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A morphism of ringed spaces $f:(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ consists of a continuous map $f:X\to Y$ and a morphism of sheaves of rings $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A propositional geometric theory has primitive propositions and sequents between formulas built with finite conjunction and arbitrary set-indexed disjunction. Start with the free frame on the primitives, identify conjunction with meet and disjunction with join, and impose each sequent as an inequality. The resulting quotient frame is $\mathcal{O}(X_T)$. In the sheaf topos $\mathrm{Sh}(X_T)$, interpret each primitive as the subterminal sheaf corresponding to its generating open. Operations on these subterminal sheaves are precisely the frame operations, so every imposed sequent is valid. This constructs the canonical model.

For a commutative ring $R$, denote the generator attached to $r$ by $U_r$. The presentation impliesThe product equality uses the multiplication implication in one direction and both factor implications in the other; the latter follow by commuting $r,s$. A ring homomorphism $\phi :R\to S$ sends these relations to the corresponding ones in $S$. It therefore gives a frame homomorphism $U_r\mapsto U_{\phi (r)}$ and hence a map $X_S\to X_R$ of locales.

If $ar+bs=1$, thenso the two opens cover. Conversely let $S=R/(r,s)$ and pull back a putative cover along $X_S\to X_R$. Both generators become $U_0$, so $1=0$ in $\mathcal{O}(X_S)$. The allowed degeneracy criterion gives $0=1$ in $S$, equivalently $1\in (r,s)$. ThusThis necessity argument never selects a prime ideal.

Finite intersections of the generating opens are again generators, so they form a basis. Assign $R[r^{-1}]$ to $U_r$. The localization of a ring carries its ordinary addition and multiplication of fractions; equality of $a/r^n$ and $b/r^m$ means that some power of $r$ annihilates the cross-multiplied difference. On $U_r\cap U_s=U_{rs}$, restriction is further localization.

We also justify restriction for any inclusion of basic opens. Adding the condition $U_t=1$ to the presentation of $X_R$ gives the open $U_t$ and the presentation of $X_{R[t^{-1}]}$: the generator for $a/t^n$ corresponds to $U_a\wedge U_t$. This is well-defined since equality of fractions, after multiplication by a power of the now-invertible $t$, makes the two membership propositions equivalent. More explicitly, $U_t=1$ implies $U_{t^{k}z}=U_z$. If $t^{k}z=0$, this gives $U_z=0$. If $x-y$ is killed by a power of $t$, then $U_{x-y}=U_{y-x}=0$ (since $U_{-1}=1$), and the additive inequalities for $x=y+(x-y)$ and $y=x+(y-x)$ give $U_x=U_y$. A zero element in the localized ring has false membership, multiplication gives intersections, and the additive relation follows by clearing denominators. The reverse assignment on $a\in R$ proves that these maps are inverse. Thus $X_{R[t^{-1}]}\cong U_t$. If $U_t\subseteq U_r$, then in this localized theory $U_r=1$. Apply the quotient argument above, with the second generator $0$, to conclude $1\in (r)$ in $R[t^{-1}]$. Therefore $r$ is invertible there, and the universal property of localization gives the restriction $R[r^{-1}]\to R[t^{-1}]$. These restrictions compose uniquely.

It remains to glue, and the allowed basis criterion reduces this to a whole-locale cover $U_r\vee U_s=1$. Let $\alpha \in R[r^{-1}]$ and $\beta \in R[s^{-1}]$ have equal restrictions to $R[(rs{)}^{-1}]$. Choose a common initial exponent $n$, writing $\alpha =a/r^n$, $\beta =b/s^n$. Equality on the overlap supplies $k\ge 0$ withSet $N=n+k$, $A=r^{k}a$ and $B=s^{k}b$. Thenwith the last equality in $R$ itself. This clearing step is necessary when the commutative ring has zero divisors.

Since $(r,s)=R$, also $(r^N,s^N)=R$. Choose $\lambda r+\mu s=1$. For $N\ge 1$, expand $(\lambda r+\mu s{)}^{2N-1}=1$; each term contains either $r^N$ or $s^N$, producing $C,D\in R$ with $C{r}^N+D{s}^N=1$. For $N=0$ choose $C=1,D=0$. DefineIn $R[r^{-1}]$,and similarly in $R[s^{-1}]$,Thus $t$ glues the matching pair. If $t^{\prime }$ is another gluing, some $r^p$ and $s^q$ annihilate $t-t^{\prime }$. The same binomial argument gives $(r^p,s^q)=R$, so $t-t^{\prime }=0$. This proves uniqueness without cancellation or a domain assumption.

The exactness of the unit-ideal localization Čech complex proves the sheaf axiom for the required basic covers, and hence the assignment with its restrictions defines a sheaf of rings $\tilde{R}$ with $\tilde{R}(U_r)=R[r^{-1}]$. On the empty basic open, localization is the zero ring, whose underlying set has one section, as required. The construction and proof remain valid for the zero ring.

Choose an open cover of $X$ by affine opens $U_i=\mathrm{Spec}{B}_i$. Since $X$ is a quasi-compact topological space, finitely many suffice. Let $f_i,a_i\in B_i$ be the restrictions of $f,a$. Part (a) identifies $U_i\cap X_f$ with $D(f_i)$, whose ring of regular functions is the localization of a ring $(B_i{)}_{f_i}$. The vanishing of $a$ there meansso $f_i^{n_i}{a}_i=0$ for some $n_i\ge 0$. Taking $n>0$ at least as large as every $n_i$ gives $(f^{n}a){|}_{U_i}=0$ on every member of the finite cover. The local identity axiom for the sheaf of rings ${\mathcal{O}}_X$ therefore gives

Let $X={\bigcup }_{i=1}^r{U}_i$ be the stated finite affine cover. By the description of sections on a principal open subscheme, after increasing denominators separately we may writefor some $a_i\in \Gamma (U_i,{\mathcal{O}}_X)$. Multiplying the numerators by powers of $f$ lets us use one exponent $n$ for every $i$.

On $W_{ij}=U_i\cap U_j$, the section $a_i-a_j$ vanishes after restriction to $(W_{ij}{)}_f$. Each $W_{ij}$ is quasi-compact, so part (b) supplies $m_{ij}$ with $f^{m_{ij}}(a_i-a_j)=0$ on $W_{ij}$. Choose one $m$ valid for all the finitely many pairs. The sections $f^m{a}_i$ now agree on every overlap, and the gluing axiom of a sheaf of rings produces $a\in \Gamma (X,{\mathcal{O}}_X)$ with $a{|}_{U_i}=f^m{a}_i$. On $X_f$,Thus some power of $f$ times $b$ extends to a global regular function on $X$.

In the classical convention over an algebraically closed field $k$, a prevariety is an irreducible topological space with a sheaf of rings which has a finite open cover by affine varieties, with compatible local identifications. A prevariety need not be separated. A classical algebraic variety is a separated prevariety: its diagonal morphism has closed image. Some authors allow reducible prevarieties and varieties; the separation criterion and product construction remain the same under that convention.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space $X$ equipped with a sheaf of rings ${\mathcal{O}}_X$. A morphism consists of a continuous map $f:X\to Y$ and a compatible morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.