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The structure morphism of a scheme $X$ is its unique morphism $X\to \mathrm{Spec}\mathbb{Z}$. More generally, an $S$-scheme comes equipped with a specified morphism to the base scheme $S$.

The structure sheaf ${\mathcal{O}}_X$ records the regular functions on every open subset of a scheme $X$. Its stalk ${\mathcal{O}}_{X,x}$ is the local ring of functions defined near $x$.

The scheme $C^{(d)}$ parametrizes effective degree-$d$ Cartier divisors on a smooth algebraic curve. Near a Cartier divisor $\sum m_i{P}_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the characteristic of a field divides $d!$. For projective $C$ it is projective.

The tensor product of two globally generated line bundles is globally generated. At a point choose generator germs arising from a global section of each factor. Their tensor is the germ of a global tensor-product section and generates the rank-one tensor-product stalk. Iteration gives the same property for every positive tensor power, on an arbitrary scheme.

Among quasi-compact morphisms of schemes, universal closedness is equivalent to the existence part of the valuative criterion: for every valuation ring $D$ with fraction field $K$, every compatible pair $\mathrm{Spec}K\to X$, $\mathrm{Spec}D\to S$ extends to $\mathrm{Spec}D\to X$. All valuation rings must be tested without further hypotheses. A universally closed morphism is automatically a quasi-compact morphism.

For a morphism of schemes $S^{\prime }\to S$ and an $S^{\prime }$-scheme $X$, the Weil restriction is the functor defined on $S$-schemes byWhen this functor is represented by a scheme, that representing object is also called the Weil restriction. Representability needs hypotheses; it is not part of a formal guarantee for every $X$ and every base map.^[ref]

For $\mathbb{C}/\mathbb{R}$, restricting the affine line gives ${\mathrm{Res}}_{\mathbb{C}/\mathbb{R}}{\mathbb{A}}_{\mathbb{C}}^1\cong {\mathbb{A}}_{\mathbb{R}}^2$. Indeed, a complex coordinate is expressed as $a+bi$, and for any real algebra $R$ the corresponding points are $R{\otimes }_{\mathbb{R}}\mathbb{C}\cong R^2$. The construction expresses geometry over an extension field using coordinates over the smaller field, while retaining its full point functor.

The Zariski closure of a subset $S$ of an algebraic variety or a scheme is the smallest Zariski-closed subset containing $S$. For $S\subseteq k^n$, define the ideal $I(S)$ of all polynomials vanishing on $S$. ThenThis description follows because an intersection of polynomial zero sets is again the common zero set of their equations.^[ref]

Every infinite subset of the complex affine line is Zariski dense: a nonzero one-variable polynomial has only finitely many roots of a polynomial. Thus $S=\{1,1/2,1/3,\dots \}$ has Zariski closure all of $\mathbb{C}$, although its closure in the ordinary Euclidean topology is only $S\cup \{0\}$. In algebraic geometry, taking a closure retains all polynomial consequences, rather than all ordinary sequential limits.

A Zariski cover of a scheme is a family of open immersions whose images cover its underlying topological space. Such families define the Zariski topology as a Grothendieck topology on the category of schemes; a cover need not be finite.

A contravariant set-valued functor on schemes is a Zariski sheaf if compatible local elements on any Zariski cover glue uniquely. Explicitly, $F(T)$ is the equalizer of ${\prod }_{i}F(U_i)\rightrightarrows {\prod }_{i,j}F(U_i{\times }_T{U}_j)$. The functor of points of every scheme is a Zariski sheaf because morphisms of schemes glue over open covers.