Lemma 33.44.4 (0AYU)—The Stacks project

Lemma 33.44.4. Let $k$ be a field. Let $f : X' \to X$ be a birational morphism of proper schemes of dimension $\leq 1$ over $k$. Then

\[ \deg (f^*\mathcal{E}) = \deg (\mathcal{E}) \]

for every finite locally free sheaf of constant rank. More generally it suffices if $f$ induces a bijection between irreducible components of dimension $1$ and isomorphisms of local rings at the corresponding generic points.

Proof. The morphism $f$ is proper (Morphisms, Lemma 29.41.7) and has fibres of dimension $\leq 0$. Hence $f$ is finite (Cohomology of Schemes, Lemma 30.21.2). Thus

\[ Rf_*f^*\mathcal{E} = f_*f^*\mathcal{E} = \mathcal{E} \otimes _{\mathcal{O}_ X} f_*\mathcal{O}_{X'} \]

Since $f$ induces an isomorphism on local rings at generic points of all irreducible components of dimension $1$ we see that the kernel and cokernel

\[ 0 \to \mathcal{K} \to \mathcal{O}_ X \to f_*\mathcal{O}_{X'} \to \mathcal{Q} \to 0 \]

have supports of dimension $\leq 0$. Note that tensoring this with $\mathcal{E}$ is still an exact sequence as $\mathcal{E}$ is locally free. We obtain

\begin{align*} \chi (X, \mathcal{E}) - \chi (X', f^*\mathcal{E}) & = \chi (X, \mathcal{E}) - \chi (X, f_*f^*\mathcal{E}) \\ & = \chi (X, \mathcal{E}) - \chi (X, \mathcal{E} \otimes f_*\mathcal{O}_{X'}) \\ & = \chi (X, \mathcal{K} \otimes \mathcal{E}) - \chi (X, \mathcal{Q} \otimes \mathcal{E}) \\ & = n\chi (X, \mathcal{K}) - n\chi (X, \mathcal{Q}) \\ & = n\chi (X, \mathcal{O}_ X) - n\chi (X, f_*\mathcal{O}_{X'}) \\ & = n\chi (X, \mathcal{O}_ X) - n\chi (X', \mathcal{O}_{X'}) \end{align*}

which proves what we want. The first equality as $f$ is finite, see Cohomology of Schemes, Lemma 30.2.4. The second equality by projection formula, see Cohomology, Lemma 20.54.2. The third by additivity of Euler characteristics, see Lemma 33.33.2. The fourth by Lemma 33.33.3. $\square$

The Stacks project

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