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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.

A ring homomorphism is understood to preserve $1$. Start with $\phi :A\to B$. Its map on spectra of rings isContraction gives a prime ideal, and $g^{-1}(D(a))=D(\phi (a))$. Thus $g$ is continuous for the Zariski topology. On each principal open subscheme, the structure sheaf map is the ring homomorphismThese maps commute with restrictions, so they define a sheaf morphism ${\mathcal{O}}_X\to g_{\ast }{\mathcal{O}}_Y$. At $\mathfrak{q}$, with $\mathfrak{p}={\phi }^{-1}(\mathfrak{q})$, its stalk map is $A_{\mathfrak{p}}\to B_{\mathfrak{q}}$. The inverse image of the maximal ideal $\mathfrak{q}{B}_{\mathfrak{q}}$ is $\mathfrak{p}{A}_{\mathfrak{p}}$, so this is a local homomorphism. We have constructed a morphism of locally ringed spaces.

Conversely, let $(g,g^{\#})$ be a morphism of locally ringed spaces. Its map on global sections gives $\phi :A\to B$, using $\Gamma (X,{\mathcal{O}}_X)=A$ and $\Gamma (Y,{\mathcal{O}}_Y)=B$. Fix $\mathfrak{q}\in Y$ and put $\mathfrak{p}=g(\mathfrak{q})$. Compatibility with the stalk maps and the local homomorphism property giveConsequently $\mathfrak{p}={\phi }^{-1}(\mathfrak{q})$, so the underlying map is forced. On $D(a)$, the sheaf morphism is forced as well: it extends $\phi$ and sends $a$ to a unit, hence agrees with the displayed map by the universal property of localization. The principal open subschemes form a basis, so the entire sheaf morphism is determined. Taking global sections of the construction recovers $\phi$. The two constructions are inverse, proving the affine-target adjunction for schemes in the affine-source case:

To describe the real affine plane scheme points, write $R=\mathbb{R}[t_1,t_2]$. It is a unique factorization domain of Krull dimension two. Its points are exactly the following prime ideals:

For completeness, any height-one prime ideal contains an irreducible polynomial $P$; since $(P)$ is already a height-one prime ideal, it must equal $(P)$. Every remaining nonzero prime ideal has height two and is maximal, by Krull dimension. For a nonreal pair, evaluation generates all of $\mathbb{C}$ over $\mathbb{R}$, so its kernel is maximal. Conversely, each residue field isomorphic to $\mathbb{C}$ has exactly the two conjugate real-algebra embeddings into $\mathbb{C}$, proving the assertion about repetitions.

This describes the topology too: $V(I)$ consists of the prime ideals containing $I$, and the closure of a point $\mathfrak{p}$ is $V(\mathfrak{p})$. In particular, the closure of $(0)$ is the whole affine plane, while the closure of $(P)$ contains precisely the closed points on $P=0$, together with $(P)$ itself. The spectrum is much larger than the set ${\mathbb{R}}^2$. For example, $(t_1^2+1)$ is a height-one point although its curve has no real points. Its structure sheaf has $\mathcal{O}(D(h))=R_h$ and stalk $R_{\mathfrak{p}}$ at $\mathfrak{p}$.

The induced morphism of schemesis contraction of prime ideals, with the structure sheaf maps given above. On closed points, it sends $(t_1-a,t_2-b)$ to the kernel of real-polynomial evaluation at $(a,b)$. A real closed point has one complex point above it; a nonreal closed point has two, interchanged by complex conjugation. The source generic point maps to the target generic point. The source height-one points are generated by irreducible complex polynomials $Q$. Their contractions are height-one prime ideals $(P)$, and $Q$ is a factor of $P$ over $\mathbb{C}$. An irreducible real $P$ either stays irreducible over $\mathbb{C}$ or splits into two distinct conjugate irreducible factors. Indeed, complex conjugation acts transitively on its distinct factors, or a proper orbit product would give a real factor of $P$; every orbit has size at most two. Repeated factors are excluded by separability in characteristic zero. Thus one or two height-one points lie above $(P)$.

The complexification fibres of a real scheme give a uniform description of all scheme-theoretic fibres, including the nonclosed points, is especially useful. The extension of coordinate rings isa free $R$-module with basis $1,s$. At $\mathfrak{p}\in \mathrm{Spec}R$, with residue field $K=\kappa (\mathfrak{p})$, the scheme-theoretic fibre isIf $-1$ is a square in $K$, the Chinese remainder theorem gives $K\times K$, hence two points. Otherwise it is a quadratic field extension, hence one point. The polynomial has no repeated root in characteristic zero, so all these scheme-theoretic fibres are reduced schemes. This also proves surjectivity. Conjugation acts on each two-point fibre by exchanging its points and fixes each one-point fibre. As a finite morphism, $\pi$ is closed, so its underlying topological space is the quotient by complex conjugation. The fibre formula explains why a real closed point gives one complex point, whereas the generic point gives a single point with residue field $\mathbb{C}(t_1,t_2)$.

The point to retain in the direct-image theorem is quasi-compactness of inverse images of affine opens and of overlaps; no separation assumption has been supplied. The underlying space of a Noetherian scheme is Noetherian, so every open subset of $X$ is quasi-compact. Fix an affine open subscheme $V=\mathrm{Spec}A$ of $Y$, put $W=f^{-1}V$, and choose a finite open cover by affine open subschemes $W={\bigcup }_i{U}_i$. For each pair $i,j$, choose a finite affine cover $(W_{ij\ell }{)}_{\ell }$ of $U_i\cap U_j$. Empty overlaps contribute no terms.

Write $M=\Gamma (W,\mathcal{F})$. The sheaf gluing axiom gives the exact sequencewhere $\delta$ is the difference of the two restrictions to each overlap chart. All terms are $A$-modules through $f$. For $a\in A$, the inverse image of $D(a)$ cuts each $U_i$ and $W_{ij\ell }$ by a principal open subscheme. Because $\mathcal{F}$ is a quasi-coherent sheaf, sections on these smaller affine charts are the corresponding module localizations at $a$. Exactness of localization and its commutation with finite products now identify the localized equalizer with the equalizer for the restricted cover. ThusThe isomorphisms respect restrictions, proving $(f_{\ast }\mathcal{F}){|}_V\cong \tilde{M}$ on the basis of principal open subschemes. Since $V$ was arbitrary, $f_{\ast }\mathcal{F}$ is a quasi-coherent sheaf. The finite covers of overlaps are what allow the proof to work for nonseparated Noetherian schemes; this is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism.

For a quasi-coherent sheaf which is not coherent but has coherent direct image, use $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ andThis infinite negative-twist sum with zero global sections is quasi-coherent: on each standard affine chart it is the sheaf associated with a direct sum of free rank-one modules. Its stalk at every point, modulo the maximal ideal, is an infinite-dimensional vector space. A finitely generated module would have a finite-dimensional quotient, so $\mathcal{F}$ is not a coherent sheaf.

Nevertheless, $\Gamma ({\mathbb{P}}^1,\mathcal{O}(-1))=0$. This follows from cohomology of twisting sheaves on projective space, or directly by gluing on the two standard affine charts: if $z=t_1/t_0$, a section is a polynomial $p(z)$ on the first chart and $q(z^{-1})$ on the second with $p(z)=z^{-1}q(z^{-1})$, which forces both to vanish. Global sections commute with this direct sum: they are the kernel of the difference map for the two-chart cover, and direct sums commute with that finite equalizer of modules. Hencewhich is coherent on $\mathrm{Spec}k$. Both schemes in this example are Noetherian.

For a finite morphism, such an example is impossible. On an affine open subscheme $V=\mathrm{Spec}A$ of the target, its inverse image is $\mathrm{Spec}B$, with $B$ a finite $A$-module. Write $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$. Its direct image corresponds to $M$ considered as an $A$-module. If the direct image is coherent, $M$ is finitely generated over $A$. The same generators also generate it over $B$, because $A$ acts through $B$. Since $B$ is Noetherian, $\tilde{M}$ is coherent. Conversely, a finite set of $B$-generators combined with a finite set of $A$-generators of $B$ gives finitely many $A$-generators of $M$. Thus coherence reflected by finite direct image gives the stronger equivalence

Finally, let $Y=\mathrm{Spec}k[x,y]$, let $X=Y\setminus \{(x,y)\}$ be the punctured affine plane, and let $j:X\hookrightarrow Y$ be the open immersion. Both are integral schemes, but $j$ is not an isomorphism of schemes because it omits a point. The cover $X=D(x)\cup D(y)$ givesinside the field of fractions $k(x,y)$. Indeed, in a reduced fraction, membership in the first localization forces every denominator factor to be associated to $x$, while membership in the second forces it to be associated to $y$. Unique factorization and coprimality force the denominator to be a unit. By the theorem just proved, $j_{\ast }{\mathcal{O}}_X$ is quasi-coherent on the affine $Y$, hence determined by this module of global sections. The natural map ${\mathcal{O}}_Y\to j_{\ast }{\mathcal{O}}_X$ corresponds to the identity of $k[x,y]$, soa coherent sheaf. This is a concrete case of codimension-two extension of regular functions on a normal variety.

Let $\mathfrak{m}$ be the maximal ideal of the local ring $A$. Every open subset of $\mathrm{Spec}A$ containing the closed point $\mathfrak{m}$ is the whole spectrum of a commutative ring: it contains a principal open subscheme $D(a)$ with $a\notin \mathfrak{m}$, and that $a$ is a unit, so $D(a)=\mathrm{Spec}A$.

Given $f:\mathrm{Spec}A\to {\mathbb{P}}_{\mathbb{Z}}^n$, choose a standard affine open subscheme $D_{+}(x_i)$ containing $f(\mathfrak{m})$. Its preimage is consequently all of $\mathrm{Spec}A$. The affine-target adjunction for schemes expresses $f$ in this chart by elements $b_j\in A$ for $j\ne i$, the images of $x_j/x_i$. It is represented by homogeneous coordinates with $a_i=1$ and $a_j=b_j$.

Conversely, a tuple $(a_0,\dots ,a_n)$ with some $a_i\in A^{\times }$ defines a morphism of schemes into $D_{+}(x_i)$ by $x_j/x_i\mapsto a_j/a_i$. Choosing another unit entry gives the same morphism of schemes, since the usual projective space transition functions identify the ratios. Multiplying all entries by one unit does not change any ratio. If two such tuples define the same morphism of schemes, choose a unit entry $a_i$ in the first and a unit entry $a_j^{\prime }$ in the second. In the second chart, the function $x_i/x_j$ pulls back to $a_i^{\prime }/a_j^{\prime }$. Because the whole map lies in $D_{+}(x_i)$, this ratio is a unit, so $a_i^{\prime }$ is a unit too. Equality in this chart gives $a_j/a_i=a_j^{\prime }/a_i^{\prime }$ for every $j$, hence $a_j^{\prime }=(a_i^{\prime }/a_i)a_j$.

Thus the correspondence is exactlyThis is the projective coordinates over a local ring description.

For a general ring, the key open-neighbourhood argument fails. Even a tuple generating the unit ideal need not have any unit entry. For example, over $A=k\times k$ the pair $((1,0),(0,1))$ defines a map to ${\mathbb{P}}^1$ whose two points have images $[1:0]$ and $[0:1]$. Neither coordinate is a unit, and no common unit multiple changes that fact. The map lies in no single standard chart. More generally, maps into projective space correspond to invertible sheaf quotients of $A^{n+1}$; the quotient need not be a free rank-one module outside the local ring case.

Put $A=k[t_1,t_2]$. The punctured affine plane is covered by the principal open subschemes $D(t_1)$ and $D(t_2)$. Since $A$ is an integral domain, restriction identifies its global sections withThis intersection is $A$: if $a/t_1^r=b/t_2^s$, then $t_2^{s}a=t_1^{r}b$, and unique factorization forces $t_1^r$ to divide $a$. Therefore the inclusion $j:X\hookrightarrow \mathrm{Spec}A$ induces the identity on global regular functions.

If $X$ were an affine scheme, affine scheme reconstruction from global sections would make $j$ an isomorphism of schemes, because its global-section map is an isomorphism. But its image omits the closed point $(t_1,t_2)$. The punctured affine plane is therefore not affine. This is a useful example of global regular functions failing to distinguish a nonaffine open subscheme from its affine ambient space.

On the principal open subscheme $D(a)\subseteq \mathrm{Spec}A$, the induced morphism of sheaves is the ring homomorphismIf $\alpha$ is injective, this map is injective by exactness of localization: it is the localization of the injective $A$-module map $A\to B$. Explicitly, if the displayed fraction vanishes, some power of $\alpha (a)$ annihilates $\alpha (b)$; injectivity gives $a^{m}b=0$, so the original fraction also vanishes. The principal opens form an open basis, so the kernel sheaf is zero on a basis and hence zero everywhere. Conversely, an injective morphism of sheaves induces an injection on global sections. On the whole affine scheme these sections are $A\to B$, exactly $\alpha$. Thus

For an affine open subscheme $U=\mathrm{Spec}R$, quasi-coherence gives $\mathcal{I}{|}_U=\tilde{J}$ for an ideal $J\subseteq R$. Define $Z_U=\mathrm{Spec}(R/J)$ with its usual closed immersion into $U$. On a principal open subscheme $D(a)$, its restriction is $\mathrm{Spec}(R_a/J_a)$, because localization commutes with taking this quotient. These local constructions therefore agree on overlaps and glue.

The resulting underlying closed set is $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, locally $V(J)$. With inclusion $i:Z\hookrightarrow X$, its structure sheaf isAt a point corresponding to $\mathfrak{p}\supseteq J$, its stalk is $R_{\mathfrak{p}}/J_{\mathfrak{p}}$, the local ring of $\mathrm{Spec}(R/J)$. Thus the glued ringed space is a scheme and $i$ is a closed immersion, with ideal sheaf of a closed subscheme exactly $\mathcal{I}$. This is the construction of a closed subscheme from a quasi-coherent ideal. If $\mathcal{I}={\mathcal{O}}_X$, the construction gives the empty scheme; if $\mathcal{I}=0$, it gives $X$.

Write the affine scheme as $U=\mathrm{Spec}B$. A point $x\in U$ corresponds to a prime ideal $\mathfrak{p}\subset B$, and its local ring is ${\mathcal{O}}_{X,x}\cong B_{\mathfrak{p}}$ with maximal ideal $\mathfrak{p}{B}_{\mathfrak{p}}$. HenceThe last condition defines the principal open subscheme $D(\bar{f})$, soEvery point of the scheme $X$ has such an affine neighbourhood. Thus $X_f$ is locally open, and hence is open in the Zariski topology.

Let $X={\bigcup }_{i=1}^r{U}_i$ be the stated finite affine cover. By the description of sections on a principal open subscheme, after increasing denominators separately we may writefor some $a_i\in \Gamma (U_i,{\mathcal{O}}_X)$. Multiplying the numerators by powers of $f$ lets us use one exponent $n$ for every $i$.

On $W_{ij}=U_i\cap U_j$, the section $a_i-a_j$ vanishes after restriction to $(W_{ij}{)}_f$. Each $W_{ij}$ is quasi-compact, so part (b) supplies $m_{ij}$ with $f^{m_{ij}}(a_i-a_j)=0$ on $W_{ij}$. Choose one $m$ valid for all the finitely many pairs. The sections $f^m{a}_i$ now agree on every overlap, and the gluing axiom of a sheaf of rings produces $a\in \Gamma (X,{\mathcal{O}}_X)$ with $a{|}_{U_i}=f^m{a}_i$. On $X_f$,Thus some power of $f$ times $b$ extends to a global regular function on $X$.

The direct image sheaf is determined on the basis consisting of principal open subschemes $D(b)\subseteq \mathrm{Spec}B$. Their inverse images areand henceOn the other hand, localization of the restricted-scalar $B$-module givesThese isomorphisms commute with restriction to smaller principal opens, so they define the equality of quasi-coherent sheaves

Write $A=k[x,y]$. If $\dim Z=1$, its prime ideal has height one and is generated by an irreducible polynomial $f$, because $A$ is a unique factorization domain. Then $U=D(f)$ is a principal open subscheme andIf $\dim Z=0$, its complement has codimension two. Since ${\mathbb{A}}_k^2$ is normal, regular functions extend across that codimension-two subset, givingThis includes the calculation for the punctured affine plane. If $Z={\mathbb{A}}_k^2$ has dimension two, then $U$ is empty and its ring of sections is the zero ring.

Let $A=k[x_1,x_2,x_3]$ and cover the punctured affine three-space $X$ by the three principal opens $D(x_i)$. Every finite intersection is affine, so the acyclic cover theorem identifies sheaf cohomology with the cohomology ofThe augmented complex has zeroth cohomology $A$ and first cohomology zero. Its second cohomology iswith $k$-basis represented by $x_1^{-a}{x}_2^{-b}{x}_3^{-c}$ for $a,b,c\ge 1$. There are no higher Čech terms. Hence

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.

Write $A=k[x,y]$ and $\mathfrak{m}=(x,y)$. Since $f(0,0)\ne 0$, the principal open subscheme $D(f)=\mathrm{Spec}{A}_f$ contains $\mathfrak{m}$, andCover $V$ by the two affine opens $D(xf)$ and $D(yf)$, whose intersection is $D(xyf)$. The degree-zero part of the resulting Čech cohomology complex givesinside the fraction field of $A$. The last equality follows because $A_f$ is a unique factorization domain and a rational function regular after localizing at both $x$ and $y$ has no possible prime factor left in its denominator.

The same affine cover is acyclic, so its degree-one Čech group computes sheaf cohomology and givesBefore localizing at $f$, the quotienthas the $k$-basisWriting $f=c+h$ with $c=f(0,0)\in k^{\times }$ and $h\in (x,y)$, multiplication by $h$ is locally nilpotent on $Q$: for each negative monomial, a sufficiently high power of $(x,y)$ moves every term into $A_x+A_y$. Hence $c+h$ acts invertibly on $Q$ by a finite geometric series on each element. Localizing at $f$ therefore leaves $Q$ unchanged, andThe displayed infinite basis proves that this vector space is infinite-dimensional.

Write $X=\mathrm{Spec}B$, $Y=\mathrm{Spec}A$, and let the morphism correspond to a homomorphism $\phi :A\to B$. Put $I=\ker \phi$ andThe injection $A/I\hookrightarrow B$ gives $X\to Z$, and the quotient $A\to A/I$ gives a closed immersion $Z\hookrightarrow Y$, so $f$ factors through $Z$.

If $f$ factors through another closed subscheme $Z^{\prime }=\mathrm{Spec}(A/J)$, then $J\subseteq \ker \phi =I$. The resulting quotient $A/J\to A/I$ induces the unique factorization $Z\to Z^{\prime }\to Y$. Thus $Z$ is the scheme-theoretic image.

It remains to identify its underlying set. A principal open subscheme $D(a)\subseteq \mathrm{Spec}A$ misses $f(X)$ exactly when $D(\phi (a))$ is empty, equivalently when $\phi (a)$ is a nilpotent element. Hence the ideal of functions vanishing set-theoretically on $f(X)$ has radical $\sqrt{\ker \phi }$. The closure is thereforeas required.

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 Zariski topology on an affine scheme $\mathrm{Spec}A$ has closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ for ideals $I\subseteq A$. Its basic open sets are the principal open subschemes $D(f)$.