morphism of schemes

By definition, a morphism of schemes is just a morphism of locally ringed spaces. Isomorphisms are defined accordingly.

A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties).{{harvnb|Vakil|2014|loc=Exercise 6.3.C.}} Let ƒ:X→Y be a morphism of schemes. If x is a point of X, since ƒ is continuous, there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊆ V. Then ƒ: U → V is a morphism of affine schemes and thus is induced by some ring homomorphism B → A (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:X→Y is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts.

• Note: It would not be desirable to define a morphism of schemes as a morphism of ringed spaces. One trivial reason is that there is an example of a ringed-space morphism between affine schemes that is not induced by a ring homomorphism (for example,{{harvnb|Vakil|2014|loc=Exercise 6.2.E.}} a morphism of ringed spaces:
• :$\backslash operatorname\{Spec\}\; k(x)\; \backslash to\; \backslash operatorname\{Spec\}\; k[y]\_\{(y)\}\; =\; \backslash \{\; \backslash eta\; =\; (0),\; s\; =\; (y)\; \backslash \}$

:that sends the unique point to s and that comes with $k[y]\_\{(y)\}\; \backslash to\; k(x),\; \backslash ,\; y\; \backslash mapsto\; x$). More conceptually, the definition of a morphism of schemes needs to capture "Zariski-local nature" or localization of rings;{{citation |url=http://www.math.harvard.edu/~lurie/papers/DAG-V.pdf |title=Derived Algebraic Geometry V: Structured Spaces |date=22 February 2011 |at=§ 1.}} this point of view (i.e., a local-ringed space) is essential for a generalization (topos).

Let {{nowrap|f : X → Y}} be a morphism of schemes with $\backslash phi:\; \backslash mathcal\{O\}\_Y\; \backslash to\; f\_*\backslash mathcal\{O\}\_X$. Then, for each point x of X, the homomorphism on the stalks:

:$\backslash phi:\; \backslash mathcal\{O\}\_\{Y,\; f(x)\}\; \backslash to\; \backslash mathcal\{O\}\_\{X,\; x\}$

is a local ring homomorphism: i.e., $\backslash phi(\backslash mathfrak\{m\}\_\{f(x)\})\; \backslash subseteq\; \backslash mathfrak\{m\}\_x$ and so induces an injective homomorphism of residue fields

:$\backslash phi:\; k(f(x))\; \backslash hookrightarrow\; k(x)$.

(In fact, φ maps th n-th power of a maximal ideal to the n-th power of the maximal ideal and thus induces the map between the (Zariski) cotangent spaces.)

For each scheme X, there is a natural morphism

:$\backslash theta:\; X\; \backslash to\; \backslash operatorname\{Spec\}\; \backslash Gamma(X,\; \backslash mathcal\{O\}\_X),$

which is an isomorphism if and only if X is affine; θ is obtained by gluing U → target which come from restrictions to open affine subsets U of X. This fact can also be stated as follows: for any scheme X and a ring A, there is a natural bijection:

:$\backslash operatorname\{Mor\}(X,\; \backslash operatorname\{Spec\}(A))\; \backslash cong\; \backslash operatorname\{Hom\}(A,\; \backslash Gamma(X,\; \backslash mathcal\; O\_X)).$

(Proof: The map $\backslash phi\; \backslash mapsto\; \backslash operatorname\{Spec\}(\backslash phi)\; \backslash circ\; \backslash theta$ from the right to the left is the required bijection. In short, θ is an adjunction.)

Moreover, this fact (adjoint relation) can be used to characterize an affine scheme: a scheme X is affine if and only if for each scheme S, the natural map

:$\backslash operatorname\{Mor\}(S,\; X)\; \backslash to\; \backslash operatorname\{Hom\}(\backslash Gamma(X,\; \backslash mathcal\{O\}\_X),\; \backslash Gamma(S,\; \backslash mathcal\{O\}\_S))$

is bijective.{{harvnb|Grothendieck|Dieudonné|1960|loc=Ch. I, Corollarie 1.6.4.}} (Proof: if the maps are bijective, then $\backslash operatorname\{Mor\}(-,\; X)\; \backslash simeq\; \backslash operatorname\{Mor\}(-,\; \backslash operatorname\{Spec\}\; \backslash Gamma(X,\; \backslash mathcal\{O\}\_X))$ and X is isomorphic to $\backslash operatorname\{Spec\}\; \backslash Gamma(X,\; \backslash mathcal\{O\}\_X)$ by Yoneda's lemma; the converse is clear.)

{{see also|Grothendieck's relative point of view}}

Fix a scheme S, called a base scheme. Then a morphism $p:\; X\; \backslash to\; S$ is called a scheme over S or an S-scheme; the idea of the terminology is that it is a scheme X together with a map to the base scheme S. For example, a vector bundle E → S over a scheme S is an S-scheme.

An S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ. Given an S-scheme $X\; \backslash to\; S$, viewing S as an S-scheme over itself via the identity map, an S-morphism $S\; \backslash to\; X$ is called a S-section or just a section.

All the S-schemes form a category: an object in the category is an S-scheme and a morphism in the category an S-morphism. (This category is the slice category of the category of schemes with the base object S.)

Let $\backslash varphi:\; B\; \backslash to\; A$ be a ring homomorphism and let

:$\backslash begin\{cases\}\; \backslash varphi^a:\; \backslash operatorname\{Spec\}\; A\; \backslash to\; \backslash operatorname\{Spec\}\; B,\; \backslash \backslash \; \backslash mathfrak\{p\}\; \backslash mapsto\; \backslash varphi^\{-1\}(\backslash mathfrak\{p\})\backslash end\{cases\}$

be the induced map. Then

• $\backslash varphi^a$ is continuous.Proof: $\{\backslash varphi^a\}^\{-1\}(D(f))\; =\; D(\backslash varphi(f))$ for all f in A.
• If $\backslash varphi$ is surjective, then $\backslash varphi^a$ is a homeomorphism onto its image.{{harvnb|Grothendieck|Dieudonné|1960|loc=Ch. I, Corollaire 1.2.4.}}
• For every ideal I of A, $\backslash overline\{\backslash varphi^a(V(I))\}\; =\; V(\backslash varphi^\{-1\}(I)).${{harvnb|Grothendieck|Dieudonné|1960|loc=Ch. I, 1.2.2.3.}}
• $\backslash varphi^a$ has dense image if and only if the kernel of $\backslash varphi$ consists of nilpotent elements. (Proof: the preceding formula with I = 0.) In particular, when B is reduced, $\backslash varphi^a$ has dense image if and only if $\backslash varphi$ is injective.

Let f: Spec A → Spec B be a morphism of schemes between affine schemes with the pullback map $\backslash varphi$: B → A. That it is a morphism of locally ringed spaces translates to the following statement: if $x\; =\; \backslash mathfrak\{p\}\_x$ is a point of Spec A,

:$\backslash mathfrak\{p\}\_\{f(x)\}\; =\; \backslash varphi^\{-1\}(\backslash mathfrak\{p\}\_x)$.

(Proof: In general, $\backslash mathfrak\{p\}\_x$ consists of g in A that has zero image in the residue field k(x); that is, it has the image in the maximal ideal $\backslash mathfrak\{m\}\_x$. Thus, working in the local rings, $g(f(x))\; =\; 0\; \backslash Rightarrow\; \backslash varphi(g)\; \backslash in\; \backslash varphi(\backslash mathfrak\{m\}\_\{f(x))\})\; \backslash subseteq\; \backslash mathfrak\{m\}\_x\; \backslash Rightarrow\; g\; \backslash in\; \backslash varphi^\{-1\}(\backslash mathfrak\{m\}\_x)$. If $g(f(x))\; \backslash ne\; 0$, then $g$ is a unit element and so $\backslash varphi(g)$ is a unit element.)

Hence, each ring homomorphism B → A defines a morphism of schemes Spec A → Spec B and, conversely, all morphisms between them arise this fashion.

{{See also|morphism of varieties#Examples}}

• Let R be a field or $\backslash Z.$ For each R-algebra A, to specify an element of A, say f in A, is to give a R-algebra homomorphism $R[t]\; \backslash to\; A$ such that $t\; \backslash mapsto\; f$. Thus, $A\; =\; \backslash operatorname\{Hom\}\_\{R-\backslash text\{alg\}\}(R[t],\; A)$. If X is a scheme over S = Spec R, then taking $A\; =\; \backslash Gamma(X,\; \backslash mathcal\{O\}\_X)$ and using the fact Spec is a right adjoint to the global section functor, we get $$\backslash Gamma(X,\; \backslash mathcal\{O\}\_X)\; =\; \backslash operatorname\{Mor\}\_S(X,\; \backslash mathbb\{A\}^1\_S)$$ where $\backslash mathbb\{A\}^1\_S\; =\; \backslash operatorname\{Spec\}(R[t])$. Note the equality is that of rings.
• Similarly, for any S-scheme X, there is the identification of the multiplicative groups: $$\backslash Gamma(X,\; \backslash mathcal\{O\}\_X^*)\; =\; \backslash operatorname\{Mor\}\_S(X,\; \backslash mathbb\{G\}\_m)$$ where $\backslash mathbb\{G\}\_m\; =\; \backslash operatorname\{Spec\}(R[t,\; t^\{-1\}])$ is the multiplicative group scheme.
• Many examples of morphisms come from families parameterized by some base space. For example, $$\backslash text\{Proj\}\backslash left(\backslash frac\{\backslash Complex[x,y][a,b,c]\}\{(ax^2\; +bxy\; +\; cy^2)\}\backslash right)\; \backslash to\; \backslash text\{Proj\}(\backslash Complex[a,b,c])\; =\; \backslash mathbb\{P\}^2\_\{a,b,c\}$$ is a projective morphism of projective varieties where the base space parameterizes quadrics in $\backslash mathbb\{P\}^1$.

Given a morphism of schemes $f:\; X\; \backslash to\; Y$ over a scheme S, the morphism $X\; \backslash to\; X\; \backslash times\_S\; Y$ to the fiber product induced by the identity $1\_X\; :\; X\; \backslash to\; X$ and f is called the graph morphism of f. The graph morphism of the identity is called the diagonal morphism.

{{see also|glossary of algebraic geometry}}

Morphisms of finite type are one of the basic tools for constructing families of varieties. A morphism $f:X\backslash to\; S$ is of finite type if there exists a cover $\backslash operatorname\{Spec\}(A\_i)\; \backslash to\; S$ such that the fibers $X\; \backslash times\_S\; \backslash operatorname\{Spec\}(A\_i)$ can be covered by finitely many affine schemes $\backslash operatorname\{Spec\}(B\_\{ij\})$ making the induced ring morphisms $A\_i\; \backslash to\; B\_\{ij\}$ into finite-type morphisms. A typical example of a finite-type morphism is a family of schemes. For example,

:$\backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Z[x,y,z]\}\{(x^n\; +\; zy^n,\; z^5-1)\}\; \backslash right)\; \backslash to\; \backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Z[z]\}\{(z^5-1)\}\; \backslash right)$

is a morphism of finite type. A simple non-example of a morphism of finite-type is $\backslash operatorname\{Spec\}(k[x\_1,x\_2,x\_3,\backslash ldots]))\; \backslash to\; \backslash operatorname\{Spec\}(k)$ where $k$ is a field. Another is an infinite disjoint union

:$\backslash coprod^\backslash infty\; X\; \backslash to\; X$

A morphism of schemes $i:Z\; \backslash to\; X$ is a closed immersion if the following conditions hold:

1. $i$ defines a homeomorphism of $Z$ onto its image
2. $i^\backslash \#:\; \backslash mathcal\{O\}\_X\; \backslash to\; i\_*\backslash mathcal\{O\}\_Z$ is surjective.

This condition is equivalent to the following: given an affine open $\backslash operatorname\{Spec\}(R)\; =\; U\; \backslash subseteq\; X$ there exists an ideal $I\; \backslash subseteq\; R$ such that $i^\{-1\}(U)\; =\; \backslash operatorname\{Spec\}(R/I)$.

Of course, any (graded) quotient $R/I$ defines a subscheme of $\backslash operatorname\{Spec\}(R)$ ($\backslash operatorname\{Proj\}(R)$). Consider the quasi-affine scheme $\backslash mathbb\{A\}^2\; -\; \backslash \{0\; \backslash \}$ and the subset of the $x$-axis contained in $X$. Then if we take the open subset $\backslash operatorname\{Spec\}(k[x,y,y^\{-1\}])$ the ideal sheaf is $(x)$ while on the affine open $\backslash operatorname\{Spec\}(k[x,y,x^\{-1\}])$ there is no ideal since the subset does not intersect this chart.

Separated morphisms define families of schemes which are analogous to Hausdorff topological spaces. For example, given a separated morphism $X\; \backslash to\; S$ in $\backslash text\{Sch\}/\backslash Complex$ the associated analytic spaces $X(\backslash Complex)^\{an\}\; \backslash to\; S(\backslash Complex)^\{an\}$ are both Hausdorff. We say a morphism of schemes $f:X\; \backslash to\; S$ is separated if the diagonal morphism $\backslash Delta\_\{X/S\}:X\; \backslash to\; X\backslash times\_SX$ is a closed immersion. In topology, an analogous condition for a space $X$ to be Hausdorff is if the diagonal set

:$\backslash Delta\; =\; \backslash \{(x,x)\; \backslash in\; X\backslash times\; X\; \backslash \}$

is a closed subset of $X\backslash times\; X$. Nevertheless, most schemes are not Hausdorff as topological spaces, as the Zariski topology is in general highly non-Hausdorff.

Most morphisms encountered in scheme theory will be separated. For example, consider the affine scheme

:$X\; =\; \backslash operatorname\{Spec\}\backslash left(\; \backslash frac\{\backslash Complex\; [x,y]\}\{(f)\}\; \backslash right)$

over $\backslash operatorname\{Spec\}(\backslash Complex).$ Since the product scheme is

:$X\; \backslash times\_\backslash Complex\; X\; =\; \backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Complex[x,y]\}\{(f)\}\backslash otimes\_\backslash Complex\; \backslash frac\{\backslash Complex[x,y]\}\{(f)\}\; \backslash right)$

the ideal defining the diagonal is generated by

:$x\backslash otimes\; 1\; -\; 1\; \backslash otimes\; x,\; y\; \backslash otimes\; 1\; -\; 1\; \backslash otimes\; y$

showing the diagonal scheme is affine and closed. This same computation can be used to show that projective schemes are separated as well.

The only time care must be taken is when gluing together a family of schemes. For example, if we take the diagram of inclusions

:$\backslash begin\{matrix\}$

\operatorname{Spec}(R[x,x^{-1}]) & &\\

&\searrow & \\

& & \operatorname{Spec}(R[x]) \\

&\nearrow & \\

\operatorname{Spec}(R[x,x^{-1}]) & &

\end{matrix}

then we get the scheme-theoretic analogue of the classical line with two-origins.

A morphism $f:X\backslash to\; S$ is called proper if it is

1. separated
2. of finite type
3. universally closed.

The last condition means that given a morphism $S\text{'}\; \backslash to\; S$ the base change morphism $S\text{'}\backslash times\_SX$ is a closed immersion. Most known examples of proper morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry.

Projective morphisms define families of projective varieties over a fixed base scheme. Note that there are two definitions: Hartshorne's which states that a morphism $f:X\backslash to\; S$ is called projective if there exists a closed immersion $X\; \backslash to\; \backslash mathbb\{P\}^n\_S\; =\; \backslash mathbb\{P\}^n\backslash times\; S$ and the EGA definition which states that a scheme $X\; \backslash in\; \backslash text\{Sch\}/S$ is projective if there is a quasi-coherent $\backslash mathcal\{O\}\_S$-module of finite type such that there is a closed immersion $X\; \backslash to\; \backslash mathbb\{P\}\_S(\backslash mathcal\{E\})$. The second definition is useful because an exact sequence of $\backslash mathcal\{O\}\_S$-modules can be used to define projective morphisms.

A projective morphism $f:X\; \backslash to\; \backslash \{\; *\backslash \}$ defines a projective scheme. For example,

:$\backslash text\{Proj\}\backslash left(\; \backslash frac\{\backslash Complex[x,y,z]\}\{(x^n\; +\; y^n\; -\; z^n)\}\; \backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(\backslash Complex\; )$

defines a projective curve of genus $(n-1)(n-1)/2$ over $\backslash Complex$.

If we let $S\; =\; \backslash mathbb\{A\}^1\_t$ then the projective morphism

:$\backslash underline\{\backslash operatorname\{Proj\}\}\_S\backslash left(\; \backslash frac\{\backslash mathcal\{O\}\_S[x\_0,x\_1,x\_2,x\_3,x\_4]\}\{\backslash left(x\_0^5\; +\; \backslash cdots\; +\; x\_4^5\; -\; t\; x\_0\; x\_1\; x\_2\; x\_3\; x\_4\backslash right)\}\; \backslash right)\; \backslash to\; S$

defines a family of Calabi-Yau manifolds which degenerate.

Another useful class of examples of projective morphisms are Lefschetz pencils: they are projective morphisms $\backslash pi:X\; \backslash to\; \backslash mathbb\{P\}^1\_k\; =\; \backslash operatorname\{Proj\}(k[s,t])$ over some field $k$. For example, given smooth hypersurfaces $X\_1,\; X\_2\; \backslash subseteq\; \backslash mathbb\{P\}^n\_k$ defined by the homogeneous polynomials $f\_1,f\_2$ there is a projective morphism

:$\backslash underline\{\backslash operatorname\{Proj\}\}\_\{\backslash mathbb\{P\}^1\}\backslash left(\; \backslash frac\{\backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^1\}[x\_0,\backslash ldots,x\_n]\}\{(sf\_1\; +\; tf\_2)\}\; \backslash right)\backslash to\; \backslash mathbb\{P\}^1$

giving the pencil.

A nice classical example of a projective scheme is by constructing projective morphisms which factor through rational scrolls. For example, take $S\; =\; \backslash mathbb\{P\}^1$ and the vector bundle $\backslash mathcal\{E\}\; =\; \backslash mathcal\{O\}\_S\; \backslash oplus\; \backslash mathcal\{O\}\_S\; \backslash oplus\; \backslash mathcal\{O\}\_S(3)$. This can be used to construct a $\backslash mathbb\{P\}^2$-bundle $\backslash mathbb\{P\}\_S(\backslash mathcal\{E\})$ over $S$. If we want to construct a projective morphism using this sheaf we can take an exact sequence, such as

:$\backslash mathcal\{O\}\_S(-d)\backslash oplus\backslash mathcal\{O\}\_S(-e)\; \backslash to\; \backslash mathcal\{E\}\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; 0$

which defines the structure sheaf of the projective scheme $X$ in $\backslash mathbb\{P\}\_S(\backslash mathcal\{E\}).$

Flat morphisms have an algebraic definition but have a very concrete geometric interpretation: flat families correspond to families of varieties which vary "continuously". For example,

:$\backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Complex[x,y,t]\}\{(xy-t))\}\; \backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(\backslash Complex[t])$

is a family of smooth affine quadric curves which degenerate to the normal crossing divisor

:$\backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Complex[x,y]\}\{(xy)\}\; \backslash right)$

at the origin.

One important property that a flat morphism must satisfy is that the dimensions of the fibers should be the same. A simple non-example of a flat morphism then is a blowup since the fibers are either points or copies of some $\backslash mathbb\{P\}^n$.

Let $f:X\; \backslash to\; S$ be a morphism of schemes. We say that $f$ is flat at a point $x\; \backslash in\; X$ if the induced morphism $\backslash mathcal\{O\}\_\{f(x)\}\; \backslash to\; \backslash mathcal\{O\}\_x$ yields an exact functor $-\backslash otimes\_\{\backslash mathcal\{O\}\_\{f(x)\}\}\; \backslash mathcal\{O\}\_x.$ Then, $f$ is flat if it is flat at every point of $X$. It is also faithfully flat if it is a surjective morphism.

Using our geometric intuition it obvious that

:$f:\backslash operatorname\{Spec\}(\backslash Complex\; [x,y]/(xy))\; \backslash to\; \backslash operatorname\{Spec\}(\backslash Complex\; [x])$

is not flat since the fiber over $0$ is $\backslash mathbb\{A\}^1$ with the rest of the fibers are just a point. But, we can also check this using the definition with local algebra: Consider the ideal $\backslash mathfrak\{p\}\; =\; (x)\; \backslash in\; \backslash operatorname\{Spec\}(\backslash Complex\; [x,y]/(xy)).$ Since $f(\backslash mathfrak\{p\})\; =\; (x)\; \backslash in\; \backslash operatorname\{Spec\}(\backslash Complex[x])$ we get a local algebra morphism

:$f\_\{\backslash mathfrak\{p\}\}:\; \backslash left(\; \backslash Complex\; [x]\; \backslash right)\_\{(x)\}\; \backslash to\; \backslash left(\backslash Complex\; [x,y]/(xy)\; \backslash right)\_\{(x)\}$

If we tensor

:$0\; \backslash to\; \backslash Complex[x]\_\{(x)\}\; \backslash overset\{\backslash cdot\; x\}\{\backslash longrightarrow\}\; \backslash Complex[x]\_\{(x)\}$

with $(\backslash Complex[x,y]/(xy))\_\{(x)\}$, the map

:$(\backslash Complex[x,y]/(xy)))\_\{(x)\}\; \backslash xrightarrow\{\backslash cdot\; x\}\; (\backslash Complex[x,y]/(xy))\_\{(x)\}$

has a non-zero kernel due the vanishing of $xy$. This shows that the morphism is not flat.

A morphism $f:X\; \backslash to\; Y$ of affine schemes is unramified if $\backslash Omega\_\{X/Y\}\; =\; 0$. We can use this for the general case of a morphism of schemes $f:X\backslash to\; Y$. We say that $f$ is unramified at $x\backslash in\; X$ if there is an affine open neighborhood $x\; \backslash in\; U$ and an affine open $V\backslash subseteq\; Y$ such that $f(U)\backslash subseteq\; V$ and $\backslash Omega\_\{U/V\}\; =\; 0.$ Then, the morphism is unramified if it is unramified at every point in $X$.

One example of a morphism which is flat and generically unramified, except for at a point, is

:$\backslash operatorname\{Spec\}\backslash left(\; \backslash frac\{\backslash Complex\; [t,x]\}\{(x^n-t)\}\; \backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(\backslash Complex[t])$

We can compute the relative differentials using the sequence

:$\backslash frac\{\backslash Complex[t,x]\}\{(x^n-t)\}\backslash otimes\_\{\backslash Complex[t]\}\backslash Complex[t]dt\; \backslash to\; \backslash left(\backslash frac\{\backslash Complex[t,x]\}\{(x^n-t)\}dt\; \backslash oplus\; \backslash frac\{\backslash Complex[t,x]\}\{(x^n-t)\}dx\backslash right)\; /\; (nx^\{n-1\}dx\; -\; dt)\; \backslash to\; \backslash Omega\_\{X/Y\}\; \backslash to\; 0$

showing

:$\backslash Omega\_\{X/Y\}\; \backslash cong\; \backslash left(\backslash frac\{\backslash Complex\; [t,x]\}\{(x^n-t)\}dx\backslash right)/(x^\{n-1\}dx)\; \backslash neq\; 0$

if we take the fiber $t\; =\; 0$, then the morphism is ramified since

:$\backslash Omega\_\{X\_0/\backslash Complex\; \}\; =\; \backslash left(\backslash frac\{\backslash Complex[x]\}\{x^\{n\}\}dx\backslash right)/(x^\{n-1\}dx)$

otherwise we have

:$\backslash Omega\_\{X\_\backslash alpha/\backslash Complex\}\; =\; \backslash left(\backslash frac\{\backslash Complex[x]\}\{(x^n-\backslash alpha)\}dx\backslash right)/(x^\{n-1\}dx)\; \backslash cong\; \backslash frac\{\backslash Complex[x]\}\{(\backslash alpha)\}dx\; \backslash cong\; 0$

showing that it is unramified everywhere else.

A morphism of schemes $f:X\; \backslash to\; Y$ is called étale if it is flat and unramfied. These are the algebro-geometric analogue of covering spaces. The two main examples to think of are covering spaces and finite separable field extensions. Examples in the first case can be constructed by looking at branched coverings and restricting to the unramified locus.

{{see also|rational point#Definition|Functor represented by a scheme}}

By definition, if X, S are schemes (over some base scheme or ring B), then a morphism from S to X (over B) is an S-point of X and one writes:

:$X(S)\; =\; \backslash \{\; f\; \backslash mid\; f:\; S\; \backslash to\; X\; \backslash text\{\; over\; \}\; B\; \backslash \}$

for the set of all S-points of X. This notion generalizes the notion of solutions to a system of polynomial equations in classical algebraic geometry. Indeed, let X = Spec(A) with $A\; =B[t\_1,\; \backslash dots,\; t\_n]/(f\_1,\; \backslash dots,\; f\_m)$. For a B-algebra R, to give an R-point of X is to give an algebra homomorphism A →R, which in turn amounts to giving a homomorphism

:$B[t\_1,\; \backslash dots,\; t\_n]\; \backslash to\; R,\; \backslash ,\; t\_i\; \backslash mapsto\; r\_i$

that kills f_i's. Thus, there is a natural identification:

:$X(\backslash operatorname\{Spec\}\; R)\; =\; \backslash \{\; (r\_1,\; \backslash dots,\; r\_n)\; \backslash in\; R^n\; |\; f\_1(r\_1,\; \backslash dots,\; r\_n)\; =\; \backslash cdots\; =\; f\_m(r\_1,\; \backslash dots,\; r\_n)\; =\; 0\; \backslash \}.$

Example: If X is an S-scheme with structure map π: X → S, then an S-point of X (over S) is the same thing as a section of π.

In category theory, Yoneda's lemma says that, given a category C, the contravariant functor

:$C\; \backslash to\; \backslash mathcal\{P\}(C)\; =\; \backslash operatorname\{Fct\}(C^\{\backslash text\{op\}\},\; \backslash mathbf\{Sets\}),\; \backslash ,\; X\; \backslash mapsto\; \backslash operatorname\{Mor\}(-,\; X)$

is fully faithful (where $\backslash mathcal\{P\}(C)$ means the category of presheaves on C). Applying the lemma to C = the category of schemes over B, this says that a scheme over B is determined by its various points.

It turns out that in fact it is enough to consider S-points with only affine schemes S, precisely because schemes and morphisms between them are obtained by gluing affine schemes and morphisms between them. Because of this, one usually writes X(R) = X(Spec R) and view X as a functor from the category of commutative B-algebras to Sets.

Example: Given S-schemes X, Y with structure maps p, q,

:$(X\; \backslash times\_S\; Y)(R)\; =\; X(R)\; \backslash times\_\{S(R)\}\; Y(R)\; =\; \backslash \{\; (x,\; y)\; \backslash in\; X(R)\; \backslash times\; Y(R)\; \backslash mid\; p(x)\; =\; q(y)\; \backslash \}$.

Example: With B still denoting a ring or scheme, for each B-scheme X, there is a natural bijection

:$\backslash mathbf\{P\}^n\_B(X)\; =$ { the isomorphism classes of line bundles L on X together with n + 1 global sections generating L. };

in fact, the sections s_i of L define a morphism $X\; \backslash to\; \backslash mathbf\{P\}^n\_B,\; \backslash ,\; x\; \backslash mapsto\; (s\_0(x)\; :\; \backslash dots\; :\; s\_n(x))$. (See also Proj construction#Global Proj.)

Remark: The above point of view (which goes under the name functor of points and is due to Grothendieck) has had a significant impact on the foundations of algebraic geometry. For example, working with a category-valued (pseudo-)functor instead of a set-valued functor leads to the notion of a stack, which allows one to keep track of morphisms between points (i.e., morphisms between morphisms).

{{main|rational map}}

A rational map of schemes is defined in the same way for varieties. Thus, a rational map from a reduced scheme X to a separated scheme Y is an equivalence class of a pair $(U,\; f\_U)$ consisting of an open dense subset U of X and a morphism $f\_U\; :\; U\; \backslash to\; Y$. If X is irreducible, a rational function on X is, by definition, a rational map from X to the affine line $\backslash mathbb\{A\}^1$ or the projective line $\backslash mathbb\{P\}^1.$

A rational map is dominant if and only if it sends the generic point to the generic point.{{harvnb|Vakil|2014|loc=Exercise 6.5.A}}

A ring homomorphism between function fields need not induce a dominant rational map (even just a rational map).{{harvnb|Vakil|2014|loc=A paragraph after Exercise 6.5.B}} For example, Spec k[x] and Spec k(x) and have the same function field (namely, k(x)) but there is no rational map from the former to the latter. However, it is true that any inclusion of function fields of algebraic varieties induces a dominant rational map (see morphism of algebraic varieties#Properties.)

• Regular embedding
• Constructible set (topology)
• Universal homeomorphism

{{reflist}}

• {{EGA | book=I}}
• {{Hartshorne AG}}
• Milne, Review of Algebraic Geometry at [http://www.jmilne.org/math/CourseNotes/iAG.pdf Algebraic Groups: The theory of group schemes of finite type over a field.]
• {{citation |last=Vakil |first=Ravi |date=30 December 2014 |url=http://math.stanford.edu/~vakil/216blog/FOAGdec3014public.pdf |title=Foundations of Algebraic Geometry |edition=Draft}}

Category:Algebraic geometry