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

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.

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 $$\mathcal O_Y(-1)$$.