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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 $\mathbb{C}(t_1,t_2)$.

An irreducible scheme is a nonempty scheme whose underlying topological space cannot be expressed as the union of two proper closed subsets. Equivalently, any two nonempty open subsets meet. A reduced scheme is one whose local rings have no nonzero nilpotent elements. Equivalently, every affine open subscheme has a reduced ring of regular functions. We may define an integral scheme as a nonempty scheme for which the coordinate ring of every nonempty affine open subscheme is an integral domain. We shall show that this is equivalent to being reduced and irreducible. The nonempty convention matters: the empty scheme is reduced but is not irreducible or integral.

For a commutative ring $A$, the spectrum of a ring is reduced exactly when $A$ is a reduced ring. One direction follows since localization preserves reducedness. Conversely, if $a$ is a nonzero nilpotent element, choose a prime ideal containing its proper annihilator. Then $a/1$ cannot vanish at that localization, contradicting reducedness of its local ring.

The spectrum of a ring is irreducible exactly when its nilradical $N=\sqrt{(0)}$ is a prime ideal. To see the essential implication directly, if $ab\in N$, then $D(a)\cap D(b)=D(ab)$ is empty. Irreducibility forces $D(a)$ or $D(b)$ to be empty, hence $a\in N$ or $b\in N$. Also $N$ is proper because the spectrum is nonempty. Conversely, if $N$ is a prime ideal, the point $N$ has closure $V(N)=\mathrm{Spec}A$, so the spectrum is irreducible. Combining the two criteria gives

Now suppose $X$ is reduced and irreducible. Every nonempty affine open subscheme is also reduced and irreducible: irreducibility passes to nonempty open subsets, because their nonempty open subsets are nonempty opens of $X$. The affine criterion makes each coordinate ring an integral domain, so $X$ is integral. Conversely, suppose all its nonempty affine coordinate rings are integral domains. Their localizations show that $X$ is reduced. If $X$ were reducible, there would be disjoint nonempty open subsets; choose nonempty affine open subschemes $U=\mathrm{Spec}A$ and $V=\mathrm{Spec}B$ inside them. Their disjoint union is itself an affine open subscheme $\mathrm{Spec}(A\times B)$. Since $A,B$ are nonzero, $(1,0)(0,1)=0$ contradicts the domain condition. Therefore

The generic point of an integral scheme $X$ is the unique point $\eta$ with $\overline{\{\eta \}}=X$. For existence, take a nonempty affine open subscheme $V=\mathrm{Spec}A$. Its point $(0)$ has closure containing $V$, which is dense in $X$, so its closure in $X$ is all of $X$. For uniqueness, both proposed generic points lie in every nonempty open subset, hence in $V$, where the only dense point is $(0)$. The function field isHere the stalk description shows that the field of fractions is independent of the choice of nonempty affine open subscheme.

Every nonempty open subscheme $U$ contains $\eta$, so taking a germ there defines $\Gamma (U,{\mathcal{O}}_X)\to K(X)$. If a section has zero germ, restrict it to any nonempty affine open subscheme $V\subseteq U$. Its image in $\mathrm{Frac}\Gamma (V,{\mathcal{O}}_X)$ is zero. Since this coordinate ring is an integral domain, the section is zero on $V$. Such affines cover $U$, and the sheaf gluing axiom makes the section zero on $U$. Thus the generic-point embedding of regular functions is

For a nonreduced reducible fibre between integral schemes, take both source and target to be the affine line over $\mathbb{C}$, and use the ring homomorphismBoth coordinate rings are integral domains, so both schemes are integral. At the target closed point $t=0$, the scheme-theoretic fibre has ringby the Chinese remainder theorem. Its underlying space has two distinct closed points, so it is reducible. The class of $x(x-1)$ is nonzero but has square zero, so it is not reduced. Even a morphism between integral schemes can have a fibre consisting of two nonreduced double points.

For an affine variety $V\subseteq {\mathbb{A}}_k^N$, one construction defines ${\mathcal{O}}_V(U)$ as the functions $U\to k$ which locally have the form $p/q$, where $p,q$ are elements of the coordinate ring $k[V]$ and $q$ does not vanish on the neighbourhood in question. Restrictions are restrictions of functions, and locality gives the sheaf axioms. In particularThe second formula identifies its local rings.

In the classical convention, an algebraic variety over $k$ is an irreducible ringed space with a sheaf of $k$-valued functions, admitting a finite open cover by spaces isomorphic to affine varieties, and satisfying the separated variety condition that its diagonal is closed in $X\times X$. Its topology is Noetherian. One can instead allow reducible reduced varieties; the arguments below still work with the denominator-clearing version of the localization argument. A morphism of varieties is a continuous map whose pullback takes local regular functions to regular functions, equivalently a morphism of locally ringed $k$-spaces. The maps on local rings are local because a function nonzero at the image point stays nonzero at the source point.

An affine variety supplies its own finite affine cover, is Noetherian, and is separated: its diagonal in $V\times V$ is cut out by the differences of corresponding coordinates. A regular map between affine varieties pulls coordinate functions back to regular functions. It is continuous because the inverse image of any polynomial zero set is the zero set of the pulled-back regular functions; on affine charts such zero sets are closed. Pulling back a locally represented fraction $p/q$ gives a regular fraction wherever its denominator is nonzero. Hence a regular map is a morphism of varieties in this definition.

For a morphism $\varphi :X\to Y$ which is an isomorphism over each member $U_i$ of an open cover of $Y$, every fibre contains exactly one point. Thus $\varphi$ is bijective. Its inverse is continuous and regular on every $U_i$, since there it is the given inverse of the local isomorphism. These inverses agree on overlaps, being inverses of the same map. They glue to a global inverse morphism of varieties. Therefore being an isomorphism is local on the target.

Let $A=\Gamma (X,{\mathcal{O}}_X)$, and let $Y$ be affine with coordinate ring $B$. Given a $k$-algebra homomorphism $\alpha :B\to A$, choose a presentation $B=k[y_1,\dots ,y_N]/I$. The maplands in $Y$ because all the equations in $I$ become zero functions. Its coordinates are global regular functions, so it is a morphism on every affine chart of $X$, and therefore globally. On a target neighbourhood where a fraction $b/c$ is defined its pullback is $\alpha (b)/\alpha (c)$, proving that the induced map on global sections is exactly $\alpha$. This also proves independence of the chosen generators and uniqueness. This is the affine-target adjunction for varieties.

For $f\in A$, the natural map $A_f\to \Gamma (X_f,{\mathcal{O}}_X)$ sends $a/f^N$ to that regular function. In the irreducible convention, if $f\ne 0$, $X_f$ is dense, so this map is injective: a global regular function vanishing there vanishes everywhere. Cover $X$ by finitely many affine charts $U_j$. A section $s$ on $X_f$ restricts on $U_j\cap X_f=D_{U_j}(f{|}_{U_j})$ to an element of $\Gamma (U_j,{\mathcal{O}}_X{)}_{f{|}_{U_j}}$. A common power $f^N$ clears all these finitely many denominators. The resulting regular sections on $U_j$ agree on the dense principal open in each overlap, hence agree there altogether and glue to a global section $a$. Thus $s=a/f^N$. If $f=0$, both sides are the zero ring of sections on the empty open. ConsequentlyThis is localization of global sections on a principal open. For reduced reducible varieties, equality on a principal open instead means that a sufficiently large power of $f$ annihilates the difference; finitely many charts and overlap refinements allow one common extra power. The same denominator-clearing proof then gives the stated localization isomorphism without a density assumption.

Now suppose ${\sum }_i{g}_i{f}_i=1$ in $A$ and every $X_{f_i}$ is affine. Each $A_{f_i}=\Gamma (X_{f_i},{\mathcal{O}}_X)$ is a finitely generated $k$-algebra. Select finitely many generators, writing them as $a_{ij}/f_i^{N_{ij}}$. Let $B\subseteq A$ be the $k$-subalgebra generated by all $f_i$, all $g_i$, and all $a_{ij}$. It is finitely generated, andThe inclusion from left to right is immediate, while the chosen generators give the reverse inclusion. This construction does not assume that $A$ was finitely generated in advance.

The reduced algebra $B$ is the coordinate ring of an affine variety $Y$ (irreducible when $X$ is). The map $B\hookrightarrow A$ produces $X\to Y$ by the preceding construction. The sets $D_Y(f_i)$ cover $Y$, since their functions generate the unit ideal already in $B$. Their inverse images are $X_{f_i}$, and on them the map is the isomorphism corresponding to $B_{f_i}=A_{f_i}$. The target-local argument above now provesThis is affineness from a unit-ideal principal affine cover. The printed sets are $X_{f_i}$; the missing index in the TeX aid is not a different hypothesis.

A sheaf of modules $\mathcal{F}$ over ${\mathcal{O}}_X$ is quasi-coherent when locally it admits a presentationwith arbitrary indexing sets $I,J$. On an affine chart, sheafifying the corresponding module presentation and using exactness of localization identifies its cokernel with a sheaf $\tilde{N}$. Thus one may equivalently require that locally on affine charts $\mathcal{F}$ is associated with a module. This does not impose finite generation, which belongs to the stronger coherent condition.

On the affine variety $X$ with coordinate ring $A$, put $M=\Gamma (X,\mathcal{F})$. Refine local module charts to a finite principal open cover $X={\bigcup }_{j}D(g_j)$ such that $\mathcal{F}{|}_{D(g_j)}\cong {\tilde{N}}_j$ for an $A_{g_j}$-module $N_j$. This is possible because principal opens form an open basis and $X$ is quasi-compact. On the overlap $D(g_j{g}_k)$ the sections are the corresponding further localization of $N_j$ or $N_k$. The sheaf gluing axiom gives an equalizerwhere the last map is the difference of restrictions. For any $f\in A$, localize this sequence. Exactness of localization and commutation with finite products give exactly the equalizer for the cover $D(f)={\bigcup }_{j}D(f{g}_j)$. HenceThe isomorphisms are canonical and commute with all basic-open restrictions. Since the sections of $\tilde{M}$ on these opens are $M_f$, they identify the two sheaves:Conversely $\tilde{M}$ is quasi-coherent for every $A$-module $M$, since a free presentation of $M$ gives the required sheaf presentation. Also $\Gamma (X,\tilde{M})=M$, and module homomorphisms sheafify, while a sheaf morphism is determined on every $D(f)$ by the localization of its map on global sections. This proves the affine module-sheaf equivalence.

For a short exact sequence of quasi-coherent sheaves, the sequence of their stalks is exact. Under this equivalence the stalk at $x$ is the localization of the global-section module at ${\mathfrak{m}}_x$. Exactness of module sequences can be checked at all maximal ideals, by localization detects zero elements applied to their homology, so the global-section modules also form a short exact sequence. Therefore global sections are exact on quasi-coherent sheaves over an affine variety. This is stronger than the left exactness of the global section functor on arbitrary sheaves, and follows from localization, not from assuming the cohomology vanishing still to be proved.

For an open inclusion $j:U\hookrightarrow X$, the requested sheaf isRestriction gives $\mathcal{F}\to {}_U\mathcal{F}$. This direct image from an open restriction is not extension by zero: its stalks outside $U$ may be nonzero.

The locally vanishing principle for sheaf cohomology says that a class $\xi \in H^i(X,\mathcal{F})$, $i>0$, is killed on a suitable neighbourhood of every point. Under the stated hypothesis that basis opens and their finite intersections have zero cohomology in degrees $1,\dots ,i-1$, these neighbourhoods can be chosen in that basis so that the image of $\xi$ underis zero. In particular its restriction in $H^i(U,\mathcal{F}{|}_U)$ is zero. This is a statement about individual classes; it does not assert that every locally vanishing class is already globally zero.

Here is a noncircular proof of vanishing of quasi-coherent cohomology on an affine scheme, applied to the variety. Induct on $i>0$, simultaneously for every affine variety and quasi-coherent sheaf. Assume all lower positive degrees vanish. The principal-open basis is closed under finite intersections, so it satisfies the principle's hypothesis. Given $\xi \in H^i(X,\mathcal{F})$, choose a finite principal cover $\mathcal{U}=(D(f_j))$ on which it restricts to zero.

Applying the Čech cochain complex to a flasque resolution gives the Čech lifting below the first possible local cohomology degree comparison segment, under lower-degree vanishing on all intersectionsFor completeness, the double complex has terms ${\check{C}}^p(\mathcal{U},{\mathcal{I}}^q)$. Its augmented rows are exact because each ${\mathcal{I}}^q$ is flasque, so its total cohomology is the cohomology of global sections of the resolution. The vertical cohomology on intersections in degrees $0<q<i$ is zero. Equivalently, start with local primitives of a cocycle representing $\xi$, take their differences on pairwise overlaps, and solve successively for primitives of those differences in degrees $i-1,i-2,\dots ,1$. The last difference is a Čech $i$-cocycle with values in $\mathcal{F}$. This identifies the kernel of the restriction map with the displayed Čech group. No vanishing in degree $i$ on the intersections has been assumed.

It remains a module calculation. Write $\mathcal{F}=\tilde{M}$. The augmented Čech complex isSince the $D(f_j)$ cover $X$, the $f_j$ generate the unit ideal. This complex is exact. To see it algebraically, use alternating cochains, with repeated indices giving zero. For a cocycle $c$, choose $N$ large enough to clear every $f_j$-denominator in $f_j^N{c}_{jI}$, as well as the finitely many cocycle relations; vanishing criterion in a module localization allows a further power to clear relations which initially hold only after localization. Since the ideal $(f_j^N)$ is still the unit ideal, choose $a_j$ with ${\sum }_j{a}_j{f}_j^N=1$. Defineusing those cleared representatives in $M_{f_I}$. The cocycle identity gives $\delta b=({\sum }_j{a}_j{f}_j^N)c=c$. The same argument with the augmentation gives gluing and uniqueness in degree zero. This is exactness of the unit-ideal localization Čech complex.

Thus ${\check{H}}^i(\mathcal{U},\mathcal{F})=0$, so the class $\xi$, whose restrictions were zero, is zero. The induction starts at $i=1$, when the lower-degree condition is empty. We have proved

Let $Y$ be the affine variety with coordinate ring $A$. Its principal opens $D(f_i)$ cover $Y$ because ${\sum }_i{g}_i{f}_i=1$. On $X_{f_i}$, the localization of global sections on a principal open identifies its coordinate ring with $A_{f_i}$, giving an isomorphismOn an overlap the two maps are induced by the same elements of $A$, or equivalently by the same identification with $A_{f_i{f}_j}$, so they agree. Glue them to $\varphi :X\to Y$. The inverses agree on the overlaps as well and glue to its inverse. At a closed point $P$, this is the map corresponding to the evaluation ring homomorphism $A\to k$, $a\mapsto a(P)$. This proves the cohomological criterion for affineness by an explicit global isomorphism.

For $U_{i_0\dots i_q}=U_{i_0}\cap \cdots \cap U_{i_q}$, the Čech cochain complex has groupsTerms in ${\delta }^2$ cancel in pairs, and Čech cohomology is $\ker \delta /\mathrm{im}\delta$. A degree-zero cocycle is exactly a family of compatible local sections. The sheaf gluing axiom gives their unique global section, proving ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathcal{F}(X)$. If the cover has affine finite intersections, a quasi-coherent sheaf has no higher cohomology on those intersections by vanishing of quasi-coherent cohomology on an affine scheme. The acyclic cover theorem then identifies all Čech groups with sheaf cohomology. In particular, a finite affine open cover of a separated variety has this property.

For the sheaf of units of the structure sheaf, a multiplicative degree-one cocycle is a family $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$ satisfying $g_{ij}{g}_{jk}=g_{ik}$, with $g_{ji}=g_{ij}^{-1}$. It glues trivial rank-one free modules into an invertible sheaf. Changing the local frames multiplies $g_{ij}$ by a coboundary, and two sets of transition data give isomorphic line bundles precisely when their cocycles differ this way. Tensoring line bundles multiplies their cocycles. Thus line bundles trivialized by an open cover give the group isomorphismFor the remaining arguments, work over the algebraically closed ground field. On the irreducible variety $V$, put $\mathcal{Q}={\mathcal{K}}^{\ast }/{\mathcal{O}}_V^{\ast }$, a quotient of sheaves of abelian groups. A global section of $\mathcal{Q}$ is locally represented by rational functions $f_i$ whose ratios are regular units. The corresponding unit cocycle $f_j/f_i$ defines an invertible sheaf. A single global rational function has trivial cocycle. Conversely, every line bundle has a nonzero rational section: choose a nonzero vector in its one-dimensional fibre at the generic point and express it in local frames. This supplies such local $f_i$. If the associated line bundle is trivial, changing frames makes all $f_i$ restrictions of one rational function. Thereforeis exact. This is the Cartier-divisor description of the Picard group. The sheaf of nonzero rational functions on an irreducible variety is flasque, since all restrictions between nonempty open sets are the identity on $k(V{)}^{\ast }$. Apply the long exact sequence in sheaf cohomology to $1\to {\mathcal{O}}_V^{\ast }\to {\mathcal{K}}^{\ast }\to \mathcal{Q}\to 1$. Since $H^1(V,{\mathcal{K}}^{\ast })=0$, its connecting map has exactly the cokernel just computed, provingFinally the Segre description of a smooth quadric surface identifies $V$ with ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The two rulings of a smooth quadric surface have classes $F_1,F_2$ generating $\mathrm{Pic}(V)\cong {\mathbb{Z}}^2$. For completeness, remove one line in each ruling: the remaining chart is the affine plane, with factorial coordinate ring $k[s,t]$ and trivial divisor class group. The localization sequence for the divisor class group makes $F_1,F_2$ generators, and their degrees on the two ruling lines prove independence. The hyperplane class, and hence the conic $C$, has bidegree $(1,1)$. By Picard-group localization on a smooth variety, the Picard group of a smooth affine quadric surface isExplicitly, the restriction of ${\mathcal{O}}_V(1,0)$ is nontrivial: if it were trivial on $U$, its rational trivialization would have divisor supported on $C$, forcing $(1,0)$ to be an integer multiple of $(1,1)$, which is impossible.

For affine varieties $X\subset {\mathbb{A}}^m$ and $Y\subset {\mathbb{A}}^n$, a morphism of affine varieties is a map whose $n$ coordinate functions are restrictions to $X$ of polynomials in its $m$ ambient coordinates. Intrinsically, its coordinates belong to the coordinate ring $A(X)$; they must satisfy the defining relations of $Y$.

Let $A=A(X)=k[x_1,\dots ,x_m]/I(X)$. Polynomial restrictions are regular functions, and a restriction is zero precisely when its class in the coordinate ring is zero. We prove surjectivity.

Using the usual irreducible-variety convention, view a regular function $f$ as an element of the fraction field of $A$. On a neighbourhood of every point it has the form $a/b$, where $a,b\in A$ and $b$ does not vanish at that point. Equality on a nonempty open subset of the irreducible variety is equality of rational functions. DefineThis is an ideal and contains a denominator nonzero at each point. Thus its zero locus is empty. The Hilbert Nullstellensatz implies $J=A$, so $1\in J$ and $f\in A$. ThereforeThe result also holds under a convention allowing reduced reducible affine varieties. One may cover by finitely many principal opens $D(b_i)$, represent $f$ in each localization, and increase powers until the local representatives glue after clearing denominators. Since these opens cover, the ideal generated by those powers is the unit ideal by the Nullstellensatz; a unit linear combination glues their numerators to one element of $A$. Irreducibility is used only for the shorter fraction-field proof, not for the assertion itself.

The semicubical parabola has coordinate ringunder $x\mapsto t^3$, $y\mapsto t^2$. The isomorphism is valid in every characteristic: after division by the monic polynomial in $x$, a remainder $P(y)+xQ(y)$ maps to a sum of distinct even powers and odd powers at least three, and can vanish only when both polynomials are zero.

Its fraction field contains $t=t^3/t^2$, which is an integral element over $A$ because it satisfies $z^2-y=0$. However $t\notin A$, since a polynomial in $t^2,t^3$ has no term of degree one. Thus $A$ is not an integrally closed domain. The ring $k[t]$ of the affine line is a principal ideal domain and therefore an integrally closed domain. This property is preserved by ring isomorphism. By coordinate-ring functoriality,The parametrization is a bijection on points over an algebraically closed field, but that does not make its inverse a morphism at the cusp.

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,

Suppose first that $\overline{f(X)}=Y$. If $g\in \ker f^{\ast }$, then $g\circ f=0$, so the zero set of the regular function $g$ contains $f(X)$ and hence its closure $Y$. Thus $g=0$ in the coordinate ring $k[Y]$, and $f^{\ast }$ is injective.

Conversely, if $\overline{f(X)}$ were a proper closed subset of $Y$, its vanishing ideal in $k[Y]$ would contain a nonzero regular function $g$. Then $g$ vanishes on $f(X)$, so $f^{\ast }(g)=0$, contradicting injectivity. ThereforeThis is the coordinate-ring criterion for a dominant morphism of affine varieties.

For an irreducible affine variety, define dimension byIrreducibility makes both coordinate rings integral domains. The injective map $f^{\ast }:k[Y]\hookrightarrow k[X]$ extends to an embedding of function fieldsAn algebraically independent family in $k(Y)$ remains algebraically independent in $k(X)$, so the dimension of an irreducible affine variety by transcendence degree gives

Subtracting the two defining equations givesConsequently the variety isEach member is irreducible. Indeed, after imposing its linear equation, its coordinate ring isThe invertible linear substitutionturns this intowhich is an integral domain. The two curves are distinct and neither contains the other, so they are precisely the irreducible components of two intersecting complex hyperbolas.

For an affine algebraic group $G$, its Lie algebra is the tangent space at the identity,Equivalently, it is the space of left-invariant derivations of the coordinate ring $\mathbb{C}[G]$. The Lie bracket is the commutator of derivations,It is antisymmetric because $[Y,X]=-[X,Y]$. Associativity of composition givesafter all six triple products cancel in pairs, proving the Jacobi identity. This construction is the Lie algebra of an affine algebraic group.