The Stacks project

Exercise 111.43.5. Let $f : X \to Y$ be a finite locally free morphism of degree $2$. Assume that $X$ and $Y$ are integral schemes and that $2$ is invertible in the structure sheaf of $Y$, i.e., $2 \in \Gamma (Y, \mathcal{O}_ Y)$ is invertible. Show that the $\mathcal{O}_ Y$-module map

\[ f^\sharp : \mathcal{O}_ Y \longrightarrow f_*\mathcal{O}_ X \]

has a left inverse, i.e., there is an $\mathcal{O}_ Y$-module map $\tau : f_*\mathcal{O}_ X \to \mathcal{O}_ Y$ with $\tau \circ f^\sharp = \text{id}$. Conclude that $H^ n(Y, \mathcal{O}_ Y) \to H^ n(X, \mathcal{O}_ X)$ is injective1.

[1] There does exist a finite locally free morphism $X \to Y$ between integral schemes of degree $2$ where the map $H^1(Y, \mathcal{O}_ Y) \to H^1(X, \mathcal{O}_ X)$ is not injective.

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View Exercise 111.43.5 as pdf