Varieties and Schemes

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

Last updates: 12 February 2012

Varieties

An affine algebraic variety over $\overline{𝔽}$ is a set $$X=\left\{\right(x_1,\dots ,x_n\left)\right|f_{\alpha }(x_1,\dots ,x_n)=0\text{for all}{f}_{\alpha }\in S\}$$ where $S$ is a set of polynomials in $\overline{𝔽}[t_1,\dots ,t_n]$. By definition, these are the closed sets in the Zariski topology on ${\overline{𝔽}}^n$. Let $U$ be an open set of $X$ and define ${𝒪}_X\left(U\right)$ to be the set of functions $f:U\to \overline{𝔽}$ that are regular at every point of $x\in U$, i.e. if $x\in U$ then there exists a neighborhood $U_{\alpha }\subseteq U$ of $x$ and functions $g,h\in \overline{𝔽}[t_1,\dots ,t_n]$ such that $$\text{if}y\in U_{\alpha }\text{then}h\left(y\right)\ne 0\text{and}f\left(y\right)=\frac{g\left(y\right)}{h\left(y\right)}.$$ Then ${𝒪}_X$ is a sheaf on $X$ and $(X,{𝒪}_X)$ is a ringed space. The sheaf ${𝒪}_X$ is the structure sheaf of the affine algebraic variety $X$.

A variety is a ringed space $(X,𝒪)$ such that

1. $X$ has a finite open covering $\left\{U_{\alpha }\right\}$ such that each $(U_{\alpha },𝒪{|}_{U_{\alpha }})$ is isomorphic to an affine algebraic variety,
2. $(X,𝒪)$ satisfies the separation axiom, i.e. $${\Delta }_X=\left\{\right(x,x\left)\right|x\in X\}\text{is closed in}X,$$ where the topology on $X\times X$ is the Zariski topology.

HW: (Show that the Zariski topology on $X\times X$ is, in general, finer than the product topology on $X\times X$.

A prevariety is a ringed space which satisfies (a).

Schemes

Let $A$ be a finitely generated commutative $\overline{𝔽}$-algebra and let $$X={\mathrm{Hom}}_{\overline{𝔽}\mathrm{-alg}}(A,\overline{𝔽}).$$ By definition, the closed sets of $X$ in the Zariski topology are the sets $$C_J=\left\{M\in X\right|J\subseteq M\}\text{for}J\subseteq A,$$ where we identify the points of $X$ with the maximal ideals in $A$. Let $U$ be an open set of $X$ and let $${𝒪}_X\left(U\right)=\left\{\frac{g}{h}\right|g,h\in A,x\left(h\right)\ne 0\text{for all}x\in U\}.$$ Then ${𝒪}_X$ is a sheaf on $X$, the pair $(X,{𝒪}_X)$ is a ringed space and the space $X$ is an affine $\overline{𝔽}$-scheme.

An $\overline{𝔽}$-scheme is a ringed space $(X,{𝒪}_X)$ such that

1. For each $x\in X$ the stalk ${𝒪}_{X,x}$ is a local ring,
2. $X$ has a finite open covering $\left\{U_{\alpha }\right\}$ such that each $(U_{\alpha },𝒪{|}_{U_{\alpha }})$ is isomorphic to an affine $\overline{𝔽}$-scheme,
3. $(X,{𝒪}_X)$ is reduced, i.e. for each $x\in X$ the local ring ${𝒪}_{X,x}$ has no nonzero nilpotent elements,
4. $(X,{𝒪}_X)$ satisfies the separation axiom, i.e. $${\Delta }_X=\left\{\right(x,x\left)\right|x\in X\}\text{is closed in}X.$$

A prevariety is a ringed space which satisfies (a),(b) and (c). An $\overline{𝔽}$-space is a ringed space which satisfies (a).

Notes and References

These notes are a retyped version of page 3 and 4 of Chapter 4 of Representation Theory: Lecture Notes of Arun Ram from 17 January 2003 (Book2003/chap41.17.03.tex).

References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

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