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:
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\(f\) is surjective \(\iff f^*\) is injective.
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\(f\) is injective \(\iff f^*\) is surjective.
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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 \(a, b\in k\).
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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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- True. This follows because if \(p, q\in A(Y)\), then \begin{align*} f* p &= f^* q \\ &\implies (p\circ f) = (q\circ f) && \text{by definition}\\ &\implies p = q ,\end{align*} where in the last implication we’ve used the fact that \(f\) is surjective iff \(f\) admits a right-inverse.
Which of the following are isomorphic as ringed spaces over \({\mathbb{C}}\)?
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\(\mathbb{A}^{1} \backslash\{1\}\)
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\(V\left(x_{1}^{2}+x_{2}^{2}\right) \subset \mathbb{A}^{2}\)
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\(V\left(x_{2}-x_{1}^{2}, x_{3}-x_{1}^{3}\right) \backslash\{0\} \subset \mathbb{A}^{3}\)
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\(V\left(x_{1} x_{2}\right) \subset \mathbb{A}^{2}\)
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\(V\left(x_{2}^{2}-x_{1}^{3}-x_{1}^{2}\right) \subset \mathbb{A}^{2}\)
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\(V\left(x_{1}^{2}-x_{2}^{2}-1\right) \subset \mathbb{A}^{2}\)
Show that
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Every morphism \(f:{\mathbb{A}}^1\setminus\left\{{0}\right\}\to {\mathbb{P}}^1\) can be extended to a morphism \(\widehat{f}: {\mathbb{A}}^1 \to {\mathbb{P}}^1\).
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Not every morphism \(f:{\mathbb{A}}^2\setminus\left\{{0}\right\}\to {\mathbb{P}}^1\) can be extended to a morphism \(\widehat{f}: {\mathbb{A}}^2 \to {\mathbb{P}}^1\).
- Every morphism \({\mathbb{P}}^1\to {\mathbb{A}}^1\) is constant.
Show that
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Every isomorphism \(f:{\mathbb{P}}^1\to {\mathbb{P}}^1\) is of the form \begin{align*} f(x) = {ax+b \over cx+d} && a,b,c,d\in k .\end{align*} where \(x\) is an affine coordinate on \({\mathbb{A}}^1\subset {\mathbb{P}}^1\).
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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 \(i\).
There is a bijection \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*}
Does the above bijection hold if
- \(X\) is an arbitrary prevariety but \(Y\) is still affine?
- \(Y\) is an arbitrary prevariety but \(X\) is still affine?