The Stacks project

Exercise 111.60.1 (Definitions). Provide brief definitions of the italicized concepts.

1. the left adjoint of a functor $F : \mathcal{A} \to \mathcal{B}$,

2. the transcendence degree of an extension $L/K$ of fields,

3. a regular function on a classical affine variety $X \subset k^ n$,

4. a sheaf on a topological space,

5. a local ring, and

6. a morphism of schemes $f : X \to Y$ being affine.

Exercise 111.60.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. Yoneda lemma,

2. Mayer-Vietoris,

3. dimension and cohomology,

4. Hilbert polynomial, and

5. duality for projective space.

Exercise 111.60.3. Let $k$ be an algebraically closed field. Consider the closed subset $X$ of $k^5$ with Zariski topology and coordinates $x_1, x_2, x_3, x_4, x_5$ given by the equations

What is the dimension of $X$ and why?

Exercise 111.60.4. Let $k$ be a field. Let $X = \mathbf{P}^1_ k$ be the projective space of dimension $1$ over $k$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. For $d \in \mathbf{Z}$ denote $\mathcal{E}(d) = \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(d)$ the $d$th Serre twist of $\mathcal{E}$ and $h^ i(X, \mathcal{E}(d)) = \dim _ k H^ i(X, \mathcal{E}(d))$.

1. Why is there no $\mathcal{E}$ with $h^0(X, \mathcal{E}) = 5$ and $h^0(X, \mathcal{E}(1)) = 4$?

2. Why is there no $\mathcal{E}$ with $h^1(X, \mathcal{E}(1)) = 5$ and $h^1(X, \mathcal{E}) = 4$?

3. For which $a \in \mathbf{Z}$ can there exist a vector bundle $\mathcal{E}$ on $X$ with

   \[ \begin{matrix} h^0(X, \mathcal{E})\phantom{(1)} = 1 & h^1(X, \mathcal{E})\phantom{(1)} = 1 \\ h^0(X, \mathcal{E}(1)) = 2 & h^1(X, \mathcal{E}(1)) = 0 \\ h^0(X, \mathcal{E}(2)) = 4 & h^1(X, \mathcal{E}(2)) = a \end{matrix} \]

Partial answers are welcomed and encouraged.

Exercise 111.60.5. Let $X$ be a topological space which is the union $X = Y \cup Z$ of two closed subsets $Y$ and $Z$ whose intersection is denoted $W = Y \cap Z$. Denote $i : Y \to X$, $j : Z \to X$, and $k : W \to X$ the inclusion maps.

1. Show that there is a short exact sequence of sheaves

   \[ 0 \to \underline{\mathbf{Z}}_ X \to i_*(\underline{\mathbf{Z}}_ Y) \oplus j_*(\underline{\mathbf{Z}}_ Z) \to k_*(\underline{\mathbf{Z}}_ W) \to 0 \]

   where $\underline{\mathbf{Z}}_ X$ denotes the constant sheaf with value $\mathbf{Z}$ on $X$, etc.

2. What can you conclude about the relationship between the cohomology groups of $X$, $Y$, $Z$, $W$ with $\mathbf{Z}$-coefficients?

Exercise 111.60.6. Let $k$ be a field. Let $A = k[x_1, x_2, x_3, \ldots ]$ be the polynomial ring in infinitely many variables. Denote $\mathfrak m$ the maximal ideal of $A$ generated by all the variables. Let $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $.

1. Show $H^1(U, \mathcal{O}_ U) = 0$. Hint: Čech cohomology computation.

2. What is your guess for $H^ i(U, \mathcal{O}_ U)$ for $i \geq 1$?

Exercise 111.60.7. Let $A$ be a local ring. Let $a \in A$ be a nonzerodivisor. Let $I, J \subset A$ be ideals such that $IJ = (a)$. Show that the ideal $I$ is principal, i.e., generated by one element (which will turn out to be a nonzerodivisor).