Last update: 3 June 2013
We shall put a sheaf of rings on $X=\text{Spec}\left(A\right)$ (where $A$ is any commutative ring) in such a way that the stalk of the sheaf at $x\in X$ is the local ring ${A}_{x}$ (i.e. the local ring of $A$ with respect to the prime ideal ${j}_{x}\text{).}$ For this we use the open sets $D\left(f\right)$ $(f\in A)$ and the rings of fractions ${A}_{f}$ (Chapter 3, Ex. 4 and Prop. (3.6)).
Suppose $f,g$ in $A$ are such that $D\left(f\right)\supseteq D\left(g\right)\text{.}$ Then $r\left(f\right)\supseteq r\left(g\right),$ so that ${g}^{n}=sf$ for some $s\in A$ and some $n>0\text{.}$ Define a ring homomorphism
$${\rho}_{g,f}:{A}_{f}\u27f6{A}_{g}$$as follows: ${\rho}_{g,g}(a/{f}^{m})=a{s}^{m}/{g}^{mn}\in {A}_{g}\text{.}$ Verify that ${\rho}_{g,f}$ is a well-defined ring homomorphism depending only on $f$ and $g$ (and not on the particular equation ${g}^{n}=sf$ chosen), and that if $D\left(f\right)\supseteq D\left(g\right)\supseteq D\left(h\right)$ then ${\rho}_{h,g}\circ {\rho}_{g,f}={\rho}_{h,f}\text{.}$ Then the assignment $D\left(f\right)\mapsto {A}_{f}$ (and the homomorphisms ${\rho}_{g,f}\text{)}$ forms a presheaf on the basis $\mathcal{B}={\left(D\left(f\right)\right)}_{f\in A}\text{.}$ This presheaf on $\mathcal{B}$ determines a presheaf on $X,$ denoted by ${\mathcal{O}}_{X}$ or by $\stackrel{\sim}{A}\text{.}$
Proposition (5.1).
(i) | The stalk of ${\mathcal{O}}_{X}$ at $x\in X$ is isomorphic to ${A}_{x}\text{.}$ |
(ii) | ${\mathcal{O}}_{X}$ is a sheaf on $X,$ and hence $\Gamma (D\left(f\right),{\mathcal{O}}_{X})\cong {A}_{f}$ for all $f\in A\text{.}$ In particular $\Gamma (X,{\mathcal{O}}_{X})\cong A\text{.}$ |
Proof. | |
(i) is a straightforward verification: $x\in D\left(f\right)$ if and only if $f\notin {j}_{x},$ and $a/{f}^{n}\in {A}_{f}$ maps to $a/{f}^{n}\in {A}_{{j}_{x}}={A}_{x}\text{.}$ Check that this gives an isomorphism of $\underrightarrow{\text{lim}}\hspace{0.17em}{A}_{f}$ onto ${A}_{x}\text{.}$ (ii) does require proof. We have to show that the condition of (4.3) is satisfied. First, by (3.6) $D\left(f\right)$ is canonically homeomorphic to $\text{Spec}\left({A}_{f}\right)\text{;}$ also it is easily checked that the presheaf ${\stackrel{\sim}{A}}_{f}$ on $\text{Spec}\left({A}_{f}\right)$ constructed as above is canonically isomorphic to $\stackrel{\sim}{A}|D\left(f\right)\text{.}$ Hence it is enough to show that $\Gamma (X,{\mathcal{O}}_{X})=A,$ i.e. that if ${\left(D\left({f}_{i}\right)\right)}_{i\in I}$ is any covering of $X$ by basic open sets, and if ${s}_{i}\in {A}_{{f}_{i}}$ are such that the images of ${s}_{i},{s}_{j}$ in ${A}_{g}$ are the same for all $g\in A$ such that $D\left(g\right)\subseteq D\left({f}_{i}\right)\cap D\left({f}_{j}\right),$ then there exists a unique $s\in A$ whose image in ${A}_{{f}_{i}}$ is ${s}_{i},$ for all $i\in I\text{.}$ Uniqueness: if $s,s\prime \in A$ are solutions of this problem, then $t=s-s\prime $ has zero image in each ${A}_{{f}_{i}},$ hence for each $i\in I$ we have $t{f}_{i}^{{n}_{i}}=0$ for some ${n}_{i}>0\text{.}$ Since the $D\left({f}_{i}\right)$ cover $X$ and since $D\left({f}_{i}\right)=D\left({f}_{i}^{{n}_{i}}\right),$ the ideal generated by the ${f}_{i}^{{n}_{i}}$ is the whole of $A\text{.}$ Consequently we have an equation of the form $1=\sum {a}_{i}{f}_{i}^{{n}_{i}}$ $({a}_{i}\in A),$ and hence $t=\sum {a}_{i}t{f}_{i}^{{n}_{i}}=0\text{.}$ Therefore $s=s\prime \text{.}$ Existence: $X$ is quasi-compact by (3.3), hence there is a finite subset $J$ of $I$ such that $X=\underset{i\in J}{\cup}D\left({f}_{i}\right)\text{.}$ Say ${s}_{i}={z}_{i}/{f}_{i}^{{n}_{i}}$ $(i\in J),$ where ${z}_{i}\in A\text{.}$ Since $J$ is finite we may suppose that all the ${m}_{i}$ are equal: say ${s}_{i}={z}_{i}/{f}_{i}^{m}$ $(i\in J)\text{.}$ For each pair $i,j$ in $J$ the images of ${s}_{i}$ and ${s}_{j}$ in ${A}_{{f}_{i}{f}_{j}}$ are the same, so that $${z}_{i}{f}_{j}^{m}/{\left({f}_{i}{f}_{j}\right)}^{m}={z}_{j}{f}_{i}^{m}/{\left({f}_{i}{f}_{j}\right)}^{m}\phantom{\rule{1em}{0ex}}\text{in}\hspace{0.17em}{A}_{{f}_{i}{f}_{j}},$$i.e. $$({z}_{i}{f}_{j}^{m}-{z}_{j}{f}_{i}^{m}){\left({f}_{i}{f}_{j}\right)}^{{m}_{ij}}=0\phantom{\rule{1em}{0ex}}\text{in}\hspace{0.17em}A,$$for some integer ${m}_{ij}\text{.}$ Again, we may assume that all the ${m}_{ij}$ are equal, say ${m}_{ij}=n$ for all $i,j\in J\text{;}$ then, multiplying each ${z}_{i}$ by ${f}_{i}^{n},$ we reduce to the case $n=0,$ i.e. $${z}_{i}{f}_{j}^{m}={z}_{j}{f}_{i}^{m}\text{.}$$Now the $D\left({f}_{i}\right)=D\left({f}_{i}^{m}\right)$ $(i\in J)$ cover $X,$ hence the ideal generated by the ${f}_{i}^{m}$ is the whole of $A,$ so that we have an equation of the form $$1=\sum _{i\in J}{g}_{i}msfim\phantom{\rule{1em}{0ex}}({g}_{i}\in A)\text{.}$$Put ${s}_{J}=\sum _{i\in J}{g}_{i}{z}_{i}\text{;}$ then $${s}_{J}{f}_{k}^{m}=\sum _{i}{g}_{i}{z}_{i}{f}_{k}^{m}=\sum _{i}{g}_{i}{z}_{j}{f}_{i}^{m}={z}_{j}\phantom{\rule{1em}{0ex}}\text{in}A,$$so that the image of ${s}_{J}$ in ${A}_{{f}_{j}}$ is ${a}_{j}/{f}_{j}^{m}={s}_{j}$ (for all $j\in J\text{).}$ On the face of it, ${s}_{J}$ depends on the finite subset $J\text{;}$ but if $J\prime \supseteq J$ is another finite subset of $I,$ we construct ${s}_{J},$ satisfying the same conditions as ${s}_{J},$ and by the uniqueness of the solution ${s}_{J},$ must therefore be equal to ${s}_{J}\text{.}$ $\square $ |
Thus, starting from an arbitrary commutative ring $A,$ we have constructed a topological space $X=\text{Spec}\left(A\right)$ and a sheaf of local rings ${\mathcal{O}}_{X}$ (or $\stackrel{\sim}{A}\text{)}$ on $X\text{.}$ This is the basic construction on which all else is founded. The ringed space $(X,{\mathcal{O}}_{X})$ is called the affine scheme of the ring $A\text{.}$
Let $A,B$ be rings, $X=\text{Spec}\left(A\right),$ $Y=\text{Spec}\left(B\right),$ and let $\phi :B\to A$ be a ring homomorphism. We have seen in Chapter 3 that $\phi $ defines an associated continuous mapping ${}^{a}\phi :X\to Y\text{.}$ In fact $\phi $ defines a morphism of ringed spaces $({}^{a}\phi ,\stackrel{\sim}{\phi}):(X,{\mathcal{O}}_{X})\to (Y,{\mathcal{O}}_{Y}),$ as follows. Let $g\in B,$ then $D\left(g\right)$ is a basic open set in $Y,$ and we have ${}^{a}{\phi}^{-1}\left(D\left(g\right)\right)=D\left(\phi \left(g\right)\right)$ by (3.5). How $\phi $ induces a homomorphism ${B}_{g}\to {A}_{\phi \left(g\right)},$ namely $b/{g}^{n}$ is mapped to $\phi \left(b\right)/\phi {\left(g\right)}^{n}\text{;}$ hence by (5.1) (ii) $\phi $ induces a homomorphism
$${\stackrel{\sim}{\phi}}_{D\left(g\right)}:\Gamma (D\left(g\right),\mathcal{O})\u27f6\gamma ({}^{a}{\phi}^{-1}\left(D\left(g\right)\right),{\mathcal{O}}_{X})\text{.}$$Clearly the ${\stackrel{\sim}{\phi}}_{D\left(g\right)}$ are compatible with the restriction homomorphisms, hence we have $\stackrel{\sim}{\phi}:{\mathcal{O}}_{Y}\to {\mathcal{O}}_{X}$ as required. $\stackrel{\sim}{\phi}$ induces a homomorphism of the stalks: if $y={}^{a}\phi \left(x\right),$ we have ${\stackrel{\sim}{\phi}}_{x}^{\#}:{\mathcal{O}}_{Y,y}\to {\mathcal{O}}_{X,x}\text{.}$ Now ${\mathcal{O}}_{Y,y}\cong {B}_{y},$ ${\mathcal{O}}_{X,x}\cong {A}_{x},$ and the map ${B}_{y}\to {A}_{x}$ is the obvious one: $b/s\in {B}_{y}$ is mapped to $\phi \left(b\right)/\phi \left(s\right)\in {A}_{x}\text{.}$
If $P,Q$ are local rings, $m$ and $n$ their respective maximal ideals, a homomorphism $f:P\to Q$ is said to be local if the following equivalent conditions are satisfied:
(i) | $f\left(m\right)\subseteq n$ (i.e. the image of a non-unit is a non-unit); |
(ii) | ${f}^{-1}\left(n\right)=m$ (i.e. the inverse image of a unit is a unit). |
If so, then $f$ induces a field monomorphism $P/m\to Q/n\text{.}$
Now in the case in point, the homomorphism ${B}_{y}\to {A}_{x}$ is local, for the maximal ideal of ${A}_{x}$ is ${j}_{x}{A}_{x}$ and that of ${B}_{y}$ is ${j}_{y}{B}_{y}={\phi}^{-1}\left({j}_{x}\right){A}_{x}\text{.}$ Hence the morphism $({}^{a}\phi ,\stackrel{\sim}{\phi}):(X,{\mathcal{O}}_{X})\to (Y,{\mathcal{O}}_{Y})$ has the property that the homomorphisms induced on the stalks are local homomorphisms.
Conversely, let $(\psi ,\theta ):(X,{\mathcal{O}}_{X})\to (Y,{\mathcal{O}}_{Y})$ be a morphism of ringed spaces (where $X=\text{Spec}\left(A\right),$ $Y=\text{Spec}\left(B\right)$ and ${\mathcal{O}}_{X},{\mathcal{O}}_{Y}$ are the structure sheaves $\stackrel{\sim}{A},\stackrel{\sim}{B}\text{)}$ such that ${\theta}_{x}^{\#}:{B}_{y}\to {A}_{x}$ is a local homomorphism for each $x\in X$ $\text{(}y=\psi \left(x\right)\text{).}$ We have then, in particular, a ring homomorphism $\theta \left(Y\right):\Gamma (Y,{\mathcal{O}}_{Y})\to \Gamma (X,{\mathcal{O}}_{X})\text{;}$ but $\Gamma (Y,{\mathcal{O}}_{Y})\cong B$ and $\Gamma (X,{\mathcal{O}}_{X})\cong A$ by (5.1), hence $(\psi ,\theta )$ determines a ring homomorphism $\phi :B\to A\text{.}$ Since ${\theta}_{x}^{\#}$ is local, it gives rise to an embedding
$${\theta}^{x}:k\left(y\right)\u27f6k\left(x\right)$$of the residue fields, such that for each $g\in B$ we have ${\theta}^{x}\left(g\left(y\right)\right)=\varphi \left(g\right)\hspace{0.17em}\left(x\right).$ Since ${\theta}^{x}$ is injective we have $g\left(y\right)=0$ if and only if $\phi \left(g\right)\hspace{0.17em}\left(x\right)=0,$ i.e. $g\in {j}_{y}$ if and only if $\phi \left(g\right)\in {j}_{x},$ so that ${j}_{y}={\phi}^{-1}\left({j}_{x}\right),$ i.e. $y={}^{a}\phi \left(x\right)\text{;}$ hence $\psi ={}^{a}\phi \text{.}$ Moreover, the diagram
$$\begin{array}{ccc}B& \stackrel{\phi}{\u27f6}& A\\ \downarrow & & \downarrow \\ {B}_{y}& \underset{{\theta}_{x}^{\#}}{\u27f6}& {A}_{x}\end{array}$$is commutative, hence ${\theta}_{x}^{\#}$ is the homomorphism of ${B}_{y}$ into ${A}_{x}$ induced by $\phi \text{;}$ but $\theta $ is uniquely determined by the homomorphisms ${\theta}_{x}^{\#},$ and therefore $(\psi ,\theta )=({}^{a}\phi ,\stackrel{\sim}{\phi})\text{.}$ We have therefore proved
Proposition (5.2). There is a one-to-one correspondence between the ring homomorphisms $B\to A$ and the morphisms $(\psi ,\theta ):(X,{\mathcal{O}}_{X})\to (Y,{\mathcal{O}}_{Y})$ such that ${\theta}_{x}^{\#}$ is a local homomorphism for each $x\in X\text{.}$
So far, this is the basic local theory. The next step is to define the global objects. By analogy with Serre's definition of an algebraic variety (Chapter 1), it is clear what the general definition should be.
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".