Affine schemes

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

Last update: 12 April 2012

Affine schemes

Let $A$ be a commutative ring.

• The spectrum, or prime spectrum, of $A$ is $$\mathrm{Spec}\left(A\right)=\left\{prime\; idealsxofA\right\}.$$
• The basic open sets are $$X_g=\{x\in \mathrm{Spec}\left(A\right)|g\notin X\},forg\in A.$$
• Let $g\in A.$ The basic ring at $g$ is the ring $$A\left[\frac{1}{g}\right]=\left\{\frac{f}{g^k}\right|f\in A,k\in {\mathbb{Z}}_{\ge 0}\}$$ with $$\frac{f_1}{g^k}=\frac{f_2}{g^l}if\; there\; exists\; n\in {\mathbb{Z}}_{>0} \; such\; that\; f_1{g}^{l+n}=f_2{g}^{k+n},$$ $$\frac{f_1}{g^k}+\frac{f_2}{g^l}=\frac{f_1{g}^l+f_2{g}^k}{g^{k+l}}and\frac{f_1}{g^k}\cdot \frac{f_2}{g^l}=\frac{f_1{f}_2}{g^{k+l}}$$ with the ring homomorphism $$\begin{array}{rrcl}\iota :& A& \to & A\left[\frac{1}{g}\right]\\ & f& \mapsto & \frac{f}{1}.\end{array}$$
• Spectrum is the contravariant functor $$\begin{array}{rrcl}\mathrm{Spec}:& \left\{\genfrac{}{}{0ex}{}{commutative}{rings}\right\}& \to & \left\{ringed\; spaces\right\}\\ & A& \mapsto & \mathrm{Spec}\left(A\right)=\left\{prime\; idealsxofA\right\}\\ & \phi :A_1\to A_2& \mapsto & \begin{array}{rrcl}\mathrm{Spec}\left(\phi \right):& \mathrm{Spec}\left(A_2\right)& \to & \mathrm{Spec}\left(A_1\right)\\ & x& \mapsto & {\phi }^{-1}\left(x\right)\end{array}\end{array}$$ where $X=\mathrm{Spec}\left(A\right)$ has topology with closed sets $$V\left(S\right)=\{x\in X|x\supseteq S\},for\; S\subseteq A,$$ and structure sheaf ${𝒪}_X$ determined by $${𝒪}_X\left(X_g\right)=A\left[\frac{1}{g}\right]and\begin{array}{rrcl}{𝒪}_X(X_k\supseteq X_g):& A\left[\frac{1}{h}\right]& \to & A\left[\frac{1}{g}\right]\\ & \frac{f}{h^m}& \mapsto & \frac{f{s}^m}{g^{mn}}\end{array}$$ when $g^n=sh$ with $s\in A$ and $n\in {\mathbb{Z}}_{>0}.$
• The topology on $X=\mathrm{Spec}\left(A\right)$ is the Zariski topology.
• An affine scheme is an element of the image of $\mathrm{Spec}.$

HW: Show that if $X_k\supseteq X_g$ then $\sqrt{\left(h\right)}=\sqrt{\left(g\right)}$ so that $g^n=sh$ for some $s\in A$ and $n\in {\mathbb{Z}}_{>0}.$

Points in $X=\mathrm{Spec}\left(A\right)$

Let $A$ be a commutative ring.

• The ring $A$ is local if $A$ has a unique maximal ideal.
• The residue field of a local ring $A$ with maximal ideal $𝔪$ is $k=\frac{A}{𝔪}.$
• Let $x$ be a prime ideal of $A.$ The localization of $A$ at $x$ is the ring $Ax$ given by $$A_x=\left\{\frac{f}{g}\right|f,g\in A,g\notin x\}$$ with $$\frac{f_1}{g_1}=\frac{f_2}{g_2}if\; there\; exists\; g\in A, g\notin x \; with\; f_1{g}_{2}g=f_2{g}_{1}g,$$ $$\frac{f_1}{g_1}+\frac{f_2}{g_2}=\frac{f_1{g}_2+f_2{g}_1}{g_1{g}_2}and\frac{f_1}{g_1}\cdot \frac{f_2}{g_2}=\frac{f_1{f}_2}{g_1{g}_2}$$ and ring homomorphism $$\begin{array}{rrcl}\iota :& A& \to & A_x\\ & f& \mapsto & \frac{f}{1}.\end{array}$$

HW: Show that if $A$ is a commutative ring and $x$ is a prime ideal of $A$ then $A_x$ is a local ring with maximal ideal ${𝔪}_x=x{A}_x.$

In summary, if $A$ is a commutative ring $$X=\mathrm{Spec}\left(A\right)andx\in X$$ then $$\begin{array}{rcl}{A}_x& =& localization\; of\; A \; at\; x=stalk\; of\; {𝒪}_x \; at\; x,\\ {𝔪}_x& =& x{A}_x \; is\; the\; maximal\; ideal\; of\; A_x,\\ k\left(x\right)& =& \frac{A_x}{{𝔪}_x} \; is\; the\; residue\; field\; of\; A_x,\end{array}$$ and the evaluation homomorphism at $x$ is $$\begin{array}{rrcccl}{\mathrm{ev}}_x:& A& \to & \frac{A}{x}& \hookrightarrow & k\left(x\right)\\ & f& \mapsto & f\left(x\right)& \mapsto & \frac{f\left(x\right)}{1},\end{array}where\; f\left(x\right)=f+x.$$

HW: Show that the stalk of ${𝒪}_x$ at $x$ is $A_x.$

HW: Show that $k\left(x\right)$ is the field of fractions of $\frac{A}{x}.$

HW: Show that $k\left(x\right)$ is the field of fractions of $A_x.$

Let $A$ be a commutative ring and let $X=\mathrm{Spec}\left(A\right).$

• A closed point of $X$ is a maximal ideal $x$ of $A.$
• The maximal ideal spectrum of $A$ is $$\left|X\right|=\mathrm{Maxspec}\left(A\right)=\left\{maximal\; idealsxofA\right\}.$$

Hence $\left|X\right|=\left\{closed\; points\; ofX\right\}.$

HW: A topological space is irreducible if every pair of non empty open sets intersect. Let $A$ be a commutative ring and $X=\mathrm{Spec}\left(A\right).$ Show that $$\begin{array}{rcl}X \; is\; irreducible& \iff & \sqrt{0}=\mathrm{nil}\left(A\right) \; is\; a\; prime\; ideal\\ & \iff & \frac{A}{\mathrm{nil}\left(A\right)} \; is\; an\; integral\; domain.\end{array}$$

HW: Let $A$ be a commutative ring and let $X=\mathrm{Spec}\left(A\right).$ Show that the irreducible components of $X$ are $V\left(x\right)$ such that $x$ is a minimal prime ideal of $A.$

HW: Let $A$ be a commutative ring and let $X=\mathrm{Spec}\left(A\right).$ Let $x\in X.$ Show that

1. $\left\{x\right\}$ is closed in $X \iff x$ is a closed point.
2. If $y\in X$ then $y\in \stackrel{\_}{\left\{x\right\}}\iff x\subseteq y.$

HW: Let $A$ be a commutative ring, $X=\mathrm{Spec}\left(A\right)$ and $$\left|X\right|=\left\{closed\; points\; inX\right\}.$$ Show that the sets $$M_f=\{y\in \left|X\right||f\notin y\},for\; f\in A,$$ are a basis for the subspace topology on $\left|X\right|$ (as a subspace of $X$ with the Zariski topology).

HW: Show that $$\begin{array}{rcl}\mathrm{Spec}\left(\mathbb{Z}\right)& =& \left\{prime\; ideals\; of\mathbb{Z}\right\}\\ & =& \left\{maximal\; ideals\; of\mathbb{Z}\right\}\\ & =& \left\{p\mathbb{Z}\right|p\in {\mathbb{Z}}_{>0}andpis\; prime\}\end{array}$$ and that the open sets in $\mathrm{Spec}\left(\mathbb{Z}\right)$ are the complements of finite sets.

HW: Show that if $𝔽$ is a field then $$\mathrm{Spec}\left(𝔽\right)=\mathrm{pt},$$ the one point topological space.

HW: Show that if $𝔽$ is a field and $\stackrel{\_}{𝔽}=𝔽$ then $$\mathrm{Spec}\left(\stackrel{\_}{𝔽}\left[t\right]\right)=\stackrel{\_}{𝔽},$$ with open sets the complements of finite sets.

HW: Show that if $𝔽$ is a field then $$\mathrm{Spec}\left(𝔽\left[t\right]\right)=\left\{p𝔽\left[t\right]\right|p\in 𝔽\left[t\right]is\; an\; irreducible\; polynomial\}.$$

HW: Show that if $𝔽$ is a field then $$\mathrm{Spec}\left(𝔽[t_1,...,t_n]\right)=\left\{\left(p\right)\right|p\in 𝔽[t_1,...,t_n]is\; an\; irreducible\; polynomial\}$$ where $\left(p\right)$ is the ideal generated by $p$ in $𝔽[t_1,...,t_n].$

HW: Show that if $𝔽$ is a field and $\stackrel{\_}{𝔽}=𝔽$ then $$\mathrm{Spec}\left(\stackrel{\_}{𝔽}[t_1,...,t_n]\right)={\stackrel{\_}{𝔽}}^n,$$ with open sets the complements of finite sets.

HW: Draw pictures of $\mathrm{Spec}\left(\mathbb{Z}\right),\mathrm{Spec}\left(ℝ\right),\mathrm{Spec}\left(ℂ\left[t\right]\right),\mathrm{Spec}\left(ℝ\left[t\right]\right),\mathrm{Spec}\left(\mathbb{Z}\left[t\right]\right).$

HW: Let $A$ be a commutative ring and let $X=\mathrm{Spec}\left(A\right)$ and $$X_g=\{x\in \mathrm{Spec}\left(A\right)|g\notin X\}for\; g\in A,$$ the basic open sets in $X.$ Show that

1. $X_g\cap X_h=X_{gh}.$
2. $X_g=\varnothing \iff g$ is nilpotent.
3. $X_g=X \iff g$ is a unit.
4. $X_g=X_h \iff \sqrt{\left(g\right)}=\sqrt{\left(h\right)}.$
5. $X_g$ is quasicompact (every open cover of $X$ has a finite subcover).

Notes and References

[AM, Ch.1, Ex.15,16] define $\mathrm{Spec}\left(A\right)$ and the Zariski topology on $\mathrm{Spec}\left(A\right).$ [AM, Ch.1, Ex.21] says that $\mathrm{Spec}$ is a functor from commutative rings to topological spaces. [AM, Ch.1, Ex.18] defines and remarks on closed points, [AM, Ch.1, Ex.19] says when $\mathrm{Spec}\left(A\right)$ is irreducible, and [AM, Ch.1, Ex.20] identifies the irreducible components of $\mathrm{Spec}\left(A\right).$

[AM, Ch.3, Ex.23,24] define the structure sheaf of $\mathrm{Spec}\left(A\right).$ In particular, [AM, Ch.3, Ex.23(v)] says that the stalk at $x$ is the local ring $A_x.$

[AM, Ch.3, Ex.27] defines the constructible topology on $\mathrm{Spec}\left(A\right).$ [AM, Ch.3, Ex.28-29] make further remarks on the constructible topology and [AM, Ch.3, Ex.30] characterizes when the constructible topology and the Zariski topology coincide. It is not yet clear to me whether this has any use and/or whether there is any connection to the étale topology. Wikipedia "Constructible sheaf" says that constructible sheaves generalize the constructible topology (see? Deligne, SGA 4$\frac{1}{2}$).

References

References?

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