# Problem Set 5 (Monday, October 26)

::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv 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 $f:{\mathbb{A}}^1\to{\mathbb{A}}^1$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = ax+b$ for some $a, b\in k$.

d.  If $f:{\mathbb{A}}^2\to{\mathbb{A}}^2$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = Ax+b$ for some $a \in \operatorname{Mat}(2\times 2, k)$ and $b\in k^2$.
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::: {.solution .proofenv .proofenv .proofenv .proofenv .proofenv}
```{=tex}
\hfill
```
a.  **True**. This follows because if $p, q\in A(Y)$, then `<span class="math display">     \begin{align*}         f* p &= f^* q \\       &\implies (p\circ f) = (q\circ f) && \text{by definition}\\       &\implies p = q        ,\end{align*}     <span>`{=html} where in the last implication we've used the fact that $f$ is surjective iff $f$ admits a right-inverse.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 4.19"}
Which of the following are isomorphic as ringed spaces over ${\mathbb{C}}$?

(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) $V\left(x_{1}^{2}-x_{2}^{2}-1\right) \subset \mathbb{A}^{2}$
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 5.7"}
Show that

a.  Every morphism $f:{\mathbb{A}}^1\setminus\left\{{0}\right\}\to {\mathbb{P}}^1$ can be extended to a morphism $\widehat{f}: {\mathbb{A}}^1 \to {\mathbb{P}}^1$.

b.  Not every morphism $f:{\mathbb{A}}^2\setminus\left\{{0}\right\}\to {\mathbb{P}}^1$ can be extended to a morphism $\widehat{f}: {\mathbb{A}}^2 \to {\mathbb{P}}^1$.

c.  Every morphism ${\mathbb{P}}^1\to {\mathbb{A}}^1$ is constant.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 5.8"}
Show that

a.  Every isomorphism $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ is of the form `<span class="math display">     \begin{align*}       f(x) = {ax+b \over cx+d} && a,b,c,d\in k     .\end{align*}     <span>`{=html} where $x$ is an affine coordinate on ${\mathbb{A}}^1\subset {\mathbb{P}}^1$.

b.  Given three distinct points $a_i \in {\mathbb{P}}^1$ and three distinct points $b_i \in {\mathbb{P}}^1$, there is a unique isomorphism $f:{\mathbb{P}}^1 \to {\mathbb{P}}^1$ such that $f(a_i) = b_i$ for all $i$.
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::: {.proposition .proofenv .proofenv .proofenv .proofenv .proofenv title="?"}
There is a bijection `<span class="math display"> \begin{align*} \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} \end{align*} <span>`{=html}
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 5.9"}
Does the above bijection hold if

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