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    $\Spec A$ as a locally ringed space

    D-VKR54

    (The spectrum)

    \(\Spec A\) is the set of prime ideals of \(A\), with closed sets \(V(J) = \ts{\mfp \st \mfp \supseteq J}\). Its structure sheaf sends \(U\) to the functions \[ \varphi : U \to \coprod_{\mfp \in U} A_\mfp \] with \(\varphi(\mfp) \in A_\mfp\), locally of the form \(f/g\) with \(g \notin \mfp\).

    \(\OO_{\Spec A}(D_f) = A_f\), and in particular \(\OO_{\Spec A}(\Spec A) = A\). The stalk at \(\mfp\) is \(A_\mfp\), a local ring.

    For a field \(k\), the affine line punctured at \(0\) and \(1\) is the distinguished open set \(D_{x(x-1)} \subseteq \AA^1_k = \Spec k[x]\), whose closed complement is \(V(x(x-1)) = \{(x), (x-1)\}\). It is the affine scheme \(\Spec R\) with \[R = k[x]_{x(x-1)} = k[x]\left[\frac{1}{x(x-1)}\right].\]

    For a ring map \(\varphi\colon A\to B\) and a prime \(\mfq\) of \(B\), \(\inverseof{\varphi}(\mfq)\) is a prime of \(A\), so \(\varphi\) induces a continuous map \(\Spec B\to\Spec A\). The preimage of a maximal ideal need not be maximal: for \(\ZZ\subseteq\QQ\), the preimage of the maximal ideal \((0)\) of \(\QQ\) is the prime \((0)\) of \(\ZZ\), which is not maximal.

    Morphisms of schemes are morphisms of locally ringed spaces: the induced maps on stalks are local homomorphisms. With this condition, ring maps \(A\to B\) correspond bijectively to morphisms \(\Spec B\to\Spec A\) [@Har10a, Proposition II.2.3]; as morphisms of ringed spaces there are more (D-AN662).

    What are the points of $\Spec A$, and what is its topology?
    What is the structure sheaf of $\Spec A$?
    What are the stalks of $\OO_{\Spec A}$?
    Construct a ring $R$ such that $\Spec R$ is $\AA^1_k$ punctured at $0$ and $1$.
    What is the value of $n \in \ZZ$ at a prime $p \in \Spec \ZZ$, and what are the zeros of $60$ as a function on $\Spec \ZZ$?
    Find a ring $R$ and $0 \neq f \in R$ whose value at every point of $\Spec R$ is zero.
    Describe all the prime ideals of $k[x,y]$.
    Show that $\sqrt{I}$ is the intersection of the prime ideals containing $I$.
    If $D(f) = \bigcup_i D(g_i)$, show that $f^n = \sum_i b_i g_i$ for some $n$ and some $b_i \in R$.
    Show that the Zariski topology on $\Spec R$ is not $T_1$ in general.

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    • D-MORIMMOpen, closed, and locally closed immersionsrelated-to
    • D-MORFIBThe fibre of a morphism as a base changerelated-to
    • D-VARREGThe sheaf of regular functions on a varietyrelated-to
    • D-SCHPROJ$\Proj$ of a graded ringuses
    • D-SCHFORMALFormal schemes, and the formal discuses
    • D-Q7Q2NFrom a fan to a toric varietyuses
    • PR-RJ3P7Going up and going down, read on spectrauses
    • PR-6TCSJProperties of $\Spec A$ read off $A$uses
    • D-QNTZYQuasicoherent and coherent sheavesuses
    • D-AN662Schemes, and morphisms of schemesuses
    • T-AFFCOMMThe affine communication lemmauses

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