## Affine schemes

Last update: 12 April 2012

## Affine schemes

Let $A$ be a commutative ring.

• The spectrum, or prime spectrum, of $A$ is $Spec(A) = { prime ideals x of A } .$
• The basic open sets are $Xg = {x∈Spec(A) | g∉X}, for g∈A.$
• Let $g\in A.$ The basic ring at $g$ is the ring $A[1g] = {fgk | f∈A, k∈ℤ≥0}$ with $f1gk + f2gl = f1gl + f2gk gk+l and f1gk ⋅ f2gl = f1f2gk+l$ with the ring homomorphism $ι: A → A[1g] f ↦ f1.$
• Spectrum is the contravariant functor $Spec: {commutativerings} → {ringed spaces} A ↦ Spec(A) = {prime ideals x of A} φ: A1→A2 ↦ Spec(φ): Spec(A2) → Spec(A1) x ↦ φ-1(x)$ where $X=\mathrm{Spec}\left(A\right)$ has topology with closed sets and structure sheaf ${𝒪}_{X}$ determined by $𝒪X(Xg) = A[1g] and 𝒪X (Xk⊇Xg): A[1h] → A[1g] fhm ↦ fsmgmn$ when ${g}^{n}=sh$ with $s\in A$ and $n\in {ℤ}_{>0}.$
• 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 $n\in {ℤ}_{>0}.$

## 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 $Ax = { fg | f,g∈A, g∉x }$ with $f1g1 + f2g2 = f1g2 + f2g1g1g2 and f1g1 ⋅ f2g2 = f1f2g1g2$ and ring homomorphism $ι: A → Ax f ↦ f1.$

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 $X = Spec(A) and x∈X$ then and the evaluation homomorphism at $x$ is

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 $|X| = Maxspec(A) = {maximal ideals x of A}.$
Hence $|X|=\left\{closed points of\phantom{\rule{.5em}{0ex}}X\right\}.$

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

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

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

HW: Show that $Spec(ℤ) = {prime ideals of ℤ} = {maximal ideals of ℤ} = {pℤ | p∈ℤ>0 and p is prime}$ and that the open sets in $\mathrm{Spec}\left(ℤ\right)$ are the complements of finite sets.

HW: Show that if $𝔽$ is a field then $Spec(𝔽) = pt,$ the one point topological space.

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

HW: Show that if $𝔽$ is a field then $Spec(𝔽[t]) = {p𝔽[t] | p∈𝔽[t] is an irreducible polynomial}.$

HW: Show that if $𝔽$ is a field then $Spec(𝔽[t1,...,tn]) = {(p) | p∈𝔽[t1,...,tn] is an irreducible polynomial}$ where $\left(p\right)$ is the ideal generated by $p$ in $𝔽\left[{t}_{1},...,{t}_{n}\right].$

HW: Show that if $𝔽$ is a field and $\stackrel{_}{𝔽}=𝔽$ then $Spec(𝔽_[t1,...,tn]) = 𝔽_n,$ with open sets the complements of finite sets.

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

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

1. ${X}_{g}\cap {X}_{h}={X}_{gh}.$
2. is nilpotent.
3. is a unit.
4. ${X}_{g}$ 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?