Lemma 82.32.2 (0ESH)—The Stacks project

Table of contents
Part 5: Topics in Geometry
Chapter 82: Chow Groups of Spaces
Section 82.32: Degrees of zero cycles
Lemma 82.32.2 (cite)

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). $\square$

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