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\begin{document}

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\section*{extension of scalars}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{general_abstract}{General abstract}\dotfill \pageref*{general_abstract} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{geometric_interpretation}{Geometric interpretation}\dotfill \pageref*{geometric_interpretation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{extension of scalars} of a [[module]] along a [[homomorphism]] of [[rings]] is the [[Isbell duality|algebraic dual]] of what geometrically is the [[pullback]] of [[bundles]] along a map of their base spaces (with respect to the discussion at \emph{\href{http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction}{modules - as generalized vector bundles}}).

Explicitly, extension of scalars along a [[ring]] [[homomorphism]] $f : R \to S$ is the operation on $R$-[[modules]] given by forming the [[tensor product of modules]] with $S$ regarded as an $R$-module via $f$.

There are similar functors for [[bimodules]] and in some other categories.

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

Let $R$ and $S$ be [[commutative rings]] and let $f \colon R\to S$ be a [[homomorphism]] of [[rings]].

We discuss \emph{extension of scalars} along $f$ first [[category theory|general abstractly]] and then explicitly in components.

\hypertarget{general_abstract}{}\subsubsection*{{General abstract}}\label{general_abstract}

Write $R$[[Mod]] and $S$[[Mod]] for the [[categories of modules]] over $R$ and $S$, respectively.

\begin{defn}
\label{RestrictionOfScalars}\hypertarget{RestrictionOfScalars}{}
Given a [[ring]] [[homomorphism]] $f : R \to S$ the \textbf{[[restriction of scalars]]} functor

\begin{displaymath}
f^* : S Mod \to R Mod
\end{displaymath}
is the [[functor]] that takes an $S$-[[module]] $N$ to the $R$-module $f^*N$ whose underlying [[abelian group]] is that of $N$ and whose $R$-[[action]] is given by

\begin{displaymath}
r \cdot n \coloneqq f(r)\cdot n \;\;\;\; for \; r \in R, n \in N
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{AdjointPair}\hypertarget{AdjointPair}{}
The [[restriction of scalars]] functor, def. \ref{RestrictionOfScalars}, is the [[right adjoint]] in a pair of [[adjoint functors]]

\begin{displaymath}
( f_! \dashv f^* )
  : 
  S Mod 
   \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}}
  R Mod
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{ExtensionByAdjoint}\hypertarget{ExtensionByAdjoint}{}
The [[left adjoint]] $f_! \colon R Mod \to S Mod$ in prop. \ref{AdjointPair} is called \textbf{extension of scalars} along $f$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A further [[right adjoint]] $f_*$ would be called \emph{[[coextension of scalars]]} along $f$.

\end{remark}
See also [[induced representation]] $\dashv$ [[restricted representation]].

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

\begin{prop}
\label{}\hypertarget{}{}
Given a [[ring]] [[homomorphism]] $f : R \to S$, the \emph{extension of scalars} [[functor]] $f_!$ of def. \ref{ExtensionByAdjoint} is the functor

\begin{displaymath}
f_! \coloneqq  S \otimes_R (-) \,:\, R Mod \to S Mod
\end{displaymath}
given by [[tensor product of modules]] with $S$ regarded as an $S$-$R$-[[bimodule]]: the left [[action]] being the canonical action of $S$ on itself, the right being the [[restriction of scalars]]-action along $f$.

Explicitly, for $N \in R Mod$

\begin{itemize}%
\item the elements of $f_! N$ are [[equivalence classes]] of pairs $(s,n) \in S \times N$ under the [[equivalence relation]] $(s \cdot f(r), n) = (s, r\cdot n)$ for all $s \in S$;


\item the left $S$-[[action]] is given by $s' \cdot(s,n) = (s' \cdot s,n)$.



\end{itemize}
\end{prop}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{geometric_interpretation}{}\subsubsection*{{Geometric interpretation}}\label{geometric_interpretation}

Under [[Isbell duality]] extension of scalars turns into a statement about [[geometry]].

By definition the category

\begin{displaymath}
Ring^{op} \underoverset{Spec}{\colon \simeq}{\to}
  Aff
\end{displaymath}
of (absolute) [[affine schemes]] is the [[opposite category]] of [[Ring]].

Hence for $f : R \to S$ a [[ring]] [[homomorphism]], we have equivalently a morphism

\begin{displaymath}
Spec(f) : Spec(S) \to Spec(R)
\end{displaymath}
of affine schemes.

An $R$-[[module]] $N$ corresponds to the collection of [[sections]] of a ``generalized [[vector bundle]]'' over $Spec(R)$: something that has a [[quasicoherent sheaf]] of sections.

The [[pullback]] of this ``bundle'' along $Spec(f)$ has sections forming the module $f_! N$.

Generally, for any [[fibered category]] like [[Mod]]$\to Aff$ we may regard the [[inverse image functor]] as the extension of scalars.

For that reason if there is some other fibered category $\mathcal{F}$ over the opposite of some algebraic category $\mathcal{A}$ whose objects are considered ``objects of scalars'' one is inclined to call the inverse image functor, the extension of scalars.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[complexification]] is extension of scalars along the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ of the [[real numbers]] into the [[complex numbers]].


\item [[localization of a module]]



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

\begin{itemize}%
\item \textbf{extension of scalars} $\dashv$ [[restriction of scalars]] $\dashv$ [[coextension of scalars]]

\end{itemize}


\end{document}