Sheaf total quotient rings

The sets $S_X(U)$ are multiplicative and restriction-compatible. Restriction preserves injectivity of multiplication at each retained stalk. At each stalk, multiplication by a product is the composite of the two multiplication maps; multiplication by $1$ is the identity. Therefore the localizations form the stated presheaf $P_X$.

Every localization map ${\mathcal{O}}_X(U)\to P_X(U)$ is injective. If $s\in S_X(U)$ and $sa=0$, then multiplication by $s_x$ gives $a_x=0$ for every $x\in U$. A germ is zero exactly when the section vanishes on some neighborhood, so $a$ vanishes locally everywhere and is zero by sheaf locality. The localization criterion in [F2] now gives the injectivity.

An integral scheme has a unique generic point. Choose a nonempty affine open $V_0=\mathrm{Spec}{A}_0$. The ring $A_0$ is a domain. Every nonempty open of $\mathrm{Spec}{A}_0$ contains a basic open $D(f)$ containing some prime; then $f\mathrm{\ne }0$ and $(0)\in D(f)$. Thus $(0)$ is dense in $V_0$. Since $V_0$ is dense in irreducible $X$, the closure of this point in $X$ is $X$, giving a generic point $\eta$. Any generic point ${\eta }^{\mathrm{\prime }}$ belongs to every nonempty open, so ${\eta }^{\mathrm{\prime }}\in V_0$. If ${\eta }^{\mathrm{\prime }}$ corresponds to a nonzero prime $\mathfrak{p}$, choose $0\mathrm{\ne }a\in \mathfrak{p}$. The nonempty open $D(a)$ contains $(0)$ and omits ${\eta }^{\mathrm{\prime }}$, contradicting density of $\{{\eta }^{\mathrm{\prime }}\}$. Hence $\eta$ is unique. In particular every nonempty open of $X$ contains $\eta$.

The sheaf map is injective on stalks and sections. If a germ represented by $a\in {\mathcal{O}}_X(U)$ maps to zero in $(P_X{)}_x$, then after shrinking to a neighborhood $V$ its image is zero in $P_X(V)$. Step 1.2 gives $a{\mathrm{\mid }}_V=0$, so $({\mathcal{O}}_X{)}_x\to (P_X{)}_x$ is injective. Sheafification preserves stalks, so the map to ${\mathcal{K}}_X$ is injective on every stalk. A section in its kernel has zero germ at every point, vanishes on a cover, and is zero by locality.

Each nonempty affine chart has $P_X(V)\cong K(X)$. Let $V=\mathrm{Spec}A$ be any nonempty affine chart. Choose the affine chart $V_0=\mathrm{Spec}{A}_0$ used in step 1.3. Its generic prime is $(0)$, so [F9, F10, F12] give $K(X)={\mathcal{O}}_{X,\eta }\cong (A_0{)}_{(0)}=\mathrm{Frac}(A_0)$, a field. By step 1.3, $\eta \in V$; let $\mathfrak{p}$ be its prime in $A$. Then $A_{\mathfrak{p}}\cong {\mathcal{O}}_{X,\eta }$ is a field. If $\mathfrak{p}\mathrm{\ne }(0)$, a nonzero element of $\mathfrak{p}$ stays nonzero in this localization because $A$ is a domain, so the maximal ideal $\mathfrak{p}{A}_{\mathfrak{p}}$ is nonzero, impossible for a field. Hence $\mathfrak{p}=(0)$ and [F9, F10, F12] give $K(X)={\mathcal{O}}_{X,\eta }\cong A_{(0)}=\mathrm{Frac}(A)$. If $a\mathrm{\ne }0$ in the domain $A$, then $a\mathrm{/}1\mathrm{\ne }0$ in each $A_{\mathfrak{p}}$: otherwise some $b\mathrm{\notin }\mathfrak{p}$ would satisfy $ba=0$. Thus all nonzero elements of $A$ act injectively on every stalk in $V$. The zero element does not act injectively, since these stalks are nonzero local rings. Consequently $S_X(V)=A\setminus \{0\}$ and $P_X(V)=\mathrm{Frac}(A)$.

Generic evaluation embeds ${\mathcal{O}}_X(U)$ and identifies $S_X(U)$. For nonempty $U$, cover it by affine opens $V$. If a section maps to zero at $\eta$, its restriction to each $V$ is zero because $\mathrm{\Gamma }(V,{\mathcal{O}}_X)$ embeds in its fraction field. Locality makes the section zero. A nonzero section cannot have zero germ at any point: that would make it zero on a nonempty neighborhood, which contains $\eta$ by step 1.3. Its germs are therefore nonzero in the domain stalks: on an affine neighborhood those stalks are localizations of a domain by [F9, F10], so they act injectively. Conversely, the zero section fails the injectivity condition at every point of nonempty $U$.

Generic evaluation sheafifies to a map ${\mathcal{K}}_X\to \underset{‾}{K(X)}$. For nonempty $U$, step 3.1 puts every denominator in $S_X(U)$ at a nonzero element of $K(X)$, so the localization universal property gives a ring map $P_X(U)\to K(X)$. Send each fraction to the constant locally constant function with that value. For $U=\mathrm{\varnothing }$, the sheaf empty cover axiom gives ${\mathcal{O}}_X(\mathrm{\varnothing })=0$; hence $P_X(\mathrm{\varnothing })=0=\underset{‾}{K(X)}(\mathrm{\varnothing })$, and use the unique ring map between these zero rings. The maps commute with restrictions, including restriction to $\mathrm{\varnothing }$, so they define a presheaf map $P_X\to \underset{‾}{K(X)}$. The sheafification universal property extends it to the stated map.

The resulting map is an isomorphism on stalks. Affine opens form a basis: inside an affine neighborhood, the basic opens refine any given neighborhood. On each nonempty affine open $V$, step 2.2 identifies $P_X(V)\to K(X)$ with the canonical fraction-field isomorphism. These affine neighborhoods are cofinal at every point, so the map induces a bijection on every stalk. The target stalk is $K(X)$ by the constant-sheaf description, and the source stalk agrees with that of $P_X$ by sheafification. The stalkwise isomorphism criterion completes the proof. Both sheaves have their unique empty-open section by the sheaf empty-cover axiom. Integrality is used only for the constant-function-field identification above.