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The points of the spectrum of a commutative ring $A$ are its prime ideals. Its Zariski topology has closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and basis $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$. The structure sheaf of a scheme can be defined as follows: a section over $U$ assigns an element of $A_{\mathfrak{q}}$ to every $\mathfrak{q}\in U$, and locally that assignment is represented by one fraction $a/s$, with $s$ outside every prime in the neighbourhood. Restrictions forget values. Local representability and equality can be checked on an open cover, so these assignments satisfy the sheaf gluing axiom. On $D(f)$ the sections identify with the localization of a ring $A_f$.

Evaluation at $\mathfrak{p}$ induces the ring homomorphism ${\mathcal{O}}_{X,\mathfrak{p}}\to A_{\mathfrak{p}}$. It is surjective because $a/s\in A_{\mathfrak{p}}$ is represented by a section on $D(s)$. For injectivity, represent a germ by $a/s$ on a neighbourhood of $\mathfrak{p}$. If its value in $A_{\mathfrak{p}}$ vanishes, there is $t\notin \mathfrak{p}$ with $ta=0$. The section is then zero after restricting to $D(st)$ inside that neighbourhood. Its germ is zero. Thus

The tensor product of sheaves is the sheafification of the presheafwith restriction maps induced by those of $\mathcal{F}$, $\mathcal{G}$, and the structure sheaf of a scheme. Sending the germ of $f\otimes g$ to $f_P\otimes g_P$ gives a canonical mapIt is surjective because a finite sum of tensors of germs has representatives on one common neighborhood. It is injective because an equality in a tensor product of modules is witnessed by finitely many additive and balancing relations; all their germs, including the scalars in ${\mathcal{O}}_{X,P}$, can be represented and their equalities made valid on a common smaller neighborhood. Equivalently, a filtered colimit of modules commutes with these tensor products. Since sheafification preserves the stalk of a sheaf, this proves the asserted isomorphism.

For the pullback of a sheaf of modules, first form the inverse image sheaf ${\varphi }^{-1}\mathcal{F}$: sheafify $U\mapsto {\mathrm{colim}}_{V\supseteq \varphi (U)}\mathcal{F}(V)$. The morphism of sheaves of rings ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ then definesExactness of a sequence of sheaf of modules morphisms is checked on every stalk of a sheaf. Applying the right exactness of the tensor product of modules to the stalk sequence over ${\mathcal{O}}_{Y,\varphi (P)}$ gives the exact stalk sequence of the pullbacks. Thus ${\varphi }^{\ast }$ preserves sequences ending in zero, as required.

A quasi-coherent sheaf is a sheaf of modules locally of the form $\tilde{N}$ for a module $N$ over the coordinate ring of an affine neighborhood. On an affine variety $V$ with ring $B$, this associated sheaf is characterized on principal opens by $\tilde{N}(D(b))=N_b$, using localization of a module. Suppose $\mathcal{F}{|}_V\cong \tilde{N}$ and $U\subseteq {\varphi }^{-1}V$ is affine, with ring $C$. The induced algebra map $B\to C$ gives a canonical mapAt a point $P\in U$, both sides identify with ${\mathcal{O}}_{X,P}{\otimes }_{B}N$: on the right use ${\mathcal{F}}_{\varphi (P)}\cong {\mathcal{O}}_{Y,\varphi (P)}{\otimes }_{B}N$ and associativity of the tensor product of modules. The map is therefore an isomorphism on every stalk of a sheaf, hence an isomorphism of sheaves. Such $U$ cover $X$, proving that the pullback is a quasi-coherent sheaf.

Finally, on an affine open $V\subseteq Y$ with ring $B$, the presheaf defining $M_Y$ has, on $D(b)\subseteq V$, the valueConsequently $M_Y{|}_V\cong \widetilde{M{\otimes }_{R}B}$, so $M_Y$ is a quasi-coherent sheaf. The algebra map for $M_X$ is the composite $R\to \Gamma (Y,{\mathcal{O}}_Y)\to \Gamma (X,{\mathcal{O}}_X)$. Restriction of functions and the tensor-product construction give a canonical sheaf morphism ${\varphi }^{\ast }{M}_Y\to M_X$. At $P$, it is the associativity isomorphismIt is therefore a sheaf of modules isomorphism. In particular,

Every restriction map of the structure sheaf of a scheme is a unital ring homomorphism. Thus $p\cdot 1=0$ in $\Gamma (X,{\mathcal{O}}_X)$ implies the same identity in $\Gamma (U,{\mathcal{O}}_X)$ for every open $U$. This proves (ii)$\Rightarrow$(i), so (i) and (ii) are equivalent.

An open immersion $j:U\to X$ is a morphism of schemes that identifies $U$, including its structure sheaf, with an open subscheme of $X$.

Let $X={\mathbb{A}}_k^2$ and $U=X\setminus \{(0,0)\}$. The inclusion is an open immersion and $X$ is an affine scheme, but $U$ is not affine. Indeed, regular functions extend across the missing codimension-two point, soWere $U$ affine, the inclusion would therefore correspond to an isomorphism of coordinate rings and would be an isomorphism $U\cong X$, a contradiction.

For an example with both schemes affine, the principal open subschemeis a nontrivial open immersion. The affine line is connected because $k[t]$ has no nontrivial idempotent elements.

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.

The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.