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\begin{document}

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\section*{affine scheme}

[[!redirects affine schemes]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relative_affine_schemes}{Relative affine schemes}\dotfill \pageref*{relative_affine_schemes} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{IsbellDuality}{Isbell duality}\dotfill \pageref*{IsbellDuality} \linebreak
\noindent\hyperlink{affine_serres_theorem}{Affine Serre's theorem}\dotfill \pageref*{affine_serres_theorem} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

An \textbf{affine scheme} is a [[scheme]] that as a [[sheaf]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative [[ring]]s (or equivalently as a sheaf on the subcategory of finitely presented rings) is [[representable functor|representable]]. In a [[ringed space]] picture an affine scheme is a [[locally ringed space]] which is locally isomorphic to the [[prime spectrum]] of a commutative ring. Affine schemes form a [[full subcategory]] $Aff\hookrightarrow Scheme$ of the category of schemes.

The correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a [[functor]] $Scheme\to Aff$. The \textbf{fundamental theorem on morphisms of schemes} (see \hyperlink{IsbellDuality}{below}) says that there is a bijection

\begin{displaymath}
CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).
\end{displaymath}
In other words, for fixed $Y$, and for varying $R$ there is a restricted functor

\begin{displaymath}
Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,
\end{displaymath}
and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff$ is [[fully faithful functor|fully faithful]]. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$. See at \emph{[[functorial geometry]]}.

There is an analogue of this theorem for relative [[noncommutative scheme]]s in the sense of Rosenberg.

\begin{remark}
\label{}\hypertarget{}{}
There is no similar equation the other way round, that is ``$Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)$''. As a mnemonic, note that with ordinary Galois connections between power sets, one is always [[Galois connection\#properties|homming into (not out of)]] the functorial construction. More geometrically, consider the example $Y = \mathbb{P}^n$ and $R = \mathbb{Z}$. Then the left hand side consists of all the $\mathbb{Z}$-valued points of $\mathbb{P}^n$ (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism $\mathbb{Z} \to \mathbb{Z}$, since $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.

\end{remark}
\hypertarget{relative_affine_schemes}{}\subsubsection*{{Relative affine schemes}}\label{relative_affine_schemes}

A \textbf{relative affine scheme} over a scheme $Y$ is a [[relative scheme]] $f:X\to Y$ isomorphic to the spectrum of a (commutative unital) algebra $A$ in the category of quasicoherent $\mathcal{O}_Y$-modules; such a ``relative'' spectrum has been introduced by Grothendieck. It is characterized by the property that for every open $V\subset Y$ the inverse image $f^{-1}V\subset X$ is an open affine subscheme of $X$ isomorphic to $Spec(A(V))$ and such open affines glue in such a way that $f^{-1}V\hookrightarrow f^{-1}W$ corresponds to the restriction morphism $A(W)\to A(V)$ of algebras.

Relative affine scheme is a concrete way to represent an [[affine morphism]] of schemes.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{IsbellDuality}{}\subsubsection*{{Isbell duality}}\label{IsbellDuality}

\begin{prop}
\label{AffineSchemesFullSubcategoryOfOppositeOfRings}\hypertarget{AffineSchemesFullSubcategoryOfOppositeOfRings}{}
\textbf{([[affine schemes]] form [[full subcategory]] of [[opposite category|opposite]] of [[rings]])}

The [[functor]]

\begin{displaymath}
\mathcal{O}
  \;\colon\;
  Schemes_{Aff}
    \longrightarrow
  Ring^{op}
\end{displaymath}
from affine schemes to their global [[rings of functions]] is a [[fully faithful functor]].

\end{prop}
(e.g. \hyperlink{Hartschorne77}{Hartschorne 77, chapter II, prop. 2.3})

\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[Isbell duality]] between [[geometry]] and [[algebra]])}

Prop. \ref{AffineSchemesFullSubcategoryOfOppositeOfRings} is the analog in [[algebraic geometry]] of similar statements of [[Isbell duality]] between [[geometry]] and [[algebra]], such as [[Gelfand duality]] or [[Milnor's exercise]].

\end{remark}
[[!include Isbell duality - table]]

\hypertarget{affine_serres_theorem}{}\subsubsection*{{Affine Serre's theorem}}\label{affine_serres_theorem}

[[affine Serre's theorem|Affine Serre's theorem]]

Given a commutative unital ring $R$ there is an equivalence of categories ${}_R Mod\to Qcoh(Spec R)$ between the category of $R$-modules and the category of quasicoherent sheaves of $\mathcal{O}_{Spec R}$-modules given on objects by $M\mapsto \tilde{M}$ where $\tilde{M}$ is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization $\tilde{M}(D_f) = R[f^{-1}]\otimes_R M$ where $D_f$ is the principal Zariski open set underlying $Spec R[f^{-1}]\subset Spec R$, and the restrictions are given by the canonical maps among the localizations. The action of $\mathcal{O}_{Spec R}$ is defined using a similar description of $\mathcal{O}_{Spec R} = \tilde{R}$. Its right adjoint (quasi)inverse functor is given by the global sections functor $\mathcal{F}\mapsto\mathcal{F}(Spec R)$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[spectral topological space]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Robin Hartshorne, \emph{Algebraic geometry}, Springer 1977


\item Demazure, Gabriel, \emph{Algebraic groups}



\end{itemize}
category: algebraic geometry



\end{document}