Affine varieties

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 May 2012

Affine varieties

Let $k$ and $\stackrel{\_}{𝕂}$ be fields with $k\subseteq \stackrel{\_}{𝕂}$ and $\stackrel{\_}{𝕂}$ algebraically closed.

• Define $$\begin{array}{rrcl}V:& \left\{\genfrac{}{}{0ex}{}{subsets\; of}{k[t_1,...,t_n]}\right\}& \to & \left\{subsets\; of{\stackrel{\_}{𝕂}}^n\right\}\\ & S& \mapsto & V\left(S\right)\end{array}$$ where $$V\left(S\right)=\{\{x_1,...,x_n)\in {\stackrel{\_}{𝕂}}^n|f(x_1,...,x_n)=0forf\in S\}.$$
• An affine $(k,\stackrel{\_}{𝕂})$ variety is an element of the image of $V.$

Let $Y=V\left(S\right)$ be an affine $(k,\stackrel{\_}{𝕂})$ variety.

• A regular function on $Y$ is $$\begin{array}{rrcl}f{|}_Y:& Y& \to & \stackrel{\_}{𝕂}\\ & (x_1,...,x_n)& \mapsto & f(x_1,...,x_n),\end{array}\text{for }f\in k[t_1,...,t_n].$$
• The coordinate ring of $Y$ is $$k\left[Y\right]=\frac{k[t_1,...,t_n]}{\sqrt{⟨S⟩}}\text{if}Y=V\left(S\right).$$
• The (Zariski) topology on $Y$ has closed sets $$V\left(S\right)=\{y\in Y|g\left(y\right)=0forg\in S\}for\; S\subseteq k\left[Y\right].$$
• The structure sheaf of $Y$ is $${𝒪}_Y\left(U\right)=\{\frac{f}{g}\in k(t_1,...,t_n)|g\left(u\right)\ne 0foru\in U\}\text{for open sets}U\text{of}Y.$$

HW: Show that $$\begin{array}{rrcl}\phi :& k[t_1,...,t_n]& \to & \left\{regular\; functions\; onY\right\}\\ & f& \mapsto & f{|}_Y\end{array}$$ has $\ker \phi =\sqrt{⟨S⟩}$.

HW: Show that, in $ℂ\left[t\right],$ $⟨t⟩=\sqrt{⟨t^2⟩}$ and $⟨t⟩\ne ⟨t^2⟩$.

HW: Let $A$ be a finitely generated $k-$algebra and let $u_1,...,u_n$ be generators of $A.$ Let $$𝔞=\ker \phi ,where\begin{array}{rrcl}\phi :& k[t_1,...,t_n]& \to & A\\ & t_j& \mapsto & u_j\end{array}\text{so that}A\simeq \frac{k[t_1,...,t_n]}{𝔞}.$$

HW: Show that $$\begin{array}{rcl}\left\{\genfrac{}{}{0ex}{}{affine(k,\stackrel{\_}{𝕂})}{varieties}\right\}& \leftrightarrow & \left\{\genfrac{}{}{0ex}{}{finitely\; generatedk-algebras}{with\; no\; nilpotent\; elements}\right\}\\ Y& \mapsto & k\left[Y\right]\\ V\left(𝔞\right)& ↤& A=\frac{k[t_1,...,t_n]}{𝔞}\end{array}$$ is a bijection.

HW: Let $Y$ be an affine $(k,\stackrel{\_}{𝕂})$ variety. Show that $$\begin{array}{rcl}Y& \leftrightarrow & \left\{\genfrac{}{}{0ex}{}{k-algebra\; homomorphisms}{\gamma :k\left[Y\right]\to \stackrel{\_}{𝕂}}\right\}\\ x& \mapsto & {\mathrm{ev}}_x\\ (\gamma \left(u_1\right),...,\gamma \left(u_n\right))& ↤& \gamma \end{array}$$ where $u_1,...,u_n$ are generators of $A=k\left[Y\right],$ is a bijection.

HW: Let $Y$ be an affine $(k,𝕂)$ variety,

1. $k\left[Y\right]$ the coordinate ring of $Y,$
2. $X=\mathrm{Spec}\left(k\left[Y\right]\right),$ and
3. $\left|X\right|=\mathrm{Maxspec}\left(k\left[Y\right]\right)$ the set of closed points of $X$.

Define $$\begin{array}{rrcl}\phi :& Y& \to & X\\ & x& \mapsto & \ker \left({\mathrm{ev}}_x\right)\end{array}where\begin{array}{rrcl}{\mathrm{ev}}_x:& k\left[Y\right]& \to & \stackrel{\_}{𝕂}\\ & f& \mapsto & f\left(x\right).\end{array}$$

1. Show that if $k=\stackrel{\_}{𝕂}$ then $\phi$ is injective and $\mathrm{im}\phi =\left|X\right|$.
2. Show that if $\stackrel{\_}{𝕂}$ has infinite transcendence degree over $k$ then $\phi$ is surjective (since, if $𝔭$ is a prime ideal of $A=k\left[Y\right]$ then the field of fractions of $\frac{A}{𝔭}$ is a finitely generated field extension of $k$ so that $\frac{A}{𝔭}$ is an algebraic extension of a pure transcendental extension of $k$ so that there exists a homomorphism $\gamma :A\to \stackrel{\_}{𝕂}$ with $\ker \gamma =𝔭$).

HW: Show that if $k=ℝ$ and $\stackrel{\_}{𝕂}=ℂ$ and $$Y=V\left(S\right)whereS=\{t^2+1\}$$ then $$Y=\{i,-i\}andk\left[Y\right]=\frac{ℝ\left[t\right]}{⟨t^2+1⟩}\simeq ℂ.$$ In this case $X=\mathrm{Spec}\left(ℂ\right)=\mathrm{pt}$ and $X=\left|X\right|$ and the map $\phi :Y\to X$ is surjective but not injective.

HW: Show that if $k=ℂ$ and $\stackrel{\_}{𝕂}=ℂ$ and $$Y=V\left(S\right)whereS=\{t^2+1\}$$ then $$Y=\{i,-i\}andk\left[Y\right]=\frac{ℂ\left[t\right]}{⟨t^2+1⟩}.$$ In this case $X=\mathrm{Spec}\left(k\left[Y\right]\right)$ has two points $x_1=⟨t+i⟩$ and $x_2=⟨t-i⟩$ and $X=\left|X\right|$. Then the map $$\begin{array}{cccc}\phi :& Y& ⟶& \left|X\right|=X\\ & i& ⟼& x_2\\ & -i& ⟼& x_1\end{array}\text{is a bijection.}$$

Notes and References

This definition of affine $(k,𝕂)$ varieties follows [Mac, Ch.1]. An alternative treatment is in [AM, Ch.1, Ex.27]. The exercise [AM, Ch.1, Ex.28] says that the affine varieties form a category. The comparison of affine $(k,𝕂)$ varieties with affine schemes is in [Mac, Ch.3, p.23]. The definition of the structure sheaf of $Y$ is given in [Mac, Ch.1, p.10].

Should the following left overs from a previous page be put somewhere?

• A projective variety is a variety that can be embedded in a projective space ${\mathbb{P}}^n$.
• A variety is complete if it satisfies:
  1. If $W$ is a variety then $\mathrm{pr}:V\times W\to W$ is a closed map (with respect to the Zariski topology).

References

[AM] M. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. MR0242802.

[Mac] I.G. Macdonald, Algebraic geometry. Introduction to Schemes, W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp. MR0238845.

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