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% \newcommand{\kring}{\mathbf{k}} % commutative ring k (e.g. field)
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
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\newcommand{\onto}{\twoheadrightarrow} % surjection
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\title{MAT 250B HW06}
\author{[add your name here]}
\date{Due Friday, 2/16/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\noindent \textbf{Reminder} 
{ \scriptsize Your homework submission \textbf{must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting.}

\paragraph{Midterm Exam}
The midterm exam covers material from 
\begin{itemize}
    \item Lectures 01--11
    \item HW01--HW05
    \item \S 6.1 -- \S 6.6 in the text
\end{itemize}
This will be a 50-minute pen-and-paper exam. As such, I will test you more on your understanding of definitions, properties, and common proof techniques; the proofs will not require a `spark of genius', but rather familiarity with the types of proofs we have already seen in class and on homeworks. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%-----------
\exercise
% flat modules
Let $k$ be a commutative ring, and let $P,Q$ be flat $k$-modules. Prove that $P \otimes_k Q$ is also a flat $k$-module.



%-----------
\exercise
% localization
\textbf{(Localization is exact)}
Let $R$ be a commutative ring, and let $D\inv R$ be its localization at a multiplicatively closed subset $D \subset R$ (containing $1$). 
Let $M \in R-\Mod$. 

Let $D\inv M$ denote the $D\inv R$-module defined by 
\[
    D\inv M = (D\times M) / \sim 
    \qquad
    (d,m) \sim (e,m)
    \text{ iff }
    x(dn - em) = 0 
    \text{ for some }
    x \in D.
\]

\bea

\item Verify that $D\inv M$ is indeed a $D\inv R$-module. Then verify that $D\inv M \cong D\inv R \otimes_R M$. 

\item Prove that $D\inv R$ is flat. Conclude that localization of modules is an exact functor. \textit{That is, clarify the meaning of the mantra ``localization is exact''.}


\ee

%-----------
\exercise
\textit{Here is a fact we've been using in class quite often.}

Let $F,G: \RMod \to \Ab$ be two covariant additive functors. Suppose that $F$ and $G$ are naturally isomorphic. 
\bea
    \item Prove that $F$ is left exact iff $G$ is left exact. \footnote{I don't think I need to tell you this, but just in case: you only need to prove one direction, since $F$ and $G$ are arbitrary.}
    \item Prove that $F$ is right exact iff $G$ is right exact.
    \item Conclude that $F$ is exact iff $G$ is exact. 
\ee

%-----------
\exercise
% naturality of adjoint functors
Recall that we have two adjoint isomorphisms.
\begin{itemize}
\item For \textbf{right modules} $A_R \in \Mod_R, C_S \in \Mod_S$:
    Let $_R B_S \in (R,S)\bimod$. Then there is an isomorphism of abelian groups
    \[
        \tau_{A,B,C}: \Hom_S(A \otimes_R B,C) \to \Hom_R(A, \Hom_S(B,C)).
    \]
\item For \textbf{left modules} $_R A \in _R\Mod, _S C \in _S\Mod$:
    Let $_S B_R \in (S,R)\bimod$. Then there is an isomorphism of abelian groups
    \[
        \tau'_{A,B,C}: \Hom_S(B \otimes_R A,C) \to \Hom_R(A, \Hom_S(B,C)).
    \]
\end{itemize}

In this exercise, you will check that the maps we obtain from $\tau_{A,B,C}$ by fixing two out of the three in $\{A,B,C\}$ are in fact natural isomorphisms between functors.
\textit{The proofs for the adjoint isomorphism of left modules is similar.}

\bea

\item Prove that (the collection of maps, one for each pair of choices $A,C$) 
    \[
        \tau_{A,-,C}: \Hom_S(A \otimes_R -,C) \to \Hom_R(A, \Hom_S(-,C))
    \]
    is a natural isomorphism. 
    
    \begin{remark}
    {This proves that
    \[
        F = A \otimes_R -
        \qquad \text{ and } \qquad
        G = \Hom_S(-,C) = \Hom_{\Mod_S}(-,C)
    \]
    are an adjoint pair of functors $\RMod \to \Ab$.
    
    The analogous proof for left modules shows that 
    \[
        F = - \otimes_R A
        \qquad \text{ and } \qquad
        G = \Hom_S(-,C) = \Hom_{_S\Mod}(-,C)
    \]
    are an adjoint pair of functors $\ModR \to \Ab$. 
    }
    \end{remark}
\item Prove that (the collection of maps, one for each pair of choices $B,C$) 
    \[
        \tau_{-,B,C}: \Hom_S(- \otimes_R B,C) \to \Hom_R(-, \Hom_S(B,C))
    \]
    is a natural isomorphism. 
\item Prove that (the collection of maps, one for each pair of choices $A,B$) 
    \[
        \tau_{A,B,-}: \Hom_S(A \otimes_R B, -) \to \Hom_R(A, \Hom_S(B,-))
    \]
    is a natural isomorphism. 
\ee


%-----------
% \exercise
% adjunction, commutative -- leave for later; avoid confusion


%-----------
% exercise
% adjunct functors and exactness -- leave for limits / colimits









\end{document}
