Affine Varieties

Affine varieties

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 November 2009

Affine varieties

Let $S \in 𝔽 [t_{1}, \dots, t_{n}]$ , $⟨S⟩$ be the the ideal of $𝔽 [t_{1}, \dots, t_{n}]$ generated by $S$ , and $\sqrt[n]{⟨S⟩} = \{f \in 𝔽 [t_{1}, \dots, t_{n}] | f^{n} \in ⟨S⟩ for some n \in ℤ_{\geq 0}\} .$

Let $\begin{array}{rcl} V & = & \{p \in {\overline{𝔽}}^{n} | f (p_{1}, \dots, p_{n}) = 0 for p \in S\} \\ = & \{p \in {\overline{𝔽}}^{n} | f (p_{1}, \dots, p_{n}) = 0 for p \in ⟨S⟩\} . \end{array}$ Then the ring of regular functions on $V$ is $𝒪_{V} = \{ϕ : V \to \overline{𝔽} | there exists g \in 𝔽 [t_{1}, \dots, t_{n}] with g (p) = ϕ (p) for all p \in V\}$ and the maps $\begin{array}{rcl} 𝒪_{V} (V) & ⟶ & \frac{𝔽 [t_{1}, \dots, t_{n}]}{\sqrt{⟨S⟩}} \\ f |_{V} & ⟼ & f \end{array}$ and $\begin{array}{rcl} V & ⟶ & {Hom}_{𝔽 -alg} (\frac{𝔽 [t_{1}, \dots, t_{n}]}{\sqrt{⟨S⟩}}, \overline{𝔽}) \\ (p_{1}, \dots, p_{n}) & ⟼ & (\begin{array}{rcl} p : \frac{𝔽 [t_{1}, \dots, t_{n}]}{\sqrt{⟨S⟩}} & \to & \overline{𝔽} \\ f & \mapsto & f (p) \end{array}) \end{array}$ are isomorphisms.

The closed sets in the Zariski topology are $V (E) = \{p \in V | f (p) = 0 for f \in E\} where E \subseteq 𝒪_{V} (V) .$

The structure sheaf $𝒪_{V}$ is given by $𝒪_{V} (U) = \{\frac{f}{g} \in 𝔽 (t_{1}, \dots, t_{n}) | g (p) \neq 0 for p \in U\}$ for each open set $U$ .

A projective variety is a variety that can be embedded in a projective space $ℙ^{n}$ .

A variety is complete if it satisfies:

If $W$ is a variety then $pr : V \times W \to W$ is a closed map (with respect to the Zariski topology).

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)