Spectrum

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

Spectrum

The spectrum functor $$\mathrm{Spec}:\left\{\text{commutative rings with identity}\right\}\to \left\{\text{ringed spaces}\right\}$$ is the contravariant functor given by $$\mathrm{Spec}\left(A\right)=\left\{x\subseteq A|x\text{is a prime ideal of}A\right\}$$ with closed sets $$V\left(E\right)=\left\{y\in \mathrm{Spec}\left(A\right)|y\supseteq E\right\}\text{for}E\subseteq A\text{,}$$ and structure sheaf ${𝒪}_X$, for $X=\mathrm{Spec}\left(A\right)$, determined by $${𝒪}_X\left(V{\left(\left\{f\right\}\right)}^c\right)=A_f\text{and}{𝒪}_X\left(V{\left(\left\{f\right\}\right)}^c\supseteq V{\left(\left\{g\right\}\right)}^c\right)={\rho }_{g,f}$$ for $f\in A$, where $$A_f=S^{-1}A\text{with}s=\left\{f^n|n\in {\mathbb{Z}}_{\ge 0}\right\}\text{and}$$ $$\begin{array}{rcl}{\rho }_{g,f}:A_f& ⟶& A_g\\ \frac{a}{f^m}& ⟼& \frac{a{s}^m}{g^{mm}}\end{array}\text{when}{g}^n=sf\text{with}s\in A\text{and}n\in {\mathbb{Z}}_{>0}$$ and, if $\varphi :A_1\to A_2$ is a homomorphism of rings then $$\begin{array}{rcl}\mathrm{Spec}\left(\varphi \right):\mathrm{Spec}\left(A_2\right)& ⟶& \mathrm{Spec}\left(A_1\right)\\ x& ⟼& {\varphi }^{-1}\left(x\right)\text{.}\end{array}$$

The topology on $\mathrm{Spec}\left(A\right)$ is the Zariski topology.

An affine scheme is an element of the image of $\mathrm{Spec}$.

The stalk of ${𝒪}_x$ at $x$ is ${𝔬}_x$[??? - is this right? not defined yet? see below], the local ring of $A$ with respect to the prime ideal $x$.

Ideals and local rings

Let $A$ be a commutative ring with identity.

A prime ideal of $A$ is an ideal $x\subseteq A$ such that $$\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$x$}\right.\text{is an integral domain.}$$

A maximal ideal of $A$ is an ideal $m\subseteq A$ such that $$A/m\text{is a field.}$$

Let $x$ be a prime ideal of $A$. The local ring of $A$ with respect to $x$ is $${𝔬}_x=\mathbf{\text{[??? - something missing here?]}}$$

The maximal ideal of ${𝔬}_x$ is $$m_x=x·{𝔬}_x\text{and}{k}_x={𝔬}_x/m_x$$ is the residue field of ${𝔬}_x$.

The evaluation of $x$ is the homomorphism $$\begin{array}{rcl}A& ⟶& A/x\subseteq k_x\\ f& ⟼& f\left(x\right)\text{.}\end{array}$$

If $f\in A$ let $S=\left\{f^n|n\in {\mathbb{Z}}_{\ge 0}\right\}$ and define $$A_f=S^{-1}=\left\{\frac{a}{f^n}|a\in A\text{,}n\in {\mathbb{Z}}_{\ge 0}\text{with}\right\}$$ $$\frac{a_1}{f^n}=\frac{a_2}{f^m}\text{if}{f}_{a_1}^m=f_{a_2}^n$$ and addition and multiplication [??? - something missing here?]

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)