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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