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Glue two copies of the affine line over a base along the principal open subset $D(t)$ by the identity. This scheme has two copies of the zero section and a canonical morphism of schemes to the original affine line. Over $\mathbb{Z}$, its functor of points assigns to a commutative ring $R$ the pairs $(r,e)$ with $e$ an idempotent in $R/(r)$. Indeed, $r$ gives the map to the affine line, while $e$ partitions $V(r)$ into the two clopen sets assigned to the zero sections. Complementing the opposite part gives two open subsets covering $\mathrm{Spec}R$, whose intersection is $D(r)$; the coordinate $r$ defines maps on them which glue. This also proves that no lifting of $e$ to an idempotent of $R$ is necessary.

An affine open cover of a scheme is an open cover by affine open subsets. Every scheme has such a cover by definition. A quasi-compact scheme has a finite affine open cover. Whether the intersections are affine is a further condition, supplied in particular by separatedness over an affine base.

An affine open subscheme is an open subset of a scheme with its restricted structure sheaf, which is itself an affine scheme. Every scheme has an open cover by such subschemes. On an affine scheme, the principal open subschemes give an open basis.

An affine scheme is a scheme isomorphic to $\mathrm{Spec}A$ for some commutative ring $A$. Homomorphisms $A\to B$ correspond contravariantly to morphisms $\mathrm{Spec}B\to \mathrm{Spec}A$.

Restricting the functor of points to affine schemes gives a fully faithful functor from schemes to covariant set-valued functors on commutative rings. Given a natural transformation, apply it to the inclusions of an affine open cover of $X$. Naturality on affine open subschemes of overlaps makes the resulting maps to $Y$ agree, so they glue. For an arbitrary map from an affine scheme to $X$, cover its inverse images of these charts by affine open subschemes and apply naturality again. This proves that the glued map induces the prescribed natural transformation, and uniqueness follows from the original cover. In particular, the restricted functor of points determines the scheme up to an isomorphism of schemes.

Algebraic geometry studies geometric objects defined by polynomial equations and the algebra of functions on them. For example, $y=x^2$ defines a parabola, while $xy=0$ defines the union of two coordinate lines. The coordinate ring $k[x,y]/(xy)$ records the second example algebraically: the images of $x$ and $y$ are nonzero but their product is zero, reflecting two components. Modern schemes extend this dictionary beyond ordinary point sets and permit geometric reasoning over general commutative rings. Applications range from elliptic curves in number theory to families of solutions depending on parameters.^[ref]

An algebraic variety over a field $k$ is, in a common modern convention, a reduced scheme that is a separated scheme of finite type over $k$. Some authors also require it to be irreducible; this wiki states that extra condition when reducible spaces matter. A classical convention over an algebraically closed field describes the same geometry using polynomial zero sets and their regular functions.

An affine variety has a coordinate ring $A$, a finitely generated reduced $k$-algebra, and its scheme model is $\mathrm{Spec}A$. General algebraic varieties glue finitely many such affine charts along open subsets. The regular functions are part of the structure: their local rings record which functions can be evaluated near each point.

For example, $x^2+y^2=1$ defines a real algebraic curve, while $y^2=x^3$ defines a cuspidal cubic. Both are one-dimensional, but the cusp has a singular local ring at the origin. The scheme defined by $x^2=0$ additionally remembers a nonzero nilpotent element; it is not a reduced scheme and so is outside this variety convention. This illustrates why an underlying zero set alone cannot describe every scheme.

On a quasi-compact scheme $X$, a line bundle $L$ is ample if each point has an affine neighborhood of the formEquivalently, the affine nonvanishing loci of global sections of positive tensor powers form a basis of a topology. This makes ampleness a condition that global sections recover enough affine local geometry.^[ref]

For a proper scheme over a field, this is equivalent to some positive tensor power being very ample: $L^{\otimes m}$ is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. A positive tensor power may be needed. On an elliptic curve, a degree-one line bundle is ample but has only one independent global section, so it cannot itself embed the curve. Its third power has degree three and gives a plane cubic embedding. On an affine scheme, every line bundle is ample under the scheme definition, even when the scheme is not proper.

Arithmetic geometry studies polynomial equations using both algebraic geometry and number theory. The coefficient arithmetic matters: a geometric object may have real points while having no integer or rational points of the required kind.

For example, $x^2+y^2=3$ has real solutions, but no solutions in the integers. A square is congruent to zero or one modulo four, so a sum of two squares cannot be congruent to three. Such polynomial congruences modulo a prime or prime power can therefore obstruct the existence of global arithmetic solutions.

In the language of schemes, an object over $\mathrm{Spec}\mathbb{Z}$ has fibres over prime numbers that are varieties over finite fields, together with a generic fibre over $\mathbb{Q}$. Comparing these fibres links the geometry of curves and abelian varieties to their arithmetic. Local solvability need not guarantee global solvability; the passage between these questions is a central source of structure and difficulty.

On an integral scheme $X$, a Cartier divisor is represented on an open cover $U_i$ by nonzero rational functions $f_i$, with $f_i/f_j$ a regular unit on every overlap. Changing the local equation by a regular unit leaves the divisor unchanged. A principal Cartier divisor uses one global nonzero rational function on all the charts.

The transition ratios glue the local rank-one modules $f_i^{-1}{\mathcal{O}}_{U_i}$ into the divisor line bundle ${\mathcal{O}}_X(D)$. If each $f_i$ is regular, it cuts out an effective Cartier divisor. Allowing poles gives signed divisor data.

On the projective line, the coordinate $t$ defines the principal Cartier divisor $[0]-[\infty ]$: it has one simple zero at zero and one simple pole at infinity. For a general scheme, the analogous definition uses invertible elements of the sheaf of total quotient rings, so nonzero elements must be distinguished from non-zero-divisors.^[ref]

The category of affine schemes has affine schemes as objects and morphisms of schemes as arrows. The contravariant functor $A\mapsto \mathrm{Spec}A$ gives an equivalencewith the opposite of the category of commutative rings. Precisely, every ring homomorphism $A\to B$ corresponds to a unique map $\mathrm{Spec}B\to \mathrm{Spec}A$, and every morphism of schemes between affine schemes arises this way.^[ref]

For example, $k[t]\to k[x]$, $t\mapsto x^2$, induces the squaring map of affine lines. A map from $\mathrm{Spec}k$ to $\mathrm{Spec}A$ instead corresponds to a $k$-algebra map $A\to k$, expressing a rational point. A Cartesian product of affine $k$-schemes corresponds to a tensor product of commutative algebras: $\mathrm{Spec}A{\times }_{\mathrm{Spec}k}\mathrm{Spec}B\cong \mathrm{Spec}(A{\otimes }_{k}B)$. Reversing arrows explains why geometric products become algebraic coproducts.

For every scheme $X$, projection $p:X\times {\mathbb{P}}^n\to X$ is a closed map, and remains so after every base change. Hence projective space is universally closed over its base. The assertion concerns the underlying scheme points, not just rational points over a non-algebraically closed field.^[ref]

The existence part of the valuative criterion explains the role of projective space. Given a valuation ring $V$ with fraction field $K$, a $K$-point $[a_0:\cdots :a_n]$ can be rescaled so that every coordinate belongs to $V$ and one coordinate is a unit. Indeed, among the finitely many nonzero coordinates choose $a_j$ whose principal fractional ideal contains all the others; then $a_i/a_j\in V$ and $a_j/a_j=1$. These coordinates define the required morphism of schemes $\mathrm{Spec}V\to {\mathbb{P}}^n$. Since projective space is also separated and of finite type, this proves that projection is a proper morphism.

On a Noetherian scheme, one can instead use finitely many homogeneous equations and the Projective Nullstellensatz to express the image as closed rank-defect conditions on graded multiplication map for a projective fiber. An arbitrary affine base need not give a finitely generated defining ideal, so that finite-equation hypothesis must not be assumed in the general statement.

A closed subscheme of $X$ is a scheme $Z$ together with a closed immersion $Z\hookrightarrow X$, usually identified with its image and its quotient structure sheaf.

For a commutative ring $R$, remove the section $[1:0:0]$ from ${\mathbb{P}}_R^2$. The two remaining standard charts have coordinate rings $R[a,b]$ and $R[a/b,b^{-1}]$ inside the Laurent polynomial ring $R[a,b,b^{-1}]$. Their intersection consists of constants: their monomial exponent regions are $i\ge 0,j\ge 0$ and $i\ge 0,j\le -i$, respectively. The Čech cohomology in degree one is free over $R$ on the missing monomials $a^i{b}^{-k}$ with $1\le k<i$. Equivalently its basis consists of the classes of $x_0^{m+n}/(x_1^m{x}_2^n)$ for $m,n\ge 1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are affine schemes, so this computes sheaf cohomology. If $R$ is nonzero, the first sheaf cohomology is nonzero and this scheme is not an affine scheme.

A complex algebraic variety is an algebraic variety over the complex numbers. In the scheme convention used here, it is a reduced separated scheme of finite type over $\mathbb{C}$; irreducibility is an additional condition when needed. Affine examples are zero sets of complex polynomials equipped with their regular functions.

For example, $xy=0$ is the union of two complex coordinate lines, while $y^2=x^3$ is an irreducible singular curve. The smooth locus of a complex variety has the structure of a complex manifold, but a singular point need not be smooth as a complex analytic space.

One must also distinguish topologies. The complex affine line is $\mathbb{C}$ as a point set; its proper Zariski-closed subsets are finite, whereas its usual analytic topology has many more closed and open subsets. Passing from an algebraic variety to its analytic space retains geometric information while changing which local functions and open sets are allowed.

For a scheme over $\mathbb{R}$, its base change of a morphism of schemes to $\mathbb{C}$ has scheme-theoretic fibre over a point $p$ equal to $\mathrm{Spec}(\kappa (p){\otimes }_{\mathbb{R}}\mathbb{C})$. This is the spectrum of $\kappa (p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the residue field, and otherwise one point with a quadratic field extension. In both cases it is reduced, because the quadratic polynomial has no repeated root in characteristic zero. The distinction applies to generic points as well as closed points.

If a nonempty scheme $X$ has no nontrivial idempotent in $\Gamma (X,{\mathcal{O}}_X)$, then its underlying topological space is connected. Indeed, a decomposition into two nonempty clopen subsets would produce the global regular function equal to zero on one component and one on the other. In particular, $\Gamma (X,{\mathcal{O}}_X)$ being a field implies connectedness.

The dimension of a scheme is the supremum of the lengths of strict chains of irreducible closed subsets of its underlying topological space. On an affine scheme it is the Krull dimension of its coordinate ring. Taking the reduced subscheme leaves the dimension unchanged.

For a Noetherian integral scheme $X$ regular in codimension one and an integer $n\ge 1$,The first summand is the pullback of base Weil divisors; the second is the hyperplane divisor in the projective fibre.

Remove $H$ to obtain $X\times {\mathbb{A}}^n$. The localization sequence for the divisor class group, together with invariance under adjoining affine-space coordinates, shows that the base classes and $H$ generate. Restriction to the generic fibre ${\mathbb{P}}_{K(X)}^n$ sends a base class to zero and $H$ to its degree-one generator, so no nonzero multiple of $H$ can be a base class. This detects the extra summand geometrically by fibre degree.

The positive-dimension hypothesis is essential: ${\mathbb{P}}^0$ is a point and $\mathrm{Cl}(X\times {\mathbb{P}}^0)=\mathrm{Cl}(X)$, with no hyperplane summand.

A morphism of schemes is étale when it is flat, unramified and locally of finite presentation. For a morphism between smooth varieties of the same dimension, an invertible differential gives this property locally. A finite étale morphism over an algebraically closed field has reduced finite fibres, and their cardinalities equal its degree when the source and target are connected.