Pushforward of the structure sheaf on $\mathbb{P_\mathbb{C}^1}$

Question

Today I had the final exam of the lesson Algebraic Geometry. There was a question was asked:

Let $X=Y=\mathbb{P_\mathbb{C}^1}$ the homogeneous coordinate $(x_0,x_1)$ and $(y_0,y_1)$, respectively. Let $f : X \to Y$ be a morphism given by $$ (x_0,x_1) \to (y_0,y_1)=(x_0^2,x_1^2). $$

• Show that $f_\ast O_X $ is a locally free sheaf of $O_Y $ -modules of rank two. ( $ O_X $ is the structure sheaf of $X$).
• Show that the induced map $i : O_Y \to f_\ast O_X $ is injective.
• Show that the cokernel of $i$ as a sheaf is isomorphic to $O_Y (−1)$.

There is a similar question with respect to the situation of pullback，but I don't even know how to deal with this problem. Hopefully someone can give me a hint.

Answer 1

It's possible to do this by brute force, using open affine covers for $X$ and $Y$: $$ U_0 := \{ [x_0 : 1] \in X \}\cong{\rm Spec}\ \mathbb C[x_0], \ \ \ \ U_1 :=\{[1 : x_1] \in X \} \cong {\rm Spec} \ \mathbb C[x_1] $$ $$ V_0 := \{ [y_0 : 1] \in Y \}\cong{\rm Spec}\ \mathbb C[y_0], \ \ \ \ V_1 :=\{[1 : y_1] \in Y \} \cong {\rm Spec} \ \mathbb C[y_1] $$ On $U_0 \cap U_1$, we identify $x_0 \in \mathbb C[x_0]_{(x_0)}$ with $x^{-1}_1 \in \mathbb C[x_1]_{(x_1)}$. We make a similar identification between $y_0$ and $y_1^{-1}$ on $V_0 \cap V_1$.

Conveniently, we have $f^{-1}(V_0) = U_0$ and $f^{-1}(V_1) = U_1$. The morphism $f$ is associated with the ring homomorphisms: $$ \mathbb C[y_0] \to \mathbb C[x_0] , \ \ \ y_0 \mapsto x_0^2$$ $$ \mathbb C[y_1] \to \mathbb C[x_1] , \ \ \ y_1 \mapsto x_1^2$$

The original structure sheaf $\mathcal O_X$ can be described as follows:

• On $U_0$: $(\mathcal O_X)|_{U_0}$ is the quasicoherent sheaf associated to the $\mathbb C[x_0]$-module $\mathbb C[x_0]$.
• On $U_1$: $(\mathcal O_X)|_{U_1}$ is the quasicoherent sheaf associated to the $\mathbb C[x_1]$-module $\mathbb C[x_1]$.
• On $U_0 \cap U_1$: the transition function is defined by identifying the element $x_0 \in \mathbb C[x_0]_{(x_0)}$ with the element $x^{-1}_1 \in \mathbb C[x_1]_{(x_1)}$.

So the pushforward $f_\star \mathcal O_X$ can be described like this:

• On $V_0$: $(f_\star \mathcal O_X)|_{V_0}$ is the quasicoherent sheaf associated with $\mathbb C[x_0]$, now viewed as a $\mathbb C[y_0]$-module, with $y_0$ viewed as $x_0^2$.
• On $V_1$: $(f_\star \mathcal O_X)|_{V_1}$ is the quasicoherent sheaf associated with $\mathbb C[x_1]$, now viewed as a $\mathbb C[y_1]$-module, with $y_1$ viewed as $x_1^2$.
• On $V_0 \cap V_1$: we identify the element $x_0 \in \mathbb C[x_0]_{(y_0)}$ with the element $x_1^{-1} \in \mathbb C[x_1]_{(y_1)}$.

Now observe that $\mathbb C[x_0]$ is a free $\mathbb C[y_0]$ module, by virtue of the $\mathbb C[y_0]$-module isomorphism $$ \mathbb C[x_0] \cong \mathbb C[y_0]. 1 \oplus \mathbb C[y_0]. x_0$$

So $(f_\star \mathcal O_X)|_{V_0}$ is a free sheaf of rank two. A similar statement is true on $V_1$. Thus $f_\star \mathcal O_X$ is a locally free sheaf on $Y$.

The sheaf morphism $i_\star \mathcal O_Y \to f_\star \mathcal O_X$ can described using module morphisms on the two affine patches. For example, on $V_0$, $i_\star$ is associated with the morphism of $\mathbb C[y_0]$-modules, $$ \mathbb C[y_0] \to \mathbb C[x_0], \ \ \ \ \ y_0 \mapsto x_0^2,$$

which is injective, hence injective on all localisations at prime ideals. As the same is true on $V_1$, we see that $i_\star$ is injective on all stalks.

Finally, we describe the cokernel of $i_\star$. On $V_0$ this cokernel is the sheaf associated with the $\mathbb C[y_0].x_0$ component of $\mathbb C[x_0] \cong \mathbb C[y_0]. 1 \oplus \mathbb C[y_0]. x_0$. On $V_1$, it is the sheaf associated with the $\mathbb C[y_1] . x_1 $ component of $\mathbb C[x_1] \cong \mathbb C[y_1]. 1 \oplus \mathbb C[y_1]. x_1$. Notice that $\mathbb C[y_0].x_0$ is a rank-one free module over $\mathbb C[y_0]$, and $\mathbb C[y_1].x_1$ is a rank-one free module over $\mathbb C[y_1]$. So the cokernel of $i_\star$ is locally free of rank one. It only remains to find the transition function. On the overlap $V_0 \cap V_1$, we identify $1. x_0 \in \mathbb C[y_0]_{(y_0)}.x_0$ with $y_1^{-1} . x_1 \in \mathbb C[y_1]_{(y_1)} . x_1$. The identification $1 \leftrightarrow y_1^{-1}$ is precisely the transition function for the invertible sheaf $\mathcal O_Y(-1)$.