## 31.17 Norms

Let $\pi : X \to Y$ be a finite morphism of schemes and let $d \geq 1$ be an integer. Let us say there exists a norm of degree $d$ for $\pi$1 if there exists a multiplicative map

$\text{Norm}_\pi : \pi _*\mathcal{O}_ X \to \mathcal{O}_ Y$

of sheaves such that

1. the composition $\mathcal{O}_ Y \xrightarrow {\pi ^\sharp } \pi _*\mathcal{O}_ X \xrightarrow {\text{Norm}_\pi } \mathcal{O}_ Y$ equals $g \mapsto g^ d$, and

2. if $f \in \mathcal{O}_ X(\pi ^{-1}V)$ is zero at $x \in \pi ^{-1}(V)$, then $\text{Norm}_\pi (f)$ is zero at $\pi (x)$.

We observe that condition (1) forces $\pi$ to be surjective. Since $\text{Norm}_\pi$ is multiplicative it sends units to units hence, given $y \in Y$, if $f$ is a regular function on $X$ defined at but nonvanishing at any $x \in X$ with $\pi (x) = y$, then $\text{Norm}_\pi (f)$ is defined and does not vanish at $y$. This holds without requiring (2); in fact, the constructions in this section will only require condition (1) and only certain vanishing properties (which are used in particular in the proof of Lemma 31.17.4) will require property (2).

Lemma 31.17.1. Let $\pi : X \to Y$ be a finite morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$. There exists an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{\pi ^{-1}(V)}$ is trivial.

Proof. Clearly we may assume $Y$ and hence $X$ affine. Since $\pi$ is finite the fibre $\pi ^{-1}(\{ y\} )$ over $y$ is finite. Since $X$ is affine, we can pick $s \in \Gamma (X, \mathcal{L})$ not vanishing in any point of $\pi ^{-1}(\{ y\} )$. This follows from Properties, Lemma 28.29.7 but we also give a direct argument. Namely, we can pick a finite set $E \subset X$ of closed points such that every $x \in \pi ^{-1}(\{ y\} )$ specializes to some point of $E$. For $x \in E$ denote $i_ x : x \to X$ the closed immersion. Then $\mathcal{L} \to \bigoplus _{x \in E} i_{x, *}i_ x^*\mathcal{L}$ is a surjective map of quasi-coherent $\mathcal{O}_ X$-modules, and hence the map

$\Gamma (X, \mathcal{L}) \to \bigoplus \nolimits _{x \in E} \mathcal{L}_ x/\mathfrak m_ x\mathcal{L}_ x$

is surjective (as taking global sections is an exact functor on the category of quasi-coherent $\mathcal{O}_ X$-modules, see Schemes, Lemma 26.7.5). Thus we can find an $s \in \Gamma (X, \mathcal{L})$ not vanishing at any point specializing to a point of $E$. Then $X_ s \subset X$ is an open neighbourhood of $\pi ^{-1}(\{ y\} )$. Since $\pi$ is finite, hence closed, we conclude that there is an open neighbourhood $V \subset Y$ of $y$ whose inverse image is contained in $X_ s$ as desired. $\square$

Lemma 31.17.2. Let $\pi : X \to Y$ be a finite morphism of schemes. If there exists a norm of degree $d$ for $\pi$, then there exists a homomorphism of abelian groups

$\text{Norm}_\pi : \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (Y)$

such that $\text{Norm}_\pi (\pi ^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$ for all invertible $\mathcal{O}_ Y$-modules $\mathcal{N}$.

Proof. We will use the correspondence between isomorphism classes of invertible $\mathcal{O}_ X$-modules and elements of $H^1(X, \mathcal{O}_ X^*)$ given in Cohomology, Lemma 20.6.1 without further mention. We explain how to take the norm of an invertible $\mathcal{O}_ X$-module $\mathcal{L}$. Namely, by Lemma 31.17.1 there exists an open covering $Y = \bigcup V_ j$ such that $\mathcal{L}|_{\pi ^{-1}V_ j}$ is trivial. Choose a generating section $s_ j \in \mathcal{L}(\pi ^{-1}V_ j)$ for each $j$. On the overlaps $\pi ^{-1}V_ j \cap \pi ^{-1}V_{j'}$ we can write

$s_ j = u_{jj'} s_{j'}$

for a unique $u_{jj'} \in \mathcal{O}^*_ X(\pi ^{-1}V_ j \cap \pi ^{-1}V_{j'})$. Thus we can consider the elements

$v_{jj'} = \text{Norm}_\pi (u_{jj'}) \in \mathcal{O}_ Y^*(V_ j \cap V_{j'})$

These elements satisfy the cocycle condition (because the $u_{jj'}$ do and $\text{Norm}_\pi$ is multiplicative) and therefore define an invertible $\mathcal{O}_ Y$-module. We omit the verification that: this is well defined, additive on Picard groups, and satisfies the property $\text{Norm}_\pi (\pi ^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$ for all invertible $\mathcal{O}_ Y$-modules $\mathcal{N}$. $\square$

Lemma 31.17.3. Let $\pi : X \to Y$ be a finite morphism of schemes. Assume there exists a norm of degree $d$ for $\pi$. For any $\mathcal{O}_ X$-linear map $\varphi : \mathcal{L} \to \mathcal{L}'$ of invertible $\mathcal{O}_ X$-modules there is an $\mathcal{O}_ Y$-linear map

$\text{Norm}_\pi (\varphi ) : \text{Norm}_\pi (\mathcal{L}) \longrightarrow \text{Norm}_\pi (\mathcal{L}')$

with $\text{Norm}_\pi (\mathcal{L})$, $\text{Norm}_\pi (\mathcal{L}')$ as in Lemma 31.17.2. Moreover, for $y \in Y$ the following are equivalent

1. $\varphi$ is zero at a point of $x \in X$ with $\pi (x) = y$, and

2. $\text{Norm}_\pi (\varphi )$ is zero at $y$.

Proof. We choose an open covering $Y = \bigcup V_ j$ such that $\mathcal{L}$ and $\mathcal{L}'$ are trivial over the opens $\pi ^{-1}V_ j$. This is possible by Lemma 31.17.1. Choose generating sections $s_ j$ and $s'_ j$ of $\mathcal{L}$ and $\mathcal{L}'$ over the opens $\pi ^{-1}V_ j$. Then $\varphi (s_ j) = f_ js'_ j$ for some $f_ j \in \mathcal{O}_ X(\pi ^{-1}V_ j)$. Define $\text{Norm}_\pi (\varphi )$ to be multiplication by $\text{Norm}_\pi (f_ j)$ on $V_ j$. An simple calculation involving the cocycles used to construct $\text{Norm}_\pi (\mathcal{L})$, $\text{Norm}_\pi (\mathcal{L}')$ in the proof of Lemma 31.17.2 shows that this defines a map as stated in the lemma. The final statement follows from condition (2) in the definition of a norm map of degree $d$. Some details omitted. $\square$

Lemma 31.17.4. Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ has an ample invertible sheaf and there exists a norm of degree $d$ for $\pi$. Then $Y$ has an ample invertible sheaf.

Proof. Let $\mathcal{L}$ be the ample invertible sheaf on $X$ given to us by assumption. We will prove that $\mathcal{N} = \text{Norm}_\pi (\mathcal{L})$ is ample on $Y$.

Since $X$ is quasi-compact (Properties, Definition 28.26.1) and $X \to Y$ surjective (by the existence of $\text{Norm}_\pi$) we see that $Y$ is quasi-compact. Let $y \in Y$ be a point. To finish the proof we will show that there exists a section $t$ of some positive tensor power of $\mathcal{N}$ which does not vanish at $y$ such that $Y_ t$ is affine. To do this, choose an affine open neighbourhood $V \subset Y$ of $y$. Choose $n \gg 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that

$\pi ^{-1}(\{ y\} ) \subset X_ s \subset \pi ^{-1}V$

by Properties, Lemma 28.29.6. Then $t = \text{Norm}_\pi (s)$ is a section of $\mathcal{N}^{\otimes n}$ which does not vanish at $x$ and with $Y_ t \subset V$, see Lemma 31.17.3. Then $Y_ t$ is affine by Properties, Lemma 28.26.4. $\square$

Lemma 31.17.5. Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine and there exists a norm of degree $d$ for $\pi$. Then $Y$ is quasi-affine.

Proof. By Properties, Lemma 28.27.1 we see that $\mathcal{O}_ X$ is an ample invertible sheaf on $X$. The proof of Lemma 31.17.4 shows that $\text{Norm}_\pi (\mathcal{O}_ X) = \mathcal{O}_ Y$ is an ample invertible $\mathcal{O}_ Y$-module. Hence Properties, Lemma 28.27.1 shows that $Y$ is quasi-affine. $\square$

Lemma 31.17.6. Let $\pi : X \to Y$ be a finite locally free morphism of degree $d \geq 1$. Then there exists a canonical norm of degree $d$ whose formation commutes with arbitrary base change.

Proof. Let $V \subset Y$ be an affine open such that $(\pi _*\mathcal{O}_ X)|_ V$ is finite free of rank $d$. Choosing a basis we obtain an isomorphism

$\mathcal{O}_ V^{\oplus d} \cong (\pi _*\mathcal{O}_ X)|_ V$

For every $f \in \pi _*\mathcal{O}_ X(V) = \mathcal{O}_ X(\pi ^{-1}(V))$ multiplication by $f$ defines a $\mathcal{O}_ V$-linear endomorphism $m_ f$ of the displayed free vector bundle. Thus we get a $d \times d$ matrix $M_ f \in \text{Mat}(d \times d, \mathcal{O}_ Y(V))$ and we can set

$\text{Norm}_\pi (f) = \det (M_ f)$

Since the determinant of a matrix is independent of the choice of the basis chosen we see that this is well defined which also means that this construction will glue to a global map as desired. Compatibility with base change is straightforward from the construction.

Property (1) follows from the fact that the determinant of a $d \times d$ diagonal matrix with entries $g, g, \ldots , g$ is $g^ d$. To see property (2) we may base change and assume that $Y$ is the spectrum of a field $k$. Then $X = \mathop{\mathrm{Spec}}(A)$ with $A$ a $k$-algebra with $\dim _ k(A) = d$. If there exists an $x \in X$ such that $f \in A$ vanishes at $x$, then there exists a map $A \to \kappa$ into a field such that $f$ maps to zero in $\kappa$. Then $f : A \to A$ cannot be surjective, hence $\det (f : A \to A) = 0$ as desired. $\square$

Lemma 31.17.7. Let $\pi : X \to Y$ be a finite surjective morphism with $X$ and $Y$ integral and $Y$ normal. Then there exists a norm of degree $[R(X) : R(Y)]$ for $\pi$.

Proof. Let $\mathop{\mathrm{Spec}}(B) \subset Y$ be an affine open subset and let $\mathop{\mathrm{Spec}}(A) \subset X$ be its inverse image. Then $A$ and $B$ are domains. Let $K$ be the fraction field of $A$ and $L$ the fraction field of $B$. Picture:

$\xymatrix{ L \ar[r] & K \\ B \ar[u] \ar[r] & A \ar[u] }$

Since $K/L$ is a finite extension, there is a norm map $\text{Norm}_{K/L} : K^* \to L^*$ of degree $d = [K : L]$; this is given by mapping $f \in K$ to $\det _ L(f : K \to K)$ as in the proof of Lemma 31.17.6. Observe that the characteristic polynomial of $f : K \to K$ is a power of the minimal polynomial of $f$ over $L$; in particular $\text{Norm}_{K/L}(f)$ is a power of the constant coefficient of the minimal polynomial of $f$ over $L$. Hence by Algebra, Lemma 10.38.6 $\text{Norm}_{K/L}$ maps $A$ into $B$. This determines a compatible system of maps on sections over affines and hence a global norm map $\text{Norm}_\pi$ of degree $d$.

Property (1) is immediate from the construction. To see property (2) let $f \in A$ be contained in the prime ideal $\mathfrak p \subset A$. Let $f^ m + b_1 f^{m - 1} + \ldots + b_ m$ be the minimal polynomial of $f$ over $L$. By Algebra, Lemma 10.38.6 we have $b_ i \in B$. Hence $b_0 \in B \cap \mathfrak p$. Since $\text{Norm}_{K/L}(f) = b_0^{d/m}$ (see above) we conclude that the norm vanishes in the image point of $\mathfrak p$. $\square$

Lemma 31.17.8. Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that $p\mathcal{O}_ X = 0$. Then for some $e > 0$ there exists a norm of degree $p^ e$ for $X_{red} \to X$ where $X_{red}$ is the reduction of $X$.

Proof. Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the ideal of nilpotent elements. Then $I^ n = 0$ for some $n$ (Algebra, Lemma 10.32.5). Pick $e$ such that $p^ e \geq n$. Then

$A/I \longrightarrow A,\quad f \bmod I \longmapsto f^{p^ e}$

is well defined. This produces a norm of degree $p^ e$ for $\mathop{\mathrm{Spec}}(A/I) \to \mathop{\mathrm{Spec}}(A)$. Now if $X$ is obtained by glueing some affine schemes $\mathop{\mathrm{Spec}}(A_ i)$ then for some $e \gg 0$ these maps glue to a norm map for $X_{red} \to X$. Details omitted. $\square$

Proposition 31.17.9. Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ has an ample invertible $\mathcal{O}_ X$-module. If

1. $\pi$ is finite locally free, or

2. $Y$ is an integral normal scheme, or

3. $Y$ is Noetherian, $p\mathcal{O}_ Y = 0$, and $X = Y_{red}$,

then $Y$ has an ample invertible $\mathcal{O}_ Y$-module.

Proof. Case (1) follows from a combination of Lemmas 31.17.6 and 31.17.4. Case (3) follows from a combination of Lemmas 31.17.8 and 31.17.4. In case (2) we first replace $X$ by an irreducible component of $X$ which dominates $Y$ (viewed as a reduced closed subscheme of $X$). Then we can apply Lemma 31.17.7. $\square$

Lemma 31.17.10. Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ is quasi-affine. If either

1. $\pi$ is finite locally free, or

2. $Y$ is an integral normal scheme

then $Y$ is quasi-affine.

Proof. Case (1) follows from a combination of Lemmas 31.17.6 and 31.17.5. In case (2) we first replace $X$ by an irreducible component of $X$ which dominates $Y$ (viewed as a reduced closed subscheme of $X$). Then we can apply Lemma 31.17.7. $\square$

[1] This is nonstandard notation.

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