >\Gamma (V,\mathcal{F})$ with $t|D(f)=0$, $f^{n}t=0$ for some $n\ge 0$.

Corollary (1.4.2). Every quasi-coherent sheaf on a quasi-compact open subset of $X$ is induced by a quasi-coherent sheaf on $X$.

Corollary (1.4.3). Every quasi-coherent ${\mathcal{O}}_X$-algebra on $X=\mathrm{Spec}(A)$ is $\cong B$ for some $A$-algebra $B$; every quasi-coherent $B$-module is $\cong \tilde{N}$ for some $B$-module $N$.

1.5. Coherent sheaves over a prime spectrum

Theorem (1.5.1). Let $A$ be Noetherian, $X=\mathrm{Spec}(A)$, $V\subset X$ open, $\mathcal{F}$ an $({\mathcal{O}}_X|V)$-module. Then the following are equivalent:

(a) $\mathcal{F}$ is coherent; (b) $\mathcal{F}$ is of finite type and quasi-coherent; (c) $\mathcal{F}\cong >\tilde{M}|V$ for some $A$-module $M$ of finite type.

Corollary (1.5.2). Under (1.5.1), ${\mathcal{O}}_X$ is a quasi-coherent sheaf of rings.

Corollary (1.5.3). Under (1.5.1), every coherent sheaf on an open subset of $X$ is induced by a coherent sheaf on $X$.

Corollary (1.5.4). Under (1.5.1), every quasi-coherent ${\mathcal{O}}_X$-module is the inductive limit of its coherent sub-${\mathcal{O}}_X$-modules.

1.6. Functorial properties of quasi-coherent sheaves

(1.6.1) Let $\varphi :A^{\prime }\to A$ be a ring homomorphism and ${}^a\varphi :X=\mathrm{Spec}(A)\to X^{\prime }=\mathrm{Spec}(A^{\prime })$ the associated map. There is a canonical homomorphism $\varphi :{\mathcal{O}}_{X^{\prime }}\to {}^a{\varphi }_{\ast }({\mathcal{O}}_X)$ of sheaves of rings: for each $f^{\prime }\in A^{\prime }$, $f=\varphi (f^{\prime })$, and ${}^a{\varphi }^{-1}(D(f^{\prime }))=D(f)$; identifying $\Gamma (D(f^{\prime }),{\mathcal{O}}_{X^{\prime }})=A_{f^{\prime }}^{\prime }$ and $\Gamma (D(f),{\mathcal{O}}_X)=A_f$, $\varphi$ induces a ring map ${\varphi }_{f^{\prime }}:A_{f^{\prime }}^{\prime }\to A_f$ (0.1.5.1), and these are compatible with restrictions. The pair $\Phi =({}^a\varphi ,\varphi )$ is a morphism of ringed spaces $(X,{\mathcal{O}}_X)\to (X^{\prime },{\mathcal{O}}_{X^{\prime }})$.

For $x\in X$ with $x^{\prime }={}^a\varphi (x)$, the stalk homomorphism ${\tilde{\varphi }}_x^{♯}:A_{x^{\prime }}^{\prime }\to A_x$ is the canonical map induced by $\varphi$.

Example (1.6.2). For $S\subset A$ multiplicative and $\varphi :A\to S^{-1}A$, ${}^a\varphi :Y=\mathrm{Spec}(S^{-1}A)\to X=\mathrm{Spec}(A)$ is a homeomorphism onto $\{x:{\mathfrak{j}}_x\cap S=\varnothing \}$, and ${\mathcal{O}}_Y$ is identified with the induced sheaf ${\mathcal{O}}_X|Y$ (since the stalk maps are isomorphisms).

Proposition (1.6.3). For every $A$-module $M$, there is a canonical functorial isomorphism $\widetilde{M_{[\varphi ]}}\stackrel{\sim }{\to }{\Phi }_{\ast }(\tilde{M})$ of ${\mathcal{O}}_{X^{\prime }}$-modules.

This also holds for algebras (with module by an algebra structure preserved).

Corollary (1.6.4). ${\Phi }_{\ast }$ is exact on quasi-coherent ${\mathcal{O}}_X$-modules.

Proposition (1.6.5). For every $A^{\prime }$-module $N^{\prime }$, setting $N=N^{\prime }{\otimes }_{A^{\prime }}{A}_{[\varphi ]}$, there is a canonical functorial isomorphism ${\Phi }^{\ast }(N^{\prime })\stackrel{\sim }{\to }N$ of ${\mathcal{O}}_X$-modules.

Proof sketch. The map $j:z^{\prime }\mapsto z^{\prime }\otimes 1$ is an $A^{\prime }$-homomorphism $N^{\prime }\to N_{[\varphi ]}$; by (1.3.8), $j:N^{\prime }\to \widetilde{N_{[\varphi ]}}$ is an ${\mathcal{O}}_{X^{\prime }}$-module homomorphism, hence $j:N^{\prime }\to {\Phi }_{\ast }(N)$ via (1.6.3). The corresponding $h=j^{♯}:{\Phi }^{\ast }(N^{\prime })\to N$ is bijective stalkwise — at any $x$ with $x^{\prime }={}^a\varphi (x)$, $h_x$ is the canonical isomorphism $N_{x^{\prime }}^{\prime }{\otimes }_{A_{x^{\prime }}^{\prime }}(A_x{)}_{[{\varphi }_{x^{\prime }}]}\stackrel{\sim }{\to }{N}_x$.

For $B^{\prime }$ an $A^{\prime }$-algebra, ${\Phi }^{\ast }({\tilde{B}}^{\prime })\stackrel{\sim }{\to }\widetilde{B^{\prime }{\otimes }_{A^{\prime }}{A}_{[\varphi ]}}$ is an ${\mathcal{O}}_X$-algebra isomorphism.

Corollary (1.6.6). Sections of ${\Phi }^{\ast }(N^{\prime })$ are generated, as $A$-module, by the canonical images of sections of $N^{\prime }$.

(1.6.7) The canonical map $\rho :N^{\prime }\to {\Phi }_{\ast }({\Phi }^{\ast }(N^{\prime }))$ is $j$; the canonical $\sigma :{\Phi }^{\ast }({\Phi }_{\ast }(M))\to M$ is $p$ where $p:M_{[\varphi ]}{\otimes }_{A^{\prime }}{A}_{[\varphi ]}\to M$ is $z\otimes a\mapsto a\cdot z$.

(1.6.8) For $N_1^{\prime },N_2^{\prime }$ $A^{\prime }$-modules with $N_1^{\prime }$ admitting a finite presentation, the canonical homomorphism ${\Phi }^{\ast }(\mathcal{H}o{m}_{A^{\prime }}(N_1^{\prime },N_2^{\prime }))\to \mathcal{H}o{m}_A({\Phi }^{\ast }(N_1^{\prime }),{\Phi }^{\ast }(N_2^{\prime }))$ is $\tilde{\gamma }$, where $\gamma :{\mathrm{Hom}}_{A^{\prime }}(N_1^{\prime },N_2^{\prime }){\otimes }_{A^{\prime }}A\to {\mathrm{Hom}}_A(N_1^{\prime }{\otimes }_{A^{\prime }}A,N_2^{\prime }{\otimes }_{A^{\prime }}A)$ is canonical.

(1.6.9) For ${\mathfrak{J}}^{\prime }\subset A^{\prime }$ and $M$ an $A$-module, ${\mathfrak{J}}^{\prime }M\cong \widetilde{{\mathfrak{J}}^{\prime }M}$; in particular ${\Phi }^{\ast }({\mathfrak{J}}^{\prime })\cdot A\cong \widetilde{{\mathfrak{J}}^{\prime }A}$, and ${\Phi }^{\ast }(\widetilde{A^{\prime }/{\mathfrak{J}}^{\prime }})\cong \widetilde{A/{\mathfrak{J}}^{\prime }A}$.

(1.6.10) For a third ring $A^{\prime \prime }$ and ${\varphi }^{\prime }:A^{\prime \prime }\to A^{\prime }$, with ${\varphi }^{\prime \prime }=\varphi \circ {\varphi }^{\prime }$, one has ${}^a{\varphi }^{\prime \prime }={}^a{\varphi }^{\prime }\circ {}^a\varphi$ and ${\varphi }^{\prime \prime }=\varphi \circ {\varphi }^{\prime }$, so ${\Phi }^{\prime \prime }={\Phi }^{\prime }\circ \Phi$. Thus $(\mathrm{Spec}(A),A)$ is a functor from rings to ringed spaces.

1.7. Characterization of morphisms of affine schemes

Definition (1.7.1). A ringed space $(X,{\mathcal{O}}_X)$ is an affine scheme if it is isomorphic to $(\mathrm{Spec}(A),\tilde{A})$ for some ring $A$; we then call $\Gamma (X,{\mathcal{O}}_X)\cong A$ the ring of the affine scheme, written $A(X)$.

By abuse of language, the affine scheme $\mathrm{Spec}(A)$ always denotes the ringed space $(\mathrm{Spec}(A),\tilde{A})$.

(1.7.2) For affine schemes $X=\mathrm{Spec}(A)$ and $Y=\mathrm{Spec}(B)$, every ring homomorphism $\varphi :B\to A$ gives a morphism $\Phi =({}^a\varphi ,\varphi )=\mathrm{Spec}(\varphi ):X\to Y$. Moreover, $\varphi =\Gamma (\varphi )$ recovers $\varphi$ from $\Phi$.

Theorem (1.7.3). Let $(X,{\mathcal{O}}_X)$, $(Y,{\mathcal{O}}_Y)$ be affine schemes. For a morphism $(\psi ,\theta ):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ of ringed spaces to be of the form $({}^a\varphi ,\tilde{\varphi })$ for some $\varphi :A(Y)\to A(X)$, it is necessary and sufficient that for every $x\in X$, ${\theta }_x^{♯}:{\mathcal{O}}_{\psi (x)}\to {\mathcal{O}}_x$ be a local homomorphism.

Proof. Necessity: ${\varphi }_x^{♯}$ is induced from $\varphi$ by passing to fractions, so is local. Sufficiency: let $A=A(X)$, $B=A(Y)$, $\varphi =\Gamma (\theta ):B\to A$. The local hypothesis on ${\theta }_x^{♯}$ means it induces a monomorphism ${\theta }^x:\kappa (\psi (x))\to \kappa (x)$ with ${\theta }^x(f(\psi (x)))=\varphi (f)(x)$. Hence $f(\psi (x))=0⟺\varphi (f)(x)=0$, so ${\mathfrak{j}}_{\psi (x)}={\mathfrak{j}}_{{}^a\varphi (x)}$, i.e. $\psi ={}^a\varphi$. The commutativity of stalks then forces ${\theta }_x^{♯}={\varphi }_x^{♯}$ (0.1.5.1), hence $\theta =\tilde{\varphi }$ (by (0.3.7.1)).

A morphism satisfying (1.7.3) is called a morphism of affine schemes.

Corollary (1.7.4). $\mathrm{Hom}((X,{\mathcal{O}}_X),(Y,{\mathcal{O}}_Y))$ (morphisms of affine schemes) is canonically in bijection with ${\mathrm{Hom}}_{ring}(A(Y),A(X))$. Thus the functors $A\mapsto (\mathrm{Spec}(A),\tilde{A})$ and $(X,{\mathcal{O}}_X)\mapsto \Gamma (X,{\mathcal{O}}_X)$ give an equivalence between commutative rings and the opposite category of affine schemes.

Corollary (1.7.5). If $\varphi :B\to A$ is surjective, $({}^a\varphi ,\tilde{\varphi })$ is a monomorphism of ringed spaces.

1.8. Morphisms from locally ringed spaces to affine schemes

A remark of J. Tate generalizes (1.7.3) and (2.2.4):

Proposition (1.8.1). Let $(S,{\mathcal{O}}_S)$ be an affine scheme and $(X,{\mathcal{O}}_X)$ a locally ringed space. There is a canonical bijection between ring homomorphisms $\Gamma (S,{\mathcal{O}}_S)\to \Gamma (X,{\mathcal{O}}_X)$ and morphisms of ringed spaces $(\psi ,\theta ):(X,{\mathcal{O}}_X)\to (S,{\mathcal{O}}_S)$ such that for every $x\in X$, ${\theta }_x^{♯}:{\mathcal{O}}_{\psi (x)}\to {\mathcal{O}}_x$ is local.

Proof sketch. A morphism of ringed spaces always gives $\rho :\mathrm{Hom}((X,{\mathcal{O}}_X),(S,{\mathcal{O}}_S))\to \mathrm{Hom}(\Gamma (S,{\mathcal{O}}_S),\Gamma (X,{\mathcal{O}}_X))$ via $(\psi ,\theta )\mapsto \Gamma (\theta )$. Conversely, set $A=\Gamma (S,{\mathcal{O}}_S)$ and let $\varphi :A\to \Gamma (X,{\mathcal{O}}_X)$ be a ring homomorphism. For $x\in X$, $\{f\in A:\varphi (f)(x)=0\}$ is prime (since $\kappa (x)$ is a field), giving ${}^a\varphi (x)\in S=\mathrm{Spec}(A)$. ${}^a{\varphi }^{-1}(D(f))=X_{\varphi (f)}$ (by (0.5.5.2)), so ${}^a\varphi$ is continuous. Define $\varphi :{\mathcal{O}}_S\to {}^a{\varphi }_{\ast }({\mathcal{O}}_X)$ by sending $s/f\in A_f=\Gamma (D(f),{\mathcal{O}}_S)$ to $(\varphi (s)|X_{\varphi (f)})(\varphi (f)|X_{\varphi (f)}{)}^{-1}$. Then ${\varphi }_x^{♯}(s_y/f_y)=(\varphi (s{)}_x)(\varphi (f{)}_x{)}^{-1}$ where $y={}^a\varphi (x)$; $s_y\in {\mathfrak{m}}_y⟺\varphi (s{)}_x\in {\mathfrak{m}}_x$, so ${\tilde{\varphi }}_x^{♯}$ is local. The two maps $\rho$ and $\sigma$ are mutually inverse.

(1.8.2) For locally ringed spaces $X$, $Y$, we always consider morphisms $(\psi ,\theta )$ whose stalk maps ${\theta }_x^{♯}$ are local; with this definition, locally ringed spaces form a category. Write $\mathrm{Hom}(X,Y)$ for morphisms in this category and ${\mathrm{Hom}}_{rs}(X,Y)$ for morphisms of ringed spaces (without the locality condition). The map $\rho :{\mathrm{Hom}}_{rs}(X,Y)\to \mathrm{Hom}(\Gamma (Y,{\mathcal{O}}_Y),\Gamma (X,{\mathcal{O}}_X))$ restricts to a functorial ${\rho }^{\prime }:\mathrm{Hom}(X,Y)\to \mathrm{Hom}(\Gamma (Y,{\mathcal{O}}_Y),\Gamma (X,{\mathcal{O}}_X))$.

Corollary (1.8.3). A locally ringed space $Y$ is an affine scheme if and only if ${\rho }^{\prime }:\mathrm{Hom}(X,Y)\to \mathrm{Hom}(\Gamma (Y,{\mathcal{O}}_Y),\Gamma (X,{\mathcal{O}}_X))$ is bijective for every locally ringed space $X$.

(1.8.4) Let $S=\mathrm{Spec}(A)$ and $(S^{\prime },A^{\prime })$ the ringed space whose underlying space is a point and whose sheaf is the simple sheaf defined by $A$. Let $\pi :S\to S^{\prime }$ be the unique map, and $\iota :A^{\prime }\to {\mathcal{O}}_S$ the $\pi$-morphism from the constant sections. This gives a canonical morphism $i=(\pi ,\iota ):(S,{\mathcal{O}}_S)\to (S^{\prime },A^{\prime })$. For an $A$-module $M$, write $M^{\prime }$ for the simple sheaf on $S^{\prime }$; then $i_{\ast }(\tilde{M})=M^{\prime }$.

Lemma (1.8.5). Under (1.8.4), the canonical homomorphism $i^{\ast }(i_{\ast }(M))\to M$ is an isomorphism for every $A$-module $M$.

Corollary (1.8.6). For a ringed space $(X,{\mathcal{O}}_X)$ and $u:X\to S$, there is a canonical functorial isomorphism $u^{\ast }(\tilde{M})\stackrel{\sim }{\to }{u}^{\ast }(i^{\ast }(M^{\prime }))$ of ${\mathcal{O}}_X$-modules.

Corollary (1.8.7). Under (1.8.6), there is a canonical functorial isomorphism

$${\mathrm{Hom}}_{{\mathcal{O}}_S}(\tilde{M},u_{\ast }(\mathcal{F}))\stackrel{\sim }{\to }{\mathrm{Hom}}_A(M,\Gamma (X,\mathcal{F})).\text{\hspace{1em}(1.8.7.1)}$$

(1.8.8) With the notation of (1.8.4), a morphism $(\psi ,\theta ):X\to S^{\prime }$ of ringed spaces is the same as a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$. By (1.8.1), there is a canonical bijection $\mathrm{Hom}(X,S)\stackrel{\sim }{\to }\mathrm{Hom}(X,S^{\prime })$ (the right-hand side meaning morphisms of ringed spaces). For locally ringed spaces X, Y, the analogous setup yields $\rho :{\mathrm{Hom}}_{rs}(X,Y)\to \mathrm{Hom}(X,Y^{\prime })$. By (1.8.3), affine schemes are characterized among locally ringed spaces as those for which the restriction ${\rho }^{\prime }:\mathrm{Hom}(X,Y)\to \mathrm{Hom}(X,Y^{\prime })$ is bijective for every locally ringed space $X$.

In the next chapter, this is generalized to associate to any ringed space $Z$ a locally ringed space $\mathrm{Spec}(Z)$, the starting point for a "relative" theory of preschemes over a ringed space.

(1.8.9) Pairs $(X,\mathcal{F})$ with $X$ a locally ringed space and $\mathcal{F}$ an ${\mathcal{O}}_X$-module form a category, with morphisms $(u,h):(X,\mathcal{F})\to (Y,\mathcal{G})$ where $u:X\to Y$ is a morphism of locally ringed spaces and $h:\mathcal{G}\to \mathcal{F}$ a $u$-morphism. The map $(u,h)\mapsto ({\rho }^{\prime }(u),\Gamma (h))$ is functorial.

Corollary (1.8.10). For $Y$ a locally ringed space and $\mathcal{G}$ an ${\mathcal{O}}_Y$-module, $Y$ is an affine scheme and $\mathcal{G}$ is quasi-coherent if and only if for every pair $(X,\mathcal{F})$, the canonical map (1.8.9.1) is bijective.

Remark (1.8.11). Theorem (1.7.3), Corollary (1.7.4), and Proposition (2.2.4) are special cases of (1.8.1); the definition in (1.6.1) likewise. Corollary (1.8.7) implies (1.6.3) and (1.6.4); (1.6.5) and (1.6.6) follow from (1.8.6).