Dtyjlui

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 $mathcal O_Y(-1)$.