109.52 Schemes, Final Exam, Spring 2009

These were the questions in the final exam of a course on Schemes, in the Spring of 2009 at Columbia University.

Exercise 109.52.1. Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point. Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.

1. Show that $x$ is a closed point of $X$.

2. Show that $H^0(X, \mathcal{F})$ is not zero.

3. Show that $\mathcal{F}$ is generated by global sections.

4. Show that $H^ p(X, \mathcal{F}) = 0$ for $p > 0$.

Remark 109.52.2. Let $k$ be a field. Let $\mathbf{P}^2_ k = \text{Proj}(k[X_0, X_1, X_2])$. Any invertible sheaf on $\mathbf{P}^2_ k$ is isomorphic to $\mathcal{O}_{\mathbf{P}^2_ k}(n)$ for some $n \in \mathbf{Z}$. Recall that

$\Gamma (\mathbf{P}^2_ k, \mathcal{O}_{\mathbf{P}^2_ k}(n)) = k[X_0, X_1, X_2]_ n$

is the degree $n$ part of the polynomial ring. For a quasi-coherent sheaf $\mathcal{F}$ on $\mathbf{P}^2_ k$ set $\mathcal{F}(n) = \mathcal{F} \otimes _{\mathcal{O}_{\mathbf{P}^2_ k}} \mathcal{O}_{\mathbf{P}^2_ k}(n)$ as usual.

Exercise 109.52.3. Let $k$ be a field. Let $\mathcal{E}$ be a vector bundle on $\mathbf{P}^2_ k$, i.e., a finite locally free $\mathcal{O}_{\mathbf{P}^2_ k}$-module. We say $\mathcal{E}$ is split if $\mathcal{E}$ is isomorphic to a direct sum invertible $\mathcal{O}_{\mathbf{P}^2_ k}$-modules.

1. Show that $\mathcal{E}$ is split if and only if $\mathcal{E}(n)$ is split.

2. Show that if $\mathcal{E}$ is split then $H^1({\mathbf{P}^2_ k}, \mathcal{E}(n)) = 0$ for all $n \in \mathbf{Z}$.

3. Let

$\varphi : \mathcal{O}_{\mathbf{P}^2_ k} \longrightarrow \mathcal{O}_{\mathbf{P}^2_ k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_ k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_ k}(1)$

be given by linear forms $L_0, L_1, L_2 \in \Gamma (\mathbf{P}^2_ k, \mathcal{O}_{\mathbf{P}^2_ k}(1))$. Assume $L_ i \not= 0$ for some $i$. What is the condition on $L_0, L_1, L_2$ such that the cokernel of $\varphi$ is a vector bundle? Why?

4. Given an example of such a $\varphi$.

5. Show that $\mathop{\mathrm{Coker}}(\varphi )$ is not split (if it is a vector bundle).

Remark 109.52.4. Freely use the following facts on dimension theory (and add more if you need more).

1. The dimension of a scheme is the supremum of the length of chains of irreducible closed subsets.

2. The dimension of a finite type scheme over a field is the maximum of the dimensions of its affine opens.

3. The dimension of a Noetherian scheme is the maximum of the dimensions of its irreducible components.

4. The dimension of an affine scheme coincides with the dimension of the corresponding ring.

5. Let $k$ be a field and let $A$ be a finite type $k$-algebra. If $A$ is a domain, and $x \not= 0$, then $\dim (A) = \dim (A/xA) + 1$.

Exercise 109.52.5. Let $k$ be a field. Let $X$ be a projective, reduced scheme over $k$. Let $f : X \to \mathbf{P}^1_ k$ be a morphism of schemes over $k$. Assume there exists an integer $d \geq 0$ such that for every point $t \in \mathbf{P}^1_ k$ the fibre $X_ t = f^{-1}(t)$ is irreducible of dimension $d$. (Recall that an irreducible space is not empty.)

1. Show that $\dim (X) = d + 1$.

2. Let $X_0 \subset X$ be an irreducible component of $X$ of dimension $d + 1$. Prove that for every $t \in \mathbf{P}^1_ k$ the fibre $X_{0, t}$ has dimension $d$.

3. What can you conclude about $X_ t$ and $X_{0, t}$ from the above?

4. Show that $X$ is irreducible.

Remark 109.52.6. Given a projective scheme $X$ over a field $k$ and a coherent sheaf $\mathcal{F}$ on $X$ we set

$\chi (X, \mathcal{F}) = \sum \nolimits _{i \geq 0} (-1)^ i\dim _ k H^ i(X, \mathcal{F}).$

Exercise 109.52.7. Let $k$ be a field. Write $\mathbf{P}^3_ k = \text{Proj}(k[X_0, X_1, X_2, X_3])$. Let $C \subset \mathbf{P}^3_ k$ be a type $(5, 6)$ complete intersection curve. This means that there exist $F \in k[X_0, X_1, X_2, X_3]_5$ and $G \in k[X_0, X_1, X_2, X_3]_6$ such that

$C = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F, G))$

is a variety of dimension $1$. (Variety implies reduced and irreducible, but feel free to assume $C$ is nonsingular if you like.) Let $i : C \to \mathbf{P}^3_ k$ be the corresponding closed immersion. Being a complete intersection also implies that

$\xymatrix{ 0 \ar[r] & \mathcal{O}_{\mathbf{P}^3_ k}(-11) \ar[r]^-{ \left( \begin{matrix} -G \\ F \end{matrix} \right) } & \mathcal{O}_{\mathbf{P}^3_ k}(-5) \oplus \mathcal{O}_{\mathbf{P}^3_ k}(-6) \ar[r]^-{(F, G)} & \mathcal{O}_{\mathbf{P}^3_ k} \ar[r] & i_*\mathcal{O}_ C \ar[r] & 0 }$

is an exact sequence of sheaves. Please use these facts to:

1. compute $\chi (C, i^*\mathcal{O}_{\mathbf{P}^3_ k}(n))$ for any $n \in \mathbf{Z}$, and

2. compute the dimension of $H^1(C, \mathcal{O}_ C)$.

Exercise 109.52.8. Let $k$ be a field. Consider the rings

\begin{align*} A & = k[x, y]/(xy) \\ B & = k[u, v]/(uv) \\ C & = k[t, t^{-1}] \times k[s, s^{-1}] \end{align*}

and the $k$-algebra maps

$\begin{matrix} A \longrightarrow C, & x \mapsto (t, 0), & y \mapsto (0, s) \\ B \longrightarrow C, & u \mapsto (t^{-1}, 0), & v \mapsto (0, s^{-1}) \end{matrix}$

It is a true fact that these maps induce isomorphisms $A_{x + y} \to C$ and $B_{u + v} \to C$. Hence the maps $A \to C$ and $B \to C$ identify $\mathop{\mathrm{Spec}}(C)$ with open subsets of $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$. Let $X$ be the scheme obtained by glueing $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$ along $\mathop{\mathrm{Spec}}(C)$:

$X = \mathop{\mathrm{Spec}}(A) \amalg _{\mathop{\mathrm{Spec}}(C)} \mathop{\mathrm{Spec}}(B).$

As we saw in the course such a scheme exists and there are affine opens $\mathop{\mathrm{Spec}}(A) \subset X$ and $\mathop{\mathrm{Spec}}(B) \subset X$ whose overlap is exactly $\mathop{\mathrm{Spec}}(C)$ identified with an open of each of these using the maps above.

1. Why is $X$ separated?

2. Why is $X$ of finite type over $k$?

3. Compute $H^1(X, \mathcal{O}_ X)$, or what is its dimension?

4. What is a more geometric way to describe $X$?

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