## Varieties

An affine algebraic variety over $\stackrel{‾}{𝔽}$ is a set $X={ (x1,…, xn) | fα(x1, …,xn)=0 for all fα∈S}$ where $S$ is a set of polynomials in $\stackrel{‾}{𝔽}\left[{t}_{1},\dots ,{t}_{n}\right]$. By definition, these are the closed sets in the Zariski topology on ${\stackrel{‾}{𝔽}}^{n}$. Let $U$ be an open set of $X$ and define ${𝒪}_{X}\left(U\right)$ to be the set of functions $f:U\to \stackrel{‾}{𝔽}$ 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 $g,h\in \stackrel{‾}{𝔽}\left[{t}_{1},\dots ,{t}_{n}\right]$ such that $if y∈Uα then h(y)≠0 and f(y) = g(y) h(y) .$ Then ${𝒪}_{X}$ is a sheaf on $X$ and $\left(X,{𝒪}_{X}\right)$ is a ringed space. The sheaf ${𝒪}_{X}$ is the structure sheaf of the affine algebraic variety $X$.

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

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

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

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

## Schemes

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

An $\stackrel{‾}{𝔽}$-scheme is a ringed space $\left(X,{𝒪}_{X}\right)$ 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 $\left({U}_{\alpha },𝒪{|}_{{U}_{\alpha }}\right)$ is isomorphic to an affine $\stackrel{‾}{𝔽}$-scheme,
3. $\left(X,{𝒪}_{X}\right)$ is reduced, i.e. for each $x\in X$ the local ring ${𝒪}_{X,x}$ has no nonzero nilpotent elements,
4. $\left(X,{𝒪}_{X}\right)$ satisfies the separation axiom, i.e. $ΔX= {(x,x) | x∈X} is closed in X.$

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