Lemma 81.32.2. 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 81.8.1). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).