# Problem Set 5 (Monday, October 26)

:::{.problem title="Gathmann 4.13"}
Let $f:X\to Y$ be a morphism of affine varieties and $f^*: A(Y) \to A(X)$ the induced map on coordinate rings.
Determine if the following statements are true or false:

a. $f$ is surjective $\iff f^*$ is injective.

b. $f$ is injective $\iff f^*$ is surjective.

c. If $f:\AA^1\to\AA^1$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = ax+b$ for some $a, b\in k$.

d. If $f:\AA^2\to\AA^2$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = Ax+b$ for some $a \in \Mat(2\times 2, k)$ and $b\in k^2$.

:::

:::{.solution}
\hfill

a. **True**.
  This follows because if $p, q\in A(Y)$, then
  \[  
  f* p &= f^* q \\
  &\implies (p\circ f) = (q\circ f) && \text{by definition}\\
  &\implies p = q 
  ,\]
  where in the last implication we've used the fact that $f$ is surjective iff $f$ admits a right-inverse.
:::


:::{.problem title="Gathmann 4.19"}
Which of the following are isomorphic as ringed spaces over $\CC$?

(a) $\mathbb{A}^{1} \backslash\{1\}$

(b) $V\left(x_{1}^{2}+x_{2}^{2}\right) \subset \mathbb{A}^{2}$

(c) $V\left(x_{2}-x_{1}^{2}, x_{3}-x_{1}^{3}\right) \backslash\{0\} \subset \mathbb{A}^{3}$

(d) $V\left(x_{1} x_{2}\right) \subset \mathbb{A}^{2}$

(e) $V\left(x_{2}^{2}-x_{1}^{3}-x_{1}^{2}\right) \subset \mathbb{A}^{2}$

(f) $V\left(x_{1}^{2}-x_{2}^{2}-1\right) \subset \mathbb{A}^{2}$



:::


:::{.problem title="Gathmann 5.7"}
Show that

a. Every morphism $f:\AA^1\smz \to \PP^1$ can be extended to a morphism $\hat f: \AA^1 \to \PP^1$.

b. Not every morphism $f:\AA^2\smz \to \PP^1$ can be extended to a morphism $\hat f: \AA^2 \to \PP^1$.

c. Every morphism $\PP^1\to \AA^1$ is constant.
:::


:::{.problem title="Gathmann 5.8"}
Show that

a. Every isomorphism $f:\PP^1\to \PP^1$ is of the form 
\[  
f(x) = {ax+b \over cx+d} && a,b,c,d\in k
.\]
where $x$ is an affine coordinate on $\AA^1\subset \PP^1$.

b. Given three distinct points $a_i \in \PP^1$ and three distinct points $b_i \in \PP^1$, there is a unique isomorphism $f:\PP^1 \to \PP^1$ such that $f(a_i) = b_i$ for all $i$.
:::

:::{.proposition title="?"}
There is a bijection
\[
\begin{array}{c}
\{\text { morphisms } X \rightarrow Y\} \stackrel{1: 1}{\longleftrightarrow}\left\{K \text { -algebra homomorphisms } \mathscr{O}_{Y}(Y) \rightarrow \mathscr{O}_{X}(X)\right\} \\
f \longmapsto f^{*}
\end{array}
\]
:::

:::{.problem title="Gathmann 5.9"}
Does the above bijection hold if

a. $X$ is an arbitrary prevariety but $Y$ is still affine?
a. $Y$ is an arbitrary prevariety but $X$ is still affine?
:::