Every principal ideal domain is a UFD and hence normal by the remark above. If \(A\) is a principal ideal domain with maximal ideal \(\mathfrak m\), one defines a valuation \(v\) on \(A\) by \(v(x) = \max \left\{ n;\ x\in \mathfrak m^n \right\} \) for \(x\in A\) and extends this to \(K\) by the rule \(v(xy) = v(x)+v(y)\). Conversely, \(A\) is the discrete valuation ring for the valuation \(v\) and \(t\in A\) an element with \(v(t)=1\), one shows that the non-zero ideals of \(A\) are precisely the ideals \((t^i)\), \(i\in \mathbb {N}\).

It remains to show that every normal noetherian local domain of dimension \(1\) is a discrete valuation ring. See, e.g., [ AM ] Proposition 9.2.

(1) See  [ M2 ] Theorem 11.5. (2) Let \(a\in \mathfrak p\) be any non-zero element, and write \(a = p_1\cdot \cdots \cdot p_r\) as a product of prime elements. Since \(\mathfrak p\) is a prime ideal, there exists \(i\) with \(p_i\in \mathfrak p\), so \(0\ne (p_i) \subseteq \mathfrak p\), and since \(\mathfrak p\) is a minimal non-zero prime ideal, we obtain \(\mathfrak p = (p_i)\). (See  [ M2 ] Theorem 20.1 for a converse.)

Injectivity. If \(D\) is a Weil divisor or a Cartier divisor such that \(D\) and \(-D\) are effective, then \(D\) is trivial. It therefore suffices to show that the inverse image of the subset of effective Weil divisors under the homomorphism \(\operatorname{cyc}\) consists of effective Cartier divisors. So let \(D\) be a Cartier divisor on \(X\) such that \(\operatorname{cyc}(D)\) is effective. We can check that \(D\) is effective locally on \(X\), so we may assume that \(X = \operatorname{Spec}A\) for an integrally closed domain \(A\), and that \(D\) is principal, say given by \((X, f)\). By assumption \(f\in K(X)\) is contained in \(A_{\mathfrak p}\) for every \(\mathfrak p\in \operatorname{Spec}A\) of height \(1\), and it follows from Lemma 3.23 Part (1) that \(f\in A\), as desired.

Surjectivity. Now assume that \(X\) is locally factorial. We construct an inverse to the map \(\operatorname{cyc}\). If \(Z\subset X\) is an integral closed subscheme of \(X\) of codimension \(1\) with corresponding ideal sheaf \({\mathscr I}_Z\subseteq {\mathscr O}_X\), then for every \(x\in X\), \({\mathscr I}_{Z,x}\) is a principal ideal in \({\mathscr O}_{X,x}\) by Lemma 3.23 Part (2). Using Proposition 2.19, we find an affine open cover \((U_i)_i\) of \(X\) together with elements \(f_i\in K(X)\) such that \({\mathscr I}_{Z|U_i} = f_i{\mathscr O}_X\) (inside \({\mathscr K}_X\)), for each \(i\). We then map \([Z]\) to the Cartier divisor \((U_i, f_i)_i\) (well-defined since elements in a domain that generate the same principal ideal differ at most by a unit), and extend this construction to a map \(d\colon Z^1(X)\to \operatorname{Div}(X)\) by linearity. By construction we have \(\operatorname{cyc}\circ d = \operatorname{id}\), and this implies that \(\operatorname{cyc}\) is surjective (and hence bijective with inverse \(d\)).

We can phrase the definition of principal Weil divisor as saying that it is the image under \(\operatorname{cyc}\) of a principal Cartier divisor. It is therefore clear that we also obtain an injection (or an isomorphism, respectively, if \(X\) is locally factorial) \(\operatorname{DivCl}(X)\to \operatorname{Cl}(X)\).

Clearly (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii). To show (iii) \(\Rightarrow \) (iv) first note that the fiber over \(\eta _Y\) is either empty or equal to \(\left\{ \eta _X \right\} \). (In fact, a closed point \(x\in X\) cannot map to \(\eta _Y\), because we would then obtain chain of inclusion \(k = \kappa (x) \supseteq \kappa (\eta _Y)=K(Y) \supseteq k\), contradicting the fact that the transcendence degree of \(K(Y)\) over \(k\) is \(1\).) Fibers over closed points are closed, so either all fibers are finite or there exists a closed point \(y\in Y\) with \(f^{-1}(y)=X\), i.e., \(f\) is constant. We also see that \(f\) is dominant, if all fibers are finite, because then \(\eta _X\) cannot be mapped to a closed point in \(Y\). This shows (iv) \(\Rightarrow \) (ii). Finally, (ii) \(\Rightarrow \) (i) holds, because \(f\) is proper (Lemma 1.23).

To prove that the functor is fully faithful, let \(X\), \(Y\) be integral smooth projective curves over \(k\) and consider the map

from the set of dominant \(k\)-morphisms \(X\to Y\) to the set of \(k\)-algebra homomorphisms \(K(Y)\to K(X)\). We split up the proof into the following two lemmas (the first for injectivity, the second for surjectivity) where we can relax the assumptions a little.

We prove the essential surjectivity later on.

This holds, because \(Y\) is separated over \(k\) and \(\operatorname{Spec}(K(X))\) has dense image in \(X\). (Given two morphisms \(f, g\colon X\to Y\), their equalizer is a closed subscheme in \(X\), because \(Y\) is separated over \(k\); if \(f\) and \(g\) induce the same morphism on function fields, then the equalizer contains the generic point of \(X\), hence equals the integral scheme \(X\).)

Let \(\varphi \colon K(Y)\to K(X)\) be a \(k\)-homomorphism. It is enough to find, for every closed point \(x\) in \(X\), an open neighborhood \(U_x\) and a morphism \(U_x\to Y\) which induces \(\varphi \) on function fields. In fact, the previous lemma shows that these morphisms necessarily coincide on the intersections \(U_x\cap U_{x'}\), and we obtain the desired morphism \(X\to Y\) by gluing.

So let \(x\in X\) be a closed point. By the following lemma we find a morphism \(\operatorname{Spec}({\mathscr O}_{X, x})\to Y\) which induces \(\varphi \). Its image is contained in some affine open \(V \subset Y\) (any open neighborhood of the image of the closed point will do), so we have a ring homomorphism \(\psi \colon \Gamma (V, {\mathscr O}_Y)\to {\mathscr O}_{X, x}\). Let \(U\) be an affine open neighborhood of \(x\). Then \(x\) corresponds to a prime ideal \({\mathfrak p}\subset \Gamma (U, {\mathscr O}_X)\) and \({\mathscr O}_{X,x}\) is the localization \(\Gamma (U, {\mathscr O}_X)_{\mathfrak p}\). Since \(\Gamma (V, {\mathscr O}_Y)\) is a finitely generated \(k\)-algebra, the image of \(\psi \) is contained in a suitable localization of \(\Gamma (U, {\mathscr O}_X)\) with respect to one element not contained in the prime ideal corresponding to \(x\), i.e., we may extend the morphism \(\operatorname{Spec}({\mathscr O}_{X, x})\to Y\) to some principal open subset of \(U\) that still contains \(x\). This concludes the proof of surjectivity.

The uniqueness is clear since \(f\) is separated. To show existence, we can equivalently show that the morphism \(f'\colon Y\times _S\operatorname{Spec}(R)\to \operatorname{Spec}(R)\) has a section \(s\colon \operatorname{Spec}(R)\to Y\times _S\operatorname{Spec}(R)\), i.e., \(f'\circ s=\operatorname{id}\), that extends the morphism \(\operatorname{Spec}(K)\to Y\times _S\operatorname{Spec}(R)\) given by the diagram. Let \(y\) be the image point of \(\operatorname{Spec}(K)\). We view the closure \(\overline{\left\{ y \right\} }\) as a reduced closed subscheme of \(Y\times _S\operatorname{Spec}(R)\). Since \(f\) is proper, \(f'\) is closed, thus \(f'(\overline{\{ y\} }) = \operatorname{Spec}(R)\). So there exists \(y'\in \overline{\{ y\} }\) mapping to the closed point of \(\operatorname{Spec}(R)\). We then obtain a chain of inclusions

Since \({\mathscr O}_{\overline{\{ y\} }, y}\) is of dimension \(0\) and reduced, it is a field. Since it contains \(R\), we have \({\mathscr O}_{\overline{\{ y\} }, y}=K\). We claim that \(R={\mathscr O}_{\overline{\{ y\} }, y'}\); this immediately gives us the desired section. Now, since \(R\) is a valuation ring, for \(s\in {\mathscr O}_{\overline{\{ y\} }, y'}\), if \(s\not \in R\), then \(s^{-1}\in \mathfrak m_R\). But then \(s^{-1}\in {\mathscr O}_{\overline{\{ y\} }, y'}^\times \), contradicting the fact that the inclusion \(R \subseteq {\mathscr O}_{\overline{\{ y\} }, y'}\) is a local ring homomorphism.

Let \(K/k\) be a finitely generated field extension of transcendence degree \(1\). Let \(t\in K\setminus k\) be any element. Since \(k\) is algebraically closed, \(t\) is transcendental over \(k\). Since the transcendence degree is \(1\), the extension \(K/k(t)\) is algebraic, and since \(K/k\) is finitely generated, it is even finite.

We want to construct an integral smooth projective curve \(X\), such that \(K(X) = K\). (The embedding \(K(\mathbb {P}^1_k)=k(t) \to K\) will then correspond to a morphism \(X\to \mathbb {P}^1_k\); this points us, in some sense, in the right direction to construct \(X\).)

We define \(X\) as the “normalization of \(\mathbb {P}^1_k\) in \(K\)”. We sketch the construction here and refer to [ GW1 ] Sections (12.11), (15.7) for further details. For any affine open \(U\subset \mathbb {P}^1_k\), say \(U=\operatorname{Spec}(A)\) with \(A \subset k(t) \subseteq K\), let \(\widetilde{A}\) be the integral closure of \(A\) in \(K\). This is a Dedekind ring and the ring homomorphism \(A\to \widetilde{A}\) is finite ( [ GW1 ] Corollary 12.52 (1)). Set \(\widetilde{U} = \operatorname{Spec}(\widetilde{A})\).

Since taking integral closure in some extension field is compatible with localization, this construction “globalizes”, i.e., we can glue all the schemes \(\widetilde{U}\) for varying \(U\) to obtain a scheme \(X\). Since \(X\) is covered by the \(\widetilde{U}\), it is an integral Dedekind scheme. Moreover, by construction the morphism \(X\to \mathbb {P}^1_k\) is finite; as was remarked above, this implies that \(X\) is projective over \(k\). (Note that even though we did not prove the final statement yet, \(X\) is certainly proper over \(k\), since finite morphisms are proper, and for the fully faithfulness we never used “projective”, but only “proper”. Thus if \(X\) with \(K(X)=K\) exists at all, then it has to be the one that we constructed.)

For flatness, note that the homomorphisms between the local rings are injective, because those are subrings of the respective fields of rational functions. Since a module over a discrete valuation ring is flat if and only if it is torsion-free, we get the result.

To show that \(f\) is finite, choose an element \(t\in K(Y)\) that is transcendental over \(k\). Its image in \(K(X)\) of course is again transcendental over \(k\). By the construction used in the proof of essential surjectivity in Theorem 3.29, we obtain morphisms \(X\to Y\to \mathbb {P}^1_k\) such that \(Y\to \mathbb {P}^1_k\) is finite, in particular separated, and the composition \(X\to \mathbb {P}^1_k\) is finite. It follows by Lemma ?? (1) that \(X\to Y\) is finite.

For the final statement, we use Theorem 2.23.

It is enough to handle the case \(D = [y]\) for some closed point \(y\in Y\). We need to show that \(\deg (f^*([y])) = \deg (f) = [K(X):K(Y)]\).

Claim. We have \(\deg (f^*([y])) = \dim _k\Gamma (f^{-1}(y), {\mathscr O}_{f^{-1}(y)})\), where \(f^{-1}(y) = X\times _Y\operatorname{Spec}(\kappa (y))\) is the scheme-theoretic fiber.

Once we have proved the claim, the lemma follows, because we have \(\dim _k\Gamma (f^{-1}(y), {\mathscr O}_{f^{-1}(y)}) = [K(X):K(Y)]\), since \(f\) is finite flat (this number is the rank of the locally free module \(\Gamma (f^{-1}(V), {\mathscr O}_X)\) over \(\Gamma (V, {\mathscr O}_Y)\), \(V \subset Y\) affine open).

Proof of claim. Let \(V = \operatorname{Spec}(A)\) be an affine open neighborhood of \(y\) such that there exists \(t\in A\) with \(V(t) = \left\{ y \right\} \) as a reduced closed subscheme. (Extend a uniformizer of \({\mathscr O}_{Y,y}\) to a suitable open neighborhood.) Write \(f^{-1}(V) = \operatorname{Spec}(B)\), where \(B\) is a finite \(A\)-algebra. There exist only finitely many (closed) points in \(X\) mapping to \(y\), say \(x_1,\dots , x_r\). We have

The final equality holds because \(\operatorname{Spec}(B/tB)\) topologically is the finite space \(\left\{ x_1, \dots , x_r \right\} \) with the discrete topology, so the ring \(B/tB\) is the product of the finitely many local rings at its points. We have \(\dim _k({\mathscr O}_{X, x_i}/(t)) = v_{x_i}(t)\). The claim follows.

Identifying \(K(\mathbb {P}^1_k)= k(t)\) implies that we choose a \(0\) and \(\infty \) in \(\mathbb {P}^1_k\). In other words, the standard charts of \(\mathbb {P}^1_k\) are \(\operatorname{Spec}(k[t])\) (and \(0\) corresponds to the prime ideal \((t)\) in this chart) and \(\operatorname{Spec}(k[t^{-1}])\) (and \(\infty \) corresponds to the prime ideal \((t^{-1})\) in this chart).

We have \(\bar{f}(x)=0\) if and only if the morphism \(\operatorname{Spec}({\mathscr O}_{X,x})\to \mathbb {P}^1_k\) induced by \(\bar{f}\) factors through \(\operatorname{Spec}(k[t])\), i.e., \(f\in {\mathscr O}_{X,x}\), and maps the maximal ideal \({\mathfrak m}_x\) to \((t)\), i.e., \(v_x(f) {\gt} 0\). In this case \(v_x(f)\) is the coefficient of \(\mathop{\rm div}\nolimits (f)\) at \(x\), and also the coefficient of \(\bar{f}^*([0])\) at \(x\).

Symmetrically, we have an analogous picture for \(\infty \) in place of \(0\). Altogether, the claim follows.

By the lemma, we have \(\mathop{\rm div}\nolimits (f) = \bar{f}^*([0] - [\infty ])\), and hence

By assumption, \([x]-[y] = \mathop{\rm div}\nolimits (f)\) for some \(f\in K(X)^\times \). Since \(x\ne y\), the divisor \([x]-[y]\) is non-trivial, so \(f\) is transcendental over \(k\) and defines a dominant morphism \(\bar{f}\colon X\to \mathbb {P}^1_k\). Then \(\bar{f}^*[0] = [x]\), so \(\bar{f}\) has degree \(1\) by Proposition 3.37 and therefore is an isomorphism.

Omitted for now (maybe on the next problem sheet?).

The corollary is easy to prove with the Theorem of Riemann-Roch at hand. In fact, for (1) use the theorem with \(D=0\) the trivial divisor, for (2) use \(D=K\), and for (3) use that under the assumption there \(\deg (K-D) {\lt} 0\), whence \(\ell (K-D)=0\), as remarked before.

It is clear that we have the first short exact sequence when we think of integral closed subschemes in terms of their generic points. It is easy to check that the “restriction map” \(Z^1(X)\to Z^1(U)\) induces a homomorphism between the class groups, and this yields the second exact sequence.

We can make the gluing construction for \({\mathscr O}(d)\) explicit by identifying \(\Gamma (D_+(X_i), {\mathscr O}(d))\) with \(X_i^d R\left[\frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}\right]\), and correspondingly

(inside \(R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\)). This means that the restriction map is just the inclusion map. (If \(R\) is a domain, then this describes an embedding of \({\mathscr O}(d)\) into the constant sheaf \({\mathscr K}_{\mathbb {P}^n_R}\) with values the field of rational functions.)

With this description, since the relevant restriction maps are injective, we can identify \(\Gamma (\mathbb {P}^n_R, {\mathscr O}(d))\) with the intersection

One checks that this intersection is \(R[X_0, \dots , X_n]_d\), as claimed.

If we identify \(\operatorname{Cl}(\mathbb {P}^n_k) = \operatorname{Pic}(\mathbb {P}^n_k)\) and apply Proposition 3.48 to \(X=\mathbb {P}^n_k\) and \(U=D_+(X_0)\cong \mathbb {A}^n_k\), we obtain a surjection \(\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_k)\). The injectivity statement of the previous corollary implies that this surjection (which allows us to identify \(\operatorname{Pic}(\mathbb {P}^n_k)\) with some quotient of \(\mathbb {Z}\)) must be an isomorphism. (At this point we have not yet shown that the map \(d\mapsto {\mathscr O}(d)\) is surjective, and hence an isomorphism; this will follow from the discussion in the following section.)

Consider the divisor \([Z]\) given by \(Z\). Viewed as a Cartier divisor, it corresponds to an embedding \({\mathscr O}_{\mathbb {P}^n_k}([Z])\to {\mathscr K}_{\mathbb {P}^n_k}\) whose image contains \({\mathscr O}_{\mathbb {P}^n_k}\) since the divisor \([Z]\) is effective. Let \(f\in k[X_0, \dots , X_n]_d\) be the image of \(1\in \Gamma (\mathbb {P}^n_k, {\mathscr O})\) in \(\Gamma (\mathbb {P}^n_k, {\mathscr O}([Z])) = k[X_0, \dots , X_n]_d\) under this embedding. Since the embedding \({\mathscr O}_{\mathbb {P}^n_k}([Z])\to {\mathscr K}_{\mathbb {P}^n_k}\) is entirely determined by this image, the above discussion shows that \([Z] = [V_+(f)]\) as divisors and hence that \(Z = V_+(f)\).

A homomorphism \(\alpha \colon {\mathscr O}_S^{n+1}\twoheadrightarrow {\mathscr L}\) corresponds to \(n+1\) global sections in \(\Gamma (S, {\mathscr L})\) (the “images of the standard basis vectors”). Thus \(X_0, \dots , X_n\in \Gamma (\mathbb {P}^n_R, {\mathscr O}(1))\) give rise to a homomorphism \({\mathscr O}_{\mathbb {P}^n_R}^{n+1}\to {\mathscr O}(1)\). This homomorphism is surjective. (In fact, looking back at the construction of \({\mathscr O}(1)\) by gluing and the way how we identified the global sections of \({\mathscr O}(1)\) with \(R[X_0, \dots , X_n]_1\), under the identification \({\mathscr O}(1)_{D_+(X_i)} \cong {\mathscr O}_{D_+(X_i)}\) the restriction of \(X_i\) to \(D_+(X_i)\) corresponds to \(1\in \Gamma (D_+(X_i), {\mathscr O}_{D_+(X_i)})\) and in particular is non-zero in every fiber.)

Given a morphism \(S\to \mathbb {P}^n_R\), we can pull this homomorphism back to \(S\) and obtain an element of the right hand side in the statement of the proposition.

Conversely, given a pair \(({\mathscr L}, \alpha )\) on \(S\), we can think of the corresponding morphism \(S\to \mathbb {P}^n_R\) in terms of homogeneous coordinates (i.e., for \(K\)-valued points for some field \(K\)), as follows: Denote by \(f_0, \dots , f_n\in \Gamma (S, {\mathscr L})\) the global sections corresponding to \(\alpha \). For a point \(x\in S\), the fiber \({\mathscr L}(x)\) is a one-dimensional \(\kappa (x)\)-vector space generated by the elements \(f_0(x), \dots , f_n(x)\) (i.e., at least one of them is \(\ne 0\) – this holds since \(\alpha \) is surjective). We choose an isomorphism \({\mathscr L}(x) \cong \kappa (x)\), and hence can view the \(f_i(x)\) as elements of \(\kappa (x)\). Then the morphism \(S\to \mathbb {P}^n_S\) maps \(x\) to \((f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))\). While the individual \(f_i(x)\), as elements of \(\kappa (x)\), depend on the choice of isomorphism \({\mathscr L}(x)\cong \kappa (x)\), the point \((f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))\) is independent of this choice.

To make this rigorous, consider a pair \(({\mathscr L}, \alpha )\) as above, and for \(i\in \{ 0,\dots , n\} \) define

where \(e_i\in {\mathscr O}_S^{n+1}\) denotes the \(i\)-th standard basis vector. This defines an open cover of \(S\). By definition, composing \(\alpha \) with the injection \({\mathscr O}_S\to {\mathscr O}_S^{n+1}\) as the \(i\)-th summand induces a trivialization \({\mathscr O}_{S_i}\cong {\mathscr L}_{|S_i}\) of the restriction of \({\mathscr L}\). We obtain a morphism \(S_i\to D_+(X_i) = \operatorname{Spec}k[\frac{X_0}{X_i},\dots , \frac{X_n}{X_i}]\) by mapping \(\frac{X_j}{X_i}\) to the image of \(\alpha (e_j)\in \Gamma (S_i, {\mathscr L})\) under the isomorphism \(\Gamma (S_i, {\mathscr L}) \to \Gamma (S_i, {\mathscr O}_S)\). These morphisms can be glued, and one obtains the desired morphism \(S\to \mathbb {P}^n_R\).

To conclude the proof, one checks that the two constructions are inverse to each other.