The Stacks project

Lemma 81.32.3. Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $Z \subset X$ be a closed subspace of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \deg ( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_1) \cap [Z]_ d) \]

where the left hand side is defined in Spaces over Fields, Definition 71.18.3.

Proof. Let $Z_ i \subset Z$, $i = 1, \ldots , t$ be the irreducible components of dimension $d$. 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 Spaces over Fields, Lemma 71.18.2 it suffices to prove the lemma in case $Z = X$ is a proper integral algebraic space over $k$.

By Chow's lemma there exists a proper morphism $f : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X'$ is a scheme. See More on Morphisms of Spaces, Lemma 75.40.5. Then $X'$ is a proper scheme over $k$. After replacing $X'$ by the scheme theoretic closure of $f^{-1}(U)$ we may assume that $X'$ is integral. Then

\[ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot X') = (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) \]

by Spaces over Fields, Lemma 71.18.7 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 81.19.4. Thus we may replace $X$ by $X'$ and assume that $X$ is a proper scheme over $k$. This case was proven in Chow Homology, Lemma 42.41.4. $\square$

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