Functors of Points

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We now begin preparing to extend the language of schemes further. For this, we need the functor of points viewpoint that we saw in §Morphisms of Schemes, ⁋Definition 6. This was already defined in §Morphisms of Schemes, ⁋Definition 9: to study a scheme \(X\), we look at the collection of \(T\)-points of \(X\) for every possible test scheme \(T\). In other words, we consider the functor

\[h_X=\Hom_\Sch(-,X): \Sch^\op \rightarrow \Set\]

Our first goal is to verify, via the Yoneda lemma, that this functor determines \(X\) completely up to isomorphism, and to see how this functorial viewpoint provides natural language when working with affine space, projective space, Grassmannians, fiber products, and so on.

Before starting the main discussion, let us fix notation. For each scheme \(T\), we write \(h_X(T)=\Hom_\Sch(T,X)\) as \(X(T)\), and for a scheme morphism \(\tau: T' \rightarrow T\), \(h_X(\tau): X(T) \rightarrow X(T')\) means the composition \(h_X(\tau)(\psi)=\psi\circ \tau\). In particular, when \(T=\Spec A\), we write \(X(\Spec A)\) simply as \(X(A)\), and as defined above, an element of the set \(X(T)\) is also called a \(T\)-valued point. Under this name, \(X(\tau)\) pulls a \(T\)-point of \(X\) back along \(\tau\) to a \(T'\)-point.

Meanwhile, functoriality also exists in the \(X\) direction. Given a scheme morphism \(\varphi: X\rightarrow Y\), for a fixed test scheme \(T\) the composition

\[h_\varphi(T): X(T) \rightarrow Y(T);\qquad \psi\mapsto \varphi\circ \psi\]

is well-defined, and moreover \(h_\varphi(T')\circ h_X(\tau)=h_Y(\tau)\circ h_\varphi(T)\) holds for any \(\tau: T' \rightarrow T\). That is, \(\varphi\) induces a natural transformation \(h_\varphi: h_X \rightarrow h_Y\), and from this we know that \(X\mapsto h_X\) defines a functor

\[h_{(-)}:\Sch \rightarrow \Fun(\Sch^\op, \Set)\]

This viewpoint remains valid verbatim even if we change our object of interest to \(\Sch_{/S}\), but in this post we carry out everything in \(\Sch\) for convenience.

The Yoneda Lemma and Representability

We now show that the functor of points \(h_X\) defined by \(X\) actually carries sufficient scheme-theoretic information about \(X\). This is essentially something already covered in category theory, so here we only give a brief review.

The categorical foundation of the functor of points viewpoint is, of course, the Yoneda lemma and representability. Applying [Category Theory] §Representable Functors, ⁋Theorem 4 (Yoneda) with \(\mathcal{A}=\Sch\), we know that the functor \(h_{(-)}:\Sch \rightarrow \Fun(\Sch^\op, \Set)\) is fully faithful. This shows that a scheme \(X\) is uniquely determined up to isomorphism by \(h_X\), and that a scheme morphism is exactly the same data as a natural transformation between functors of points.

As we saw above, given a scheme morphism \(\varphi:X\rightarrow Y\), we can send a \(T\)-point \(\psi:T\rightarrow X\) to the composition \(\varphi\circ\psi:T\rightarrow Y\). In this way, \(\varphi\) gives maps \(X(T)\rightarrow Y(T)\) that are compatible across all test schemes. The key observation is that this also works in reverse. Suppose we are given such a compatible collection of maps \(\alpha_T:X(T)\rightarrow Y(T)\). The element \(\alpha_X(\id_X)\) to which \(\alpha_X\) sends the identity morphism \(\id_X:X\rightarrow X\) is an \(X\)-point of \(Y\), that is, a scheme morphism \(f:X\rightarrow Y\). By naturality, \(\alpha_T(\psi)=f\circ\psi\) holds for any \(\psi:T\rightarrow X\), so the maps at all remaining \(T\)-points are forced to be composition with \(f\). In other words, a natural transformation between functors of points is exactly the same data as a single scheme morphism.

Thus a scheme is essentially a functor \(F:\Sch\rightarrow \Fun(\Sch^\op, \Set)\), and for it to actually arise from a scheme, this functor must be a representable functor. ([Category Theory] §Representable Functors, ⁋Definition 1) On the scheme \(X\) obtained in this way, there is a universal element corresponding to \(\id_X\) in \(h_X(X)\), and any scheme morphism \(f:T\rightarrow X\) is obtained by pulling this back to \(T\), giving an element of \(F(T)\). This is the same mechanism by which, in [Algebraic Topology] §Classifying Spaces, ⁋Theorem 8 (Classification Theorem), a classifying map \(f:B\rightarrow \B G\) pulls back the universal bundle to give a principal \(G\)-bundle over \(B\); the difference is that here it is the actual scheme morphism \(f:T\rightarrow X\) itself, rather than a homotopy class, that appears.

Affine Space and Projective Space as Functors

We now examine geometric objects we already know from this viewpoint. The starting point is, of course, affine space and projective space.

Proposition 1 For the affine line \(\mathbb{A}^1=\Spec \mathbb{Z}[\x]\) over \(\mathbb{Z}\) and any scheme \(T\), the set of \(T\)-points of \(\mathbb{A}^1\) is given by

\[\mathbb{A}^1(T)\cong \Gamma(T, \mathcal{O}_T)=\mathcal{O}_T(T)\]

and this correspondence is natural in \(T\).

Proof

In the adjunction

\[\Hom_\Sch(T, \Spec A)\cong \Hom_\cRing(A, \Gamma(T, \mathcal{O}_T))\]

that we saw in §Affine Scheme, ⁋Theorem 13, set \(A=\mathbb{Z}[\x]\). Since the ring \(\mathbb{Z}[\x]\) is a free object in \(\cRing\), a ring homomorphism \(\mathbb{Z}[\x] \rightarrow \Gamma(T, \mathcal{O}_T)\) is the same as freely choosing the image \(\x\mapsto a\) of the generator \(\x\), which is exactly choosing one element \(a\in \Gamma(T, \mathcal{O}_T)\). Therefore

\[\mathbb{A}^1(T)=\Hom_\Sch(T, \Spec \mathbb{Z}[\x])\cong \Hom_\cRing(\mathbb{Z}[\x], \Gamma(T, \mathcal{O}_T))\cong \Gamma(T, \mathcal{O}_T)\]

The naturality of this correspondence means that for any \(\tau: T' \rightarrow T\), the restriction map \(\Gamma(T, \mathcal{O}_T) \rightarrow \Gamma(T', \mathcal{O}_{T'})\) commutes with the above correspondence, which follows from the naturality of the adjunction.

In the language introduced in the previous section, \(\mathbb{A}^1\) represents the global section functor \(T\mapsto\Gamma(T,\mathcal{O}_T)\). Here the universal element is the one corresponding to the identity morphism \(\id_{\mathbb{A}^1}\) in \(h_{\mathbb{A}^1}(\mathbb{A}^1)=\Hom_\Sch(\mathbb{A}^1,\mathbb{A}^1)\), and chasing through the correspondence of Proposition 1 above, we know that this corresponds to \(\x\) in \(\Gamma(\mathbb{A}^1,\mathcal{O}_{\mathbb{A}^1})=\mathbb{Z}[\x]\).

Now any scheme morphism \(f:T\rightarrow\mathbb{A}^1\) defines a (global) regular function \(f^\ast \x\) on \(T\) via the pullback map \(f^\ast:\Gamma(\mathbb{A}^1,\mathcal{O}_{\mathbb{A}^1})\rightarrow\Gamma(T,\mathcal{O}_T)\). Conversely, given any global regular function \(a\in\Gamma(T,\mathcal{O}_T)\), there is a ring homomorphism \(\mathbb{Z}[\x]\rightarrow\Gamma(T,\mathcal{O}_T)\) determined by \(\x\mapsto a\), and it is exactly this choice that gives a unique scheme morphism \(f:T\rightarrow\mathbb{A}^1\), which satisfies \(f^\ast\x=a\). Generalizing this to \(n\) generators, we obtain the following.

Proposition 2 For the affine \(n\)-space \(\mathbb{A}^n=\Spec \mathbb{Z}[\x_1,\ldots, \x_n]\) over \(\mathbb{Z}\), there exists a natural bijection

\[\mathbb{A}^n(T)\cong \Gamma(T, \mathcal{O}_T)^n\]

That is, a \(T\)-point of \(\mathbb{A}^n\) is an ordered tuple of \(n\) regular functions on \(T\).

Proof

As in the proof of Proposition 1, a ring homomorphism out of the free ring \(\mathbb{Z}[\x_1,\ldots, \x_n]\) is the same as freely choosing the images \(a_i\in \Gamma(T, \mathcal{O}_T)\) of the generators \(\x_i\), so we obtain

\[\mathbb{A}^n(T)\cong \Hom_\cRing(\mathbb{Z}[\x_1,\ldots, \x_n], \Gamma(T, \mathcal{O}_T))\cong \Gamma(T, \mathcal{O}_T)^n\]

In particular, when \(T=\Spec A\), we have \(\mathbb{A}^n(A)\cong A^n\), which agrees exactly with the classical intuition: an \(A\)-point of affine \(n\)-space is a coordinate made of \(n\) elements of \(A\). More generally, if \(T\) is not affine, then \(\Gamma(T,\mathcal{O}_T)\) may be richer, so \(\mathbb{A}^n(T)\) also carries more information than classical coordinates. On the other hand, extracting only the invertible elements with respect to multiplication from the global section functor yields the following functor.

Proposition 3 For \(\mathbb{G}_m=\Spec \mathbb{Z}[\t, \t^{-1}]\), there exists a natural bijection

\[\mathbb{G}_m(T)\cong \Gamma(T, \mathcal{O}_T)^\times\]

Here \(\Gamma(T, \mathcal{O}_T)^\times\) is the group of invertible elements of the ring \(\Gamma(T, \mathcal{O}_T)\).

Proof

Since \(\mathbb{Z}[\t, \t^{-1}]=\mathbb{Z}[\t]_\t\), ring homomorphisms \(\mathbb{Z}[\t, \t^{-1}] \rightarrow \Gamma(T, \mathcal{O}_T)\) are in bijection with those for which the image \(a\) of \(\t\) is invertible. Indeed, by the universal property of localization, ring homomorphisms out of \(\mathbb{Z}[\t]_\t\) correspond exactly to ring homomorphisms \(\mathbb{Z}[\t] \rightarrow \Gamma(T, \mathcal{O}_T)\) sending the image of \(\t\) to an invertible element, and as we saw in Proposition 1, this is choosing one invertible element \(a\in \Gamma(T, \mathcal{O}_T)^\times\).

Using this, we can obtain projective space from affine space. To this end, consider the open subscheme \(U\) obtained by removing the origin from \(\mathbb{A}^{n+1}\). First, define the scheme morphism

\[\mu:\mathbb{G}_m\times\mathbb{A}^{n+1}\rightarrow\mathbb{A}^{n+1}\]

by the following expression

\[\mathbb{Z}[\x_0,\ldots,\x_n]\rightarrow\mathbb{Z}[\t,\t^{-1},\x_0,\ldots,\x_n];\qquad \x_i\mapsto\t\x_i\]

This is the scalar multiplication action of \(\mathbb{G}_m\), and it restricts to \(U\) as well. Classically, projective space was thought of as the quotient \(U/\mathbb{G}_m\) by this action.

Examining this at the level of \(T\)-points, by Proposition 3 we have \(\mathbb{G}_m(T)=\Gamma(T,\mathcal{O}_T)^\times\), so the above action can be thought of as an invertible function \(u\) acting on a tuple \((a_0,\ldots,a_n)\in U(T)\) by

\[u\cdot(a_0,\ldots,a_n)=(ua_0,\ldots,ua_n)\]

Here each \(a_i\) is a function defined on \(T\), and the tuple \((a_0(t),\ldots, a_n(t))\) at any point \(t\) of \(T\) must not be the zero vector.

A natural expectation would be \(\mathbb{P}^n(T)=U(T)/\mathbb{G}_m(T)\), but this does not hold. Indeed, for such a tuple \((a_0,\ldots, a_n)\in U(T)\),

\[T_i=\{t\in T\mid \text{$a_i(t)$ invertible}\}\]

are open subschemes of \(T\), and since the ratios \(a_j/a_i\) are always defined on \(T_i\), the expressions

\[\mathbb{Z}[\x_0/\x_i,\ldots, \x_n/\x_i]\rightarrow \Gamma(T_i, \mathcal{O}_T);\qquad \x_j/\x_i\mapsto a_j/a_i\]

define morphisms \(T_i\rightarrow D_+(\x_i)\), which agree on the intersections and glue into a single morphism \(T\rightarrow\mathbb{P}^n\). Moreover, since the \(\mathbb{G}_m(T)\)-action does not change the ratios \(a_j/a_i\), the fact that a correspondence \(U(T)/\mathbb{G}_m(T)\rightarrow\mathbb{P}^n(T)\) is obtained is itself natural.

The problem is that this correspondence is not surjective in general, and the reason is that when the image of \(\psi: T \rightarrow \mathbb{P}^n\) spans several charts of \(\mathbb{P}^n\), the way of choosing homogeneous coordinates may differ from chart to chart. Concretely, given a morphism \(\psi:T\rightarrow\mathbb{P}^n\), the \(V_i=\psi^{-1}(D_+(\x_i))\) form an open cover of \(T\), and on each \(V_i\) the tuple normalized so that the \(i\)-th coordinate is \(1\),

\[a^{(i)}=(\psi^\ast(\x_0/\x_i),\ldots, \psi^\ast(\x_n/\x_i))\in U(V_i)\]

is defined, and the transition relation connecting two different charts is given by

\[a^{(i)}=\psi^\ast(\x_j/\x_i)\cdot a^{(j)}\]

The problem is that the factor \(\psi^\ast(\x_j/\x_i)\) here is an invertible function defined only on \(V_i\cap V_j\), whereas the factor considered when gluing the \(T_i\) above came from \(\mathbb{G}_m(T)\), that is, from global scaling factors. Thus, if we simply think \(\mathbb{P}^n(T)=U(T)/\mathbb{G}_m(T)\), we miss these \(T\)-points.

On the other hand, we happen to know well a way to package a scaling factor locally in each such case: consider a line bundle defined on \(T\). On the resulting line bundle \(\mathcal{L}\), the coordinates of the local tuples combine into \(n+1\) global sections \(s_0,\ldots, s_n\in \Gamma(T, \mathcal{L})\), and the fact that the \(i\)-th coordinate of \(a^{(i)}\) was \(1\) means that \(s_i\) generates \(\mathcal{L}\) on \(V_i\). To make these ratios well-defined, we must require the following condition.

Definition 4 By globally generating sections \(s_0,\ldots, s_n\in \Gamma(T, \mathcal{L})\) of a line bundle \(\mathcal{L}\) on a scheme \(T\), we mean that at each point \(t\in T\), the stalk \(\mathcal{L}_t\) is generated as an \(\mathcal{O}_{T,t}\)-module by the germs \((s_0)_t,\ldots, (s_n)_t\). Two data \((\mathcal{L}, s_0,\ldots, s_n)\) and \((\mathcal{L}', s_0',\ldots, s_n')\) are isomorphic if there exists an \(\mathcal{O}_T\)-module isomorphism \(\theta:\mathcal{L} \rightarrow \mathcal{L}'\) such that \(\theta(s_i)=s_i'\) for each \(i\).

This isomorphism condition records the scaling of homogeneous coordinates. In particular, when \(\mathcal{L}=\mathcal{O}_T\), the automorphisms of \(\mathcal{O}_T\) are only multiplication by an invertible function \(u\in\Gamma(T,\mathcal{O}_T)^\times\). Thus \((\mathcal{O}_T,s_0,\ldots,s_n)\) and \((\mathcal{O}_T,us_0,\ldots,us_n)\) are isomorphic data.

Under this definition, the functor of points of \(\mathbb{P}^n\) is described cleanly as follows.

Theorem 5 For the projective space \(\mathbb{P}^n=\Proj \mathbb{Z}[\x_0,\ldots, \x_n]\) over \(\mathbb{Z}\), \(\mathbb{P}^n(T)\) is naturally in bijection with the isomorphism classes of data \((\mathcal{L}, s_0,\ldots, s_n)\) consisting of a line bundle \(\mathcal{L}\) on \(T\) and globally generating sections \(s_0,\ldots, s_n\in \Gamma(T, \mathcal{L})\).

Proof

Suppose a morphism \(\psi: T \rightarrow \mathbb{P}^n\) is given. The twisting sheaf \(\mathcal{O}_{\mathbb{P}^n}(1)\) on \(\mathbb{P}^n\) is a line bundle whose global sections \(\x_0,\ldots, \x_n\) are globally generating sections, so taking pullback, we obtain a line bundle \(\mathcal{L}=\psi^\ast \mathcal{O}_{\mathbb{P}^n}(1)\) on \(T\) and sections \(s_i=\psi^\ast \x_i\). Since pullback preserves the property of being globally generating sections, \((\mathcal{L}, s_0,\ldots, s_n)\) forms the data above.

Conversely, suppose a line bundle \(\mathcal{L}\) on \(T\) and its globally generating sections \(s_0,\ldots, s_n\) are given. For each \(i\), the locus \(T_{s_i}=\{t\in T\mid (s_i)_t \text{ generates } \mathcal{L}_t\}\) where the section \(s_i\) generates is open, and since the sections globally generate \(\mathcal{L}\), \(\{T_{s_i}\}_{i=0}^n\) forms an open cover of \(T\). On \(T_{s_i}\), \(s_i\) gives a trivialization of \(\mathcal{L}\vert_{T_{s_i}}\), so \(s_j/s_i\in \Gamma(T_{s_i}, \mathcal{O}_T)\) is well-defined for each \(j\). With this, in the same manner as §Morphisms of Schemes, ⁋Example 5, we define \(T_{s_i} \rightarrow D_+(\x_i)\) and check the gluing condition on the intersections to obtain a morphism \(\psi: T \rightarrow \mathbb{P}^n\).

The fact that these two constructions are inverse to each other, and that isomorphic data give the same morphism, is confirmed from the fact that even if we transport the whole \((\mathcal{L}, s_0,\ldots, s_n)\) along an \(\mathcal{O}_T\)-module isomorphism, the \(s_j/s_i\) do not change, so we get the same gluing data. Naturality means that for \(\tau: T' \rightarrow T\), pulling back the above data agrees with composing the morphism.

Concretely, let us revisit, in this language, the \(\mathbb{K}[\epsilon]/(\epsilon^2)\)-points of \(\mathbb{P}^n_\mathbb{K}\) that we examined in §From Varieties to Schemes, ⁋Example 5. Since this is a one-point space, the only line bundle on it is the trivial line bundle, and therefore, once we fix a trivialization, \(\Gamma(T,\mathcal{L})\cong\Gamma(T,\mathcal{O}_T)=A\), so choosing a section of a line bundle on it is the same as choosing an element of \(A\). Meanwhile, the condition that these are globally generating becomes, since the stalk at the unique point of \(\Spec A\) is \(A\) itself, the condition that some \(a_i\) is invertible, which is exactly the condition \((a_0,\ldots, a_n)\in U(A)\).

Now, to see how Theorem 5 works, let us look at its isomorphism classes: since the automorphisms of \(\mathcal{O}_T\) are now only multiplication by elements of \(A^\times\), two tuples give the same \(A\)-point if and only if they are \(A^\times\)-multiples of each other. Therefore

\[\mathbb{P}^n(A)=U(A)/A^\times\]

where \(U(A)\) consists of the \((n+1)\)-tuples \((a_0, \ldots, a_n)\) with coordinates in \(A\) having at least one invertible coordinate, and \(A^\times\) acts by multiplying every component by an element of \(A^\times\).

Now let \(V=\mathbb{K}^{n+1}\), and write an element of \(U(A)\) in the form \(a=b+\epsilon c\). Then we can consider the correspondence

\[\widetilde{\rho}: U(A)\rightarrow \mathbb{P}^n(\mathbb{K});\qquad b+\epsilon c\mapsto [b]\]

Intuitively, this is the function that forgets the tangent vector direction and remembers only the base point. Computing the \(A^\times\) action on \(U(A)\) with the expanded expression as above,

\[u(b+\epsilon c)=\lambda b+\epsilon(\lambda c+\mu b)\]

so this \(\widetilde{\rho}\) descends to \(\rho: \mathbb{P}^n(A)\rightarrow \mathbb{P}^n(\mathbb{K})\).

The \(\rho\) thus obtained is in fact the map functorially induced from the ring homomorphism

\[q:A\rightarrow\mathbb{K};\qquad \epsilon\mapsto0\]

That is, if we let \(\iota:\Spec\mathbb{K}\rightarrow\Spec A\) be the morphism corresponding to this ring homomorphism, we can check that \(\rho\) sends an \(A\)-point \(\psi:\Spec A\rightarrow\mathbb{P}^n\) to the composition \(\psi\circ\iota\), and one can verify that this agrees exactly with the \(\rho\) defined above.

Therefore \(\rho^{-1}(\ell)\) is the set of \(A\)-points that restrict to \(\ell\) along \(\Spec\mathbb{K}\rightarrow\Spec A\). Let us compute this directly. First, choose a class \([b+\epsilon c]\in\rho^{-1}(\ell)\); then \(\ell=\mathbb{K}b\subseteq V\), and from this \(A\)-point we can define a \(\mathbb{K}\)-linear map

\[\phi:\ell\rightarrow V/\ell,\qquad \phi(b)=c+\ell\]

This does not depend on the choice of representative. Indeed, using the earlier computation, for \(\lambda b+\epsilon(\lambda c+\mu b)\) belonging to the same class as \(b+\epsilon c\), the value to which \(\lambda b\) is moved via \(\phi\) is

\[(\lambda c+\mu b)+\ell=\lambda(c+\ell)\]

Conversely, suppose a one-dimensional subspace \(\ell\subseteq V\) and a linear map \(\phi:\ell\rightarrow V/\ell\) are given. Choose a nonzero \(b\in\ell\) and a lift \(c\in V\) of \(\phi(b)\); then we obtain \(b+\epsilon c\in U(A)\). This is, first of all, independent of the lift of \(b\): if we choose another lift \(c'=c+\mu b\), then

\[b+\epsilon c'=(1+\epsilon\mu)(b+\epsilon c)\]

which belongs to the same \(A^\times\)-class. Similarly, choosing a basis \(b'=\lambda b\) and its lift \(c'=\lambda c+\mu b\) gives

\[b'+\epsilon c'=(\lambda+\epsilon\mu)(b+\epsilon c)\]

so an \(A^\times\)-class independent of any choice is given. One can check that this is the inverse process of the above construction, and therefore we obtain the isomorphism

\[\mathbb{P}^n(A)\cong\{(\ell,\phi)\mid \ell\in \mathbb{P}^n(\mathbb{K}),\ \phi\in \Hom_\mathbb{K}(\ell, V/\ell)\}\]

That is, \(\mathbb{P}^n(A)\) is the collection of tangent vectors at all points of \(\mathbb{P}^n\), and since \(\rho\) keeps only the base point among them, \(\rho^{-1}(\ell)\) gives the tangent space \(T_\ell\mathbb{P}^n\) of \(\mathbb{P}^n\) at \(\ell\). More generally, for any \(\mathbb{K}\)-scheme \(X\), \(X(A)\) is the set collecting the tangent spaces at all \(\mathbb{K}\)-points.

Example 6 We now concretely examine the functor defined by the projective space considered above. First, the \(\mathcal{L}\) and globally generating sections \(s_0,\ldots, s_n\) representing \(\mathbb{P}^n(T)\) can be rewritten as the surjection

\[\mathcal{O}_T^{\oplus n+1}\twoheadrightarrow \mathcal{L};\qquad e_i\mapsto s_i\]

We define an isomorphism between such surjections by the diagram

and consider the functor \(F_{n+1}\) that takes \(T\in\Sch\) and assigns these isomorphism classes. Here functoriality at the level of morphisms is given by pulling back the surjection via \(\tau:T'\rightarrow T\) to define

\[\mathcal{O}_{T'}^{\oplus n+1}\twoheadrightarrow\tau^\ast\mathcal{L}\]

That is, this correspondence defines a contravariant functor

\[F_{n+1}:\Sch^\op\rightarrow\Set\]

From this viewpoint, Theorem 5 says that there exists a bijection

\[\mathbb{P}^n(T)\cong F_{n+1}(T)\]

natural in every scheme \(T\), and that the projective space \(\mathbb{P}^n\) represents this functor. By [Category Theory] §Representable Functors, ⁋Theorem 4 (Yoneda), the universal element corresponding to \(\id_{\mathbb{P}^n}\) is the quotient bundle

\[\mathcal{O}_{\mathbb{P}^n}^{\oplus n+1}\twoheadrightarrow\mathcal{O}_{\mathbb{P}^n}(1)\]

on \(\mathbb{P}^n\), and any \(T\)-point is obtained by pulling this universal quotient back to \(T\) to get a rank \(1\) quotient.

The Grassmannian is obtained from the above functor by replacing the rank \(1\) target with a rank \(k\) target. That is, for integers \(0<k<n\), we assign to a scheme \(T\) the set

\[F_{k,n}(T)=\left\{\mathcal{O}_T^n\twoheadrightarrow\mathcal{Q}\mid \mathcal{Q}\text{ is locally free of rank }k\right\}\big/\cong\]

and to a morphism \(\tau:T'\rightarrow T\) we assign pullback. We denote by \(\Gr(k,n)\) the scheme representing this contravariant functor, so there exists a bijection

\[\Gr(k,n)(T)\cong F_{k,n}(T)\]

natural in every scheme \(T\).

When \(T=\Spec\mathbb{K}\), an element of \(F_{k,n}(T)\) is a rank \(k\) quotient space \(\mathbb{K}^n\twoheadrightarrow Q\). Since this is uniquely determined by its kernel, an \((n-k)\)-dimensional subspace \(\bar S\subseteq\mathbb{K}^n\), \(\Gr(k,n)(\mathbb{K})\) agrees with the set of such subspaces. In the convention of [Algebraic Varieties] §Grassmann Varieties, ⁋Definition 1, which classifies subspaces directly, this set is denoted \(\Gr(n-k,n)\). In particular, when \(k=1\), we classify rank \(1\) quotients, recovering \(\mathbb{P}^{n-1}\) of Theorem 5.

Showing that such a functorial definition is representable is the starting point of moduli theory; in the case of the Grassmannian, one can construct the representing scheme and the universal quotient using the fact that the quotient bundle is expressed as matrix data over standard affine charts.

Fiber Products as Functors

The functor of points viewpoint fits perfectly with the fiber product defined in §Fiber Products, ⁋Definition 1. The universal property of the fiber product \(X\times_S Y\) directly tells us, at the level of functors, how its \(T\)-points are determined for any test scheme \(T\).

Proposition 7 Suppose scheme morphisms \(X \rightarrow S\) and \(Y \rightarrow S\) are given. Then for any scheme \(T\), there exists a natural bijection

\[(X\times_S Y)(T)\cong X(T)\times_{S(T)} Y(T)\]

Here the right-hand side is the fiber product in \(\Set\), that is, the set of ordered pairs in \(X(T)\times Y(T)\) for which \(X(T) \rightarrow S(T)\) and \(Y(T) \rightarrow S(T)\) give the same value.

Proof

The universal property of §Fiber Products, ⁋Definition 1 means that a morphism from \(T\) to \(X\times_S Y\) corresponds uniquely to a pair \(\psi_X: T \rightarrow X\) and \(\psi_Y: T \rightarrow Y\) whose compositions to \(S\) agree, that is, a pair such that \(\psi_X\) and \(\psi_Y\) map to the same \(S\)-point via \(X(T) \rightarrow S(T)\) and \(Y(T) \rightarrow S(T)\). Written in the language of sets, this is

\[(X\times_S Y)(T)\cong \{(\psi_X, \psi_Y)\in X(T)\times Y(T)\mid \psi_X, \psi_Y \text{ map to the same element of } S(T)\}=X(T)\times_{S(T)} Y(T)\]

Naturality means that for \(\tau: T' \rightarrow T\), the pullbacks on both sides agree, which follows from the naturality of the universal property.

Proposition 7 lets us interpret the fiber product as the operation of taking fiber products pointwise at the level of functors. From this viewpoint, the existence proof of §Fiber Products, ⁋Theorem 8 is reinterpreted as the task of showing that the functor \(T\mapsto X(T)\times_{S(T)} Y(T)\), which is trivially defined pointwise, is representable. In particular, for the product \(X\times Y=X\times_{\Spec \mathbb{Z}} Y\), we simply have \((X\times Y)(T)\cong X(T)\times Y(T)\).

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References

[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundations of algebraic geometry. Available online.

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