## Affine varieties

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 $V: { subsets of k[t1,...,tn] } → {subsets of 𝕂_n} S ↦ V(S)$ where $V(S) = { {x1,...,xn) ∈𝕂_n | f(x1,...,xn) =0 for f∈S }.$
• An affine $\left(k,\stackrel{_}{𝕂}\right)$ variety is an element of the image of $V.$

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

• A regular function on $Y$ is
• The coordinate ring of $Y$ is $k[Y] = k[t1,...,tn] ⟨S⟩ if Y=V(S).$
• The (Zariski) topology on $Y$ has closed sets
• The structure sheaf of $Y$ is $𝒪Y(U) = { fg∈ k(t1,...,tn) | g(u) ≠ 0 for u∈U} for open sets UofY.$

HW: Show that $φ: k[t1,...,tn] → {regular functions on Y} f ↦ f|Y$ has $\mathrm{ker}\phi =\sqrt{⟨S⟩}$.

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 $𝔞=kerφ, where φ: k[t1,...,tn] → A tj ↦ uj so that A≃ k[t1,...,tn] 𝔞 .$

HW: Show that ${ affine (k,𝕂_)varieties } ↔ { finitely generated k-algebras with no nilpotent elements } Y ↦ k[Y] V(𝔞) ↤ A= k[t1,...,tn] 𝔞$ is a bijection.

HW: Let $Y$ be an affine $\left(k,\stackrel{_}{𝕂}\right)$ variety. Show that $Y ↔ { k-algebra homomorphisms γ:k[Y]→𝕂_ } x ↦ evx ( γ(u1),...,γ(un) ) ↤ γ$ where ${u}_{1},...,{u}_{n}$ are generators of $A=k\left[Y\right],$ is a bijection.

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

1. $k\left[Y\right]$ the coordinate ring of $Y,$
2. $X=\mathrm{Spec}\left(k\left[Y\right]\right),$ and
3. $|X|=\mathrm{Maxspec}\left(k\left[Y\right]\right)$ the set of closed points of $X$.
Define $φ: Y → X x ↦ ker(evx) where evx: k[Y] → 𝕂_ f ↦ f(x).$
1. Show that if $k=\stackrel{_}{𝕂}$ then $\phi$ is injective and $\mathrm{im}\phi =|X|$.
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 $\mathrm{ker}\gamma =𝔭$).

HW: Show that if $k=ℝ$ and $\stackrel{_}{𝕂}=ℂ$ and $Y=V(S) where S= {t2+1}$ then $Y = {i,-i} and k[Y] = ℝ[t] ⟨t2+1⟩ ≃ ℂ.$ In this case $X=\mathrm{Spec}\left(ℂ\right)=\mathrm{pt}$ and $X=|X|$ and the map $\phi :Y\to X$ is surjective but not injective.

HW: Show that if $k=ℂ$ and $\stackrel{_}{𝕂}=ℂ$ and $Y=V(S) where S = {t2+1}$ then $Y = {i,-i} and k[Y] = ℂ[t] ⟨t2+1⟩ .$ 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 $X=|X|$. Then the map $φ: Y⟶ |X|=X i⟼x2 -i⟼x1 is a bijection.$

## Notes and References

This definition of affine $\left(k,𝕂\right)$ 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 $\left(k,𝕂\right)$ 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 ${ℙ}^{n}$.
• A variety is complete if it satisfies:
1. If $W$ is a variety then $\mathrm{pr}:V×W\to W$ 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.