82.32 Degrees of zero cycles
This section is the analogue of Chow Homology, Section 42.41. We start with defining the degree of a zero cycle on a proper algebraic space over a field.
Definition 82.32.1. Let k be a field. Let p : X \to \mathop{\mathrm{Spec}}(k) be a proper morphism of algebraic spaces. The degree of a zero cycle on X is given by proper pushforward
p_* : \mathop{\mathrm{CH}}\nolimits _0(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) \longrightarrow \mathbf{Z}
(Lemma 82.16.3) composed with the natural isomorphism \mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) \to \mathbf{Z} which maps [\mathop{\mathrm{Spec}}(k)] to 1. Notation: \deg (\alpha ).
Let us spell this out further.
Lemma 82.32.2. Let k be a field. Let X be a proper algebraic space 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 82.8.1).
\square
Lemma 82.32.3. 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 72.18.3.
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 72.18.2 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 76.40.5. 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 72.18.7 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 82.19.4. 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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