Functor represented by a scheme

<h2 id="motivation">Motivation</h2>
See also: <a href="/facts/Rational_point/j5v4cxuE">rational point § Definition</a>
The notion is an analog of a <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, where each principal G-bundle over a space S is (up to natural <a href="/facts/Isomorphism/pj8KgkxU">isomorphisms</a>) the pullback of the universal bundle 
 
 
 
 E
 G
 →
 B
 G
 
 
 {\displaystyle EG\to BG}
 
 along some map 
 
 
 
 S
 →
 B
 G
 
 
 {\displaystyle S\to BG}
 
. To give a principal G-bundle over S is the same as to give a map (called a classifying map) from S to the classifying space 
 
 
 
 B
 G
 
 
 {\displaystyle BG}
 
.
A similar phenomenon in algebraic geometry is given by a <a href="/facts/Linear_system_of_divisors/L8zleDlj">linear system</a>: to give a morphism from a base variety S to a projective space 
 
 
 
 X
 =
 
 
 P
 
 
 n
 
 
 
 
 {\displaystyle X=\mathbb {P} ^{n}}
 
 is equivalent to giving a <a href="/facts/Basepoint-free/XRQKvXGD">basepoint-free</a> linear system (or equivalently a line bundle) on S. That is, the projective space X represents the functor which gives all line bundles over S.
<a href="/facts/Yoneda%2527s_lemma/j04vXd6x">Yoneda's lemma</a> says that a scheme X determines and is determined by its functor of points.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="functor-of-points">Functor of points</h2>
Let X be a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a>. Its functor of points is the functor<blockquote>Hom(−,X) : (Affine schemes)op ⟶ Sets</blockquote>sending an affine scheme Y to the set of <a href="/facts/Morphism_of_schemes/55CKqHHo">scheme maps</a> 
 
 
 
 Y
 →
 X
 
 
 {\displaystyle Y\to X}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
A scheme is determined up to isomorphism by its functor of points. This is a stronger version of the <a href="/facts/Yoneda_lemma/j04vXd6x">Yoneda lemma</a>, which says that a X is determined by the map Hom(−,X) : Schemesop → Sets.
Conversely, a functor F : (Affine schemes)op → Sets is the functor of points of some scheme if and only if F is a sheaf with respect to the <a href="/facts/Grothendieck_site/Przxx5sH">Zariski topology</a> on (Affine schemes), and F admits an open cover by affine schemes.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="examples">Examples</h2>
<h3>Points as characters</h3>
Let X be a scheme over the base ring B. If x is a set-theoretic point of X, then the <a href="/facts/Residue_field/e6nflEmB">residue field</a> 
 
 
 
 k
 (
 x
 )
 
 
 {\displaystyle k(x)}
 
 is the residue field of the <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> 
 
 
 
 
 
 
 O
 
 
 
 X
 ,
 x
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X,x}}
 
 (i.e., the quotient by the maximal ideal). For example, if X is an affine scheme Spec(A) and x is a prime ideal 
 
 
 
 
 
 p
 
 
 
 
 {\displaystyle {\mathfrak {p}}}
 
, then the residue field of x is the <a href="/facts/Function_field_(scheme_theory)/npJNyqYI">function field</a> of the closed subscheme 
 
 
 
 Spec
 ⁡
 (
 A
 
 /
 
 
 
 p
 
 
 )
 
 
 {\displaystyle \operatorname {Spec} (A/{\mathfrak {p}})}
 
.
For simplicity, suppose 
 
 
 
 X
 =
 Spec
 ⁡
 (
 A
 )
 
 
 {\displaystyle X=\operatorname {Spec} (A)}
 
. Then the inclusion of a set-theoretic point x into X corresponds to the ring homomorphism:

A
        →
        k
        (
        x
        )
      
    
    {\displaystyle A\to k(x)}

(which is 
 
 
 
 A
 →
 
 A
 
 
 p
 
 
 
 →
 k
 (
 
 
 p
 
 
 )
 
 
 {\displaystyle A\to A_{\mathfrak {p}}\to k({\mathfrak {p}})}
 
 if 
 
 
 
 x
 =
 
 
 p
 
 
 
 
 {\displaystyle x={\mathfrak {p}}}
 
.)
The above should be compared to the <a href="/facts/Spectrum_of_a_commutative_Banach_algebra/3BTHdpg2">spectrum of a commutative Banach algebra</a>.

<h3>Points as sections</h3>
By the universal property of <a href="/facts/Fiber_product/bC0QoX5U">fiber product</a>, each R-point of a scheme X determines a morphism of R-schemes

Spec
 ⁡
 (
 R
 )
 →
 
 X
 
 R
 
 
 
 
 =
 
 d
 e
 f
 
 
 
 X
 
 ×
 
 Spec
 ⁡
 (
 B
 )
 
 
 Spec
 ⁡
 (
 R
 )
 
 
 {\displaystyle \operatorname {Spec} (R)\to X_{R}{\overset {\mathrm {def} }{=}}X\times _{\operatorname {Spec} (B)}\operatorname {Spec} (R)}
 
;
i.e., a section of the projection 
 
 
 
 
 X
 
 R
 
 
 →
 Spec
 ⁡
 (
 R
 )
 
 
 {\displaystyle X_{R}\to \operatorname {Spec} (R)}
 
. If S is a subset of X(R), then one writes 
 
 
 
 
 |
 
 S
 
 |
 
 ⊂
 
 X
 
 R
 
 
 
 
 {\displaystyle |S|\subset X_{R}}
 
 for the set of the images of the sections determined by elements in S.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Spec of the ring of dual numbers</h3>
Let 
 
 
 
 D
 =
 Spec
 ⁡
 (
 k
 [
 t
 ]
 
 /
 
 (
 
 t
 
 2
 
 
 )
 )
 
 
 {\displaystyle D=\operatorname {Spec} (k[t]/(t^{2}))}
 
, the Spec of the <a href="/facts/Ring_of_dual_numbers/DntuRu3F">ring of dual numbers</a> over a field k and X a scheme over k. Then each 
 
 
 
 D
 →
 X
 
 
 {\displaystyle D\to X}
 
 amounts to the tangent vector to X at the point that is the image of the closed point of the map.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In other words, 
 
 
 
 X
 (
 D
 )
 
 
 {\displaystyle X(D)}
 
 is the set of tangent vectors to X.

<h2 id="universal-object">Universal object</h2>

Let 
 
 
 
 F
 
 
 {\displaystyle F}
 
 be the functor represented by a scheme 
 
 
 
 X
 
 
 {\displaystyle X}
 
. Under the <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> 
 
 
 
 F
 (
 X
 )
 ≅
 
 Hom
 
 (
 X
 ,
 X
 )
 
 
 {\displaystyle F(X)\cong {\text{Hom}}(X,X)}
 
, there is a unique element of 
 
 
 
 F
 (
 X
 )
 
 
 {\displaystyle F(X)}
 
 that corresponds to the identity map 
 
 
 
 
 
 id
 
 
 X
 
 
 :
 X
 →
 X
 
 
 {\displaystyle {\text{id}}_{X}:X\to X}
 
. This unique element is known as the <a href="/facts/Initial_and_terminal_objects/oeckvXRj">universal object</a> or the universal family (when the objects being classified are families). The universal object acts as a template from which all other elements in 
 
 
 
 F
 (
 S
 )
 
 
 {\displaystyle F(S)}
 
 for any scheme 
 
 
 
 S
 
 
 {\displaystyle S}
 
 can be derived via pullback along a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> from 
 
 
 
 S
 
 
 {\displaystyle S}
 
 to 
 
 
 
 X
 
 
 {\displaystyle X}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Moduli_space/9sqrdpwR">Moduli space</a></li>
<li><a href="/facts/Weil_restriction/k4YbXyUL">Weil restriction</a></li>
<li><a href="/facts/Rational_point/j5v4cxuE">Rational point</a></li>
<li><a href="/facts/Descent_along_torsors/mQsTxno0">Descent along torsors</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/David_Mumford/WJoJ8K2u">David Mumford</a> (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fb62130">10.1007/b62130</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-63293-X.</li>
<li>Lurie, Jacob. <a href="https://www.math.ias.edu/~lurie/282ynotes/LectureXIV-Borel.pdf">"Lecture 14: Existence of Borel Reductions (I)"</a> (PDF).</li>
<li>Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag.</li>
<li>Shafarevich, Igor R. (2013). Basic Algebraic Geometry 2. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-38010-5">10.1007/978-3-642-38010-5</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-38009-9.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.math.washington.edu/~zhang/Shanghai2011/Slides/ardakov.pdf">http://www.math.washington.edu/~zhang/Shanghai2011/Slides/ardakov.pdf</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Shafarevich 1994, Ch. VI § 4.1. - Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Shafarevich 1994, Ch. VI § 4.4. - Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">In fact, X is determined by its R-points with various rings R: in the precise terms, given schemes X, Y, any natural transformation from the functor 
 
 
 
 R
 ↦
 X
 (
 R
 )
 
 
 {\displaystyle R\mapsto X(R)}
 
 to the functor 
 
 
 
 R
 ↦
 Y
 (
 R
 )
 
 
 {\displaystyle R\mapsto Y(R)}
 
 determines a morphism of schemes X →Y in a natural way. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">The Stacks Project, 01J5 <a href="https://stacks.math.columbia.edu/tag/01J5" target="_blank">https://stacks.math.columbia.edu/tag/01J5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">The functor of points, Yoneda's lemmma, moduli spaces and universal properties (Brian Osserman), Cor. 3.6 <a href="https://web.archive.org/web/20200127110057/https://pdfs.semanticscholar.org/a20d/7974f4aaee5b13657341f206d6d70939cd4e.pdf" target="_blank">https://web.archive.org/web/20200127110057/https://pdfs.semanticscholar.org/a20d/7974f4aaee5b13657341f206d6d70939cd4e.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">This seems like a standard notation; see for example "Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9)" (PDF). <a href="http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf" target="_blank">http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Shafarevich 1994, Ch. VI § 4.1. - Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Shafarevich 1994, Ch. VI § 4.1. - Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Functor represented by a scheme open-in-new

Functor represented by a scheme