Lemma 33.44.5. Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point $\xi$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$ and let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then

$\chi (X, \mathcal{E} \otimes \mathcal{F}) = r \deg (\mathcal{E}) + n \chi (X, \mathcal{F})$

where $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi$ is the rank of $\mathcal{F}$.

Proof. Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ expressing that the formula of the lemma holds. We claim that the assumptions (1) and (2) of Cohomology of Schemes, Lemma 30.12.6 hold for $\mathcal{P}$. Namely, (1) holds because the Euler characteristic and the rank $r$ are additive in short exact sequences of coherent sheaves. And (2) holds too: If $Z = X$ then we may take $\mathcal{G} = \mathcal{O}_ X$ and $\mathcal{P}(\mathcal{O}_ X)$ is true by the definition of degree. If $i : Z \to X$ is the inclusion of a closed point we may take $\mathcal{G} = i_*\mathcal{O}_ Z$ and $\mathcal{P}$ holds by Lemma 33.33.3 and the fact that $r = 0$ in this case. $\square$

Comment #4619 by on

Since $X$ is a curve and $\xi$ is the generic point, we have $\mathcal{O}_{X, \xi} = \kappa(\xi)$ and we have $\mathcal{F}_\xi = \mathcal{F}_\xi \otimes_{\mathcal{O}_{X, \xi}} \kappa(\xi)$.

Comment #4622 by Raffaele Carbone on

Are you assuming any reducedness?

Comment #4625 by on

You can search the Stacks project for the key words "curve definition" (don't use the quotes) and then you will find Definition 33.43.1. You may have to do this several times. For example the definition of a curve refers to a variety which is defined in Definition 33.3.1. The definition of a variety refers to the notion of an integral scheme which is defined in Definition 28.3.1. This in turn relies on knowing what an integral domain is, see Definition 9.2.2. Since a nilpotent element of a ring is a zerodivisor, you conclude that integral schemes are reduced. Also, if you search for "integral reduced scheme" (don't use the quotes), then you will find Lemma 28.3.4. So, yes! I also hope this will help you and others answer similar questions.

PS: Most algebraic geometers would define a curve either as is done in the Stacks project or as what in the Stacks project would be called a geometrically integral curve.

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