Affine scheme

Builds on Locally ringed space Spectrum of a ring Commutative ring Zariski topology Localization Localization at a prime ideal Sheaf

Definition

Let $A$ be a commutative ring with identity, and let $X = \operatorname{Spec} A$ be its spectrum with the Zariski topology. For $f \in A$ , write $D(f) = \set{\mathfrak{p} \in X : f \notin \mathfrak{p}}$ ; these sets form a basis of open sets of $X$ . Let $A_f$ be the localization of $A$ at the powers of $f$ . The structure sheaf $\mathcal{O}_X$ is the unique sheaf of rings on $X$ with $\mathcal{O}_X(D(f)) = A_f$ for every $f \in A$ . Its restriction maps $A_f \to A_g$ for $D(g) \subseteq D(f)$ are the natural maps. Its stalk at $\mathfrak{p}$ is the localization at the prime ideal $A_{\mathfrak p}$ , which is a local ring.

An affine scheme is a locally ringed space isomorphic to $(\operatorname{Spec} A, \mathcal{O}_{\operatorname{Spec} A})$ for some commutative ring $A$ .

Remarks

Example: $\operatorname{Spec} \Z$ is an affine scheme; its points are $(0)$ and $(p)$ for the primes $p$ , and $\mathcal{O}(D(n)) = \Z[1/n]$ for $n \ne 0$ . Example: the affine $n$ -space over a ring $R$ is $\mathbb{A}^n_R = \operatorname{Spec} R[x_1, \ldots, x_n]$ . Non-example: for a field $k$ , the open subset $U = \mathbb{A}^2_k \setminus \set{(x, y)}$ of the plane, with the restricted sheaf, is a locally ringed space but not an affine scheme. Its ring of global functions is $k[x, y]$ , so if it were affine it would be $\operatorname{Spec} k[x, y]$ , which also contains the origin $(x, y)$ . The ring is recovered as $A \cong \mathcal{O}_X(X)$ , the global sections.

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Tag	`CTBYV71CEJ`
URI	`m2://CTBYV71CEJ`
Kind	definition
Writing	W0 stub
MSC	14A15
Defines	Affine scheme
Uses	Locally ringed space, Spectrum of a ring, Commutative ring, Zariski topology, Localization, Localization at a prime ideal, Sheaf

Affine scheme

Definition

Remarks

Reading path

Further reading

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