42.41 Degrees of zero cycles
We start defining the degree of a zero cycle on a proper scheme over a field. One approach is to define it directly as in Lemma 42.41.2 and then show it is well defined by Lemma 42.18.3. Instead we define it as follows.
Definition 42.41.1. Let $k$ be a field (Example 42.7.2). Let $p : X \to \mathop{\mathrm{Spec}}(k)$ be proper. The degree of a zero cycle on $X$ is given by proper pushforward
\[ p_* : \mathop{\mathrm{CH}}\nolimits _0(X) \to \mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) \]
(Lemma 42.20.3) combined with the natural isomorphism $\mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) = \mathbf{Z}$ which maps $[\mathop{\mathrm{Spec}}(k)]$ to $1$. Notation: $\deg (\alpha )$.
Let us spell this out further.
Lemma 42.41.2. Let $k$ be a field. Let $X$ be proper over $k$. Let $\alpha = \sum n_ i[Z_ i]$ be in $Z_0(X)$. Then
\[ \deg (\alpha ) = \sum n_ i\deg (Z_ i) \]
where $\deg (Z_ i)$ is the degree of $Z_ i \to \mathop{\mathrm{Spec}}(k)$, i.e., $\deg (Z_ i) = \dim _ k \Gamma (Z_ i, \mathcal{O}_{Z_ i})$.
Proof.
This is the definition of proper pushforward (Definition 42.12.1).
$\square$
Next, we make the connection with degrees of vector bundles over $1$-dimensional proper schemes over fields as defined in Varieties, Section 33.44.
Lemma 42.41.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $\leq 1$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module of constant rank. Then
\[ \deg (\mathcal{E}) = \deg (c_1(\mathcal{E}) \cap [X]_1) \]
where the left hand side is defined in Varieties, Definition 33.44.1.
Proof.
Let $C_ i \subset X$, $i = 1, \ldots , t$ be the irreducible components of dimension $1$ with reduced induced scheme structure and let $m_ i$ be the multiplicity of $C_ i$ in $X$. Then $[X]_1 = \sum m_ i[C_ i]$ and $c_1(\mathcal{E}) \cap [X]_1$ is the sum of the pushforwards of the cycles $m_ i c_1(\mathcal{E}|_{C_ i}) \cap [C_ i]$. Since we have a similar decomposition of the degree of $\mathcal{E}$ by Varieties, Lemma 33.44.6 it suffices to prove the lemma in case $X$ is a proper curve over $k$.
Assume $X$ is a proper curve over $k$. By Divisors, Lemma 31.36.1 there exists a modification $f : X' \to X$ such that $f^*\mathcal{E}$ has a filtration whose successive quotients are invertible $\mathcal{O}_{X'}$-modules. Since $f_*[X']_1 = [X]_1$ we conclude from Lemma 42.38.4 that
\[ \deg (c_1(\mathcal{E}) \cap [X]_1) = \deg (c_1(f^*\mathcal{E}) \cap [X']_1) \]
Since we have a similar relationship for the degree by Varieties, Lemma 33.44.4 we reduce to the case where $\mathcal{E}$ has a filtration whose successive quotients are invertible $\mathcal{O}_ X$-modules. In this case, we may use additivity of the degree (Varieties, Lemma 33.44.3) and of first Chern classes (Lemma 42.40.3) to reduce to the case discussed in the next paragraph.
Assume $X$ is a proper curve over $k$ and $\mathcal{E}$ is an invertible $\mathcal{O}_ X$-module. By Divisors, Lemma 31.15.12 we see that $\mathcal{E}$ is isomorphic to $\mathcal{O}_ X(D) \otimes \mathcal{O}_ X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D'$ on $X$ (this also uses that $X$ is projective, see Varieties, Lemma 33.43.4 for example). By additivity of degree under tensor product of invertible sheaves (Varieties, Lemma 33.44.7) and additivity of $c_1$ under tensor product of invertible sheaves (Lemma 42.25.2 or 42.39.1) we reduce to the case $\mathcal{E} = \mathcal{O}_ X(D)$. In this case the left hand side gives $\deg (D)$ (Varieties, Lemma 33.44.9) and the right hand side gives $\deg ([D]_0)$ by Lemma 42.25.4. Since
\[ [D]_0 = \sum \nolimits _{x \in D} \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{D, x}) [x] = \sum \nolimits _{x \in D} \text{length}_{\mathcal{O}_{D, x}}(\mathcal{O}_{D, x}) [x] \]
by definition, we see
\[ \deg ([D]_0) = \sum \nolimits _{x \in D} \text{length}_{\mathcal{O}_{D, x}}(\mathcal{O}_{D, x}) [\kappa (x) : k] = \dim _ k \Gamma (D, \mathcal{O}_ D) = \deg (D) \]
The penultimate equality by Algebra, Lemma 10.52.12 using that $D$ is affine.
$\square$
Finally, we can tie everything up with the numerical intersections defined in Varieties, Section 33.45.
Lemma 42.41.4. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \subset X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. Then
\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \deg ( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z]_ d) \]
where the left hand side is defined in Varieties, Definition 33.45.3. In particular,
\[ \deg _\mathcal {L}(Z) = \deg (c_1(\mathcal{L})^ d \cap [Z]_ d) \]
if $\mathcal{L}$ is an ample invertible $\mathcal{O}_ X$-module.
Proof.
We will prove this by induction on $d$. If $d = 0$, then the result is true by Varieties, Lemma 33.33.3. Assume $d > 0$.
Let $Z_ i \subset Z$, $i = 1, \ldots , t$ be the irreducible components of dimension $d$ with reduced induced scheme structure and let $m_ i$ be the multiplicity of $Z_ i$ in $Z$. Then $[Z]_ d = \sum m_ i[Z_ i]$ and $c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z]_ d$ is the sum of the cycles $m_ i c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z_ i]$. Since we have a similar decomposition for $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ by Varieties, Lemma 33.45.2 it suffices to prove the lemma in case $Z = X$ is a proper variety of dimension $d$ over $k$.
By Chow's lemma there exists a birational proper morphism $f : Y \to X$ with $Y$ H-projective over $k$. See Cohomology of Schemes, Lemma 30.18.1 and Remark 30.18.2. Then
\[ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot Y) = (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) \]
by Varieties, Lemma 33.45.7 and we have
\[ f_*(c_1(f^*\mathcal{L}_1) \cap \ldots \cap c_1(f^*\mathcal{L}_ d) \cap [Y]) = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [X] \]
by Lemma 42.26.4. Thus we may replace $X$ by $Y$ and assume that $X$ is projective over $k$.
If $X$ is a proper $d$-dimensional projective variety, then we can write $\mathcal{L}_1 = \mathcal{O}_ X(D) \otimes \mathcal{O}_ X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D' \subset X$ by Divisors, Lemma 31.15.12. By additivity for both sides of the equation (Varieties, Lemma 33.45.5 and Lemma 42.25.2) we reduce to the case $\mathcal{L}_1 = \mathcal{O}_ X(D)$ for some effective Cartier divisor $D$. By Varieties, Lemma 33.45.8 we have
\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot D) \]
and by Lemma 42.25.4 we have
\[ c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [X] = c_1(\mathcal{L}_2) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [D]_{d - 1} \]
Thus we obtain the result from our induction hypothesis.
$\square$
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