Spectrum of a commutative ring ($\mathrm{Spec}A$)

The spectrum $\mathrm{Spec}A$ of a commutative ring $A$ is the set of its prime ideals, with closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and a structure sheaf whose sections locally look like fractions.

Contraction sends the zero prime ideal to zero. A complex closed point $(x-z)$ contracts to $(x-z)$ when $z$ is real, and to $((x-\mathrm{Re}z{)}^2+(\mathrm{Im}z{)}^2)$ otherwise. Thus the map on the spectrum of a commutative ring is surjective and identifies each nonreal conjugate pair. Both spectra have only finite proper Zariski-closed sets, so this is also the topological quotient by complex conjugation.

For a reduced ring with a nonempty disconnected spectrum of a commutative ring, choose a separation $V(I)\bigsqcup V(J)$. Disjointness gives $I+J=R$; covering the spectrum gives $IJ\subseteq \sqrt{(0)}=0$. For comaximal ideals, $I\cap J=IJ$. The Chinese remainder theorem then supplies a product of two nonzero quotient rings. Equivalently, if $i+j=1$ with $i\in I$, $j\in J$, then $i,j$ are nontrivial orthogonal idempotents. This connects topology of the spectrum to ring product decomposition by an idempotent without a Noetherian assumption.

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

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.

Suppose a scheme is the scheme-theoretic union of affine closed subschemes $Y=\mathrm{Spec}A$ and $Z=\mathrm{Spec}B$, meaning their ideal sheaves have zero intersection. The intersection $W$ is closed in both, hence affine with coordinate ring $C$ and surjections $A\to C$, $B\to C$. Then the whole scheme is $\mathrm{Spec}(A{\times }_{C}B)$. Both projections of this fiber product are surjective; their kernels have zero intersection and product, and their quotient by the sum is $C$. Thus its spectrum has the same closed pieces, intersection and topology. The structure sheaf is the kernel of $i_{\ast }{\mathcal{O}}_Y\oplus j_{\ast }{\mathcal{O}}_Z\to k_{\ast }{\mathcal{O}}_W$ on both spaces, identifying their locally ringed structures. A reduced Noetherian scheme has finitely many irreducible components, so induction proves it is affine if and only if its reduced irreducible components are affine. The intersection $W$ need not be reduced.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

If finitely many $X_{f_i}$ cover $X$, the morphism ${\mathcal{O}}_X^r\to {\mathcal{O}}_X$ with coefficients $f_i$ is onto. If its kernel has vanishing first sheaf cohomology, the long exact sequence in sheaf cohomology lifts the global section $1$ to coefficients $g_i$. This turns a geometric cover into a unit ideal in the ring of global sections.

Let $X$ be an algebraic variety over an algebraically closed field $k$, and assume $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. Given a closed point $P$, choose an affine open subscheme $U$ containing it and give $Y=X\setminus U$ its reduced induced subscheme structure. Since $P\notin Y$, evaluation at $P$ is a surjective morphism of sheaveswith kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. Here $k_P$ is the skyscraper sheaf at the closed rational point, and the kernel is again a coherent ideal sheaf. The long exact sequence in sheaf cohomology therefore lifts $1\in k_P$ to a global section $f$ of ${\mathcal{I}}_Y$.

Viewed as a global regular function, $f$ vanishes on $Y$ and has $f(P)=1$. Thus $P\in X_f\subseteq U$, and $X_f=U_{f{|}_U}$ is a principal open subscheme of an affine scheme, hence affine. Such neighborhoods cover $X$: every nonempty closed subset of an algebraic variety contains a closed point. This is the local step in the cohomological criterion for affineness; the closed-point hypothesis makes the coherent skyscraper argument valid.

Let $X$ be a Noetherian scheme and suppose $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. For a coherent sheaf $\mathcal{F}\subseteq {\mathcal{O}}_X^r$, projection onto the last coordinate has image an ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ and kernel ${\mathcal{F}}^{\prime }\subseteq {\mathcal{O}}_X^{r-1}$. Both are coherent sheaves, sois a short exact sequence of sheaves. Thus $\mathcal{F}$ is a sheaf extension of the quotient ideal $\mathcal{I}$ by the lower-rank submodule ${\mathcal{F}}^{\prime }$, with this order of the two terms.

For $r=1$, the assertion is exactly the assumed vanishing for coherent ideal sheaves. For larger $r$, mathematical induction gives $H^1(X,{\mathcal{F}}^{\prime })=0$, and the long exact sequence in sheaf cohomology places $H^1(X,\mathcal{F})$ between this zero group and $H^1(X,\mathcal{I})=0$. Hence $H^1(X,\mathcal{F})=0$.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

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 open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.

For $f\in A$, the principal open subset $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ of $\mathrm{Spec}A$ is the affine scheme $\mathrm{Spec}{A}_f$.

In an affine variety with coordinate ring $A$, every dense open subset $U$ contains a dense principal open subset. Write its closed complement as $V(I)$. Density says that $I$ is contained in none of the finitely many minimal prime ideals of the reduced ring $A$. Prime avoidance supplies $s\in I$ outside all these primes. Thus $D(s)\subseteq U$, and $s$ is a non-zero-divisor. Noetherian Zariski topology also makes $U$ a finite union of principal opens; one can add $D(s)$ to this cover. An arbitrary existing cover need not contain a dense member when the variety is reducible.

The affine line over a field $k$ is ${\mathbb{A}}_k^1=\mathrm{Spec}k[t]$.

The one-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

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.

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The two-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\mathrm{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, whilewhich has basis represented by $x^{-a}{y}^{-b}$ for $a,b\ge 1$.

The punctured affine plane $U={\mathbb{A}}_k^2\setminus \{(0,0)\}$ is not an affine scheme. On its cover $D(x)\cup D(y)$, global sections are $k[x,y{]}_x\cap k[x,y{]}_y=k[x,y]$ inside the field of fractions. Coprimality of $x$ and $y$ proves this equality. If $U$ were affine, its inclusion into ${\mathbb{A}}_k^2$ would be an isomorphism by affine scheme reconstruction from global sections, contradicting the omitted point.

Affine three-space is $\mathrm{Spec}k[x_1,x_2,x_3]$.

The three-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

For $U={\mathbb{A}}_k^3\setminus \{0\}$,andThe last group has basis represented by $x_1^{-a}{x}_2^{-b}{x}_3^{-c}$ for $a,b,c\ge 1$, and all groups in degrees at least three vanish.