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> [!definition] Definition. ([[affine scheme]])
> An **affine scheme** is a [[locally ringed space]] [[isomorphism|isomorphic]] in the [[category]] $\mathsf{LRS}$ to $(\text{Spec }A, \mathcal{O}_{\text{Spec A}})$ for some [[commutative ring|commutative]] [[ring]] $A$.
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> Here, $\text{Spec }A$ denotes the [[prime ideal|spectrum]] of $A$ endowed with the [[Zariski topology on a ring spectrum|Zariski topology]], and $\mathcal{O}_{\text{Spec }A}$ denotes the [[structure sheaf on a ring spectrum|structure]] [[sheaf]] on $\text{Spec }A$.
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> [[affine scheme|Affine schemes]] and [[morphism of locally ringed spaces|morphisms of locally ringed spaces]] form a [[category]] $\mathsf{Aff}$. [[the category of affine schemes is dual to that of rings|It is dual to the category of commutative rings.]]
> ![[CleanShot 2025-03-11 at
[email protected]]]
^definition
> [!definition] Definition. (Scheme-theoretic affine space $\mathbb{A}^{n}_{R}$)
> Let $R$ be a [[ring]]. In scheme theory, we *define* (scheme-theoretic) affine $n$-space over $k$ as $\mathbb{A}^{n}_{R}:=\text{Spec }R[X_{1},\dots,X_{n}].$
In the classical setting, [[affine space|we defined]] $\mathbb{A}^{n}_{k}:=k^{n}$ for $k$ an [[algebraically closed]] [[field]], which by [[Hilbert's geometry-algebra correspondence|the Nullstellensatz]] corresponded [[maximal ideal|to]] $\text{mSpec }k[X_{1},\dots,X_{n}]$. For schemes, we 'remove the $\text{m}