Exercise 2.4

Question

Let $M_i$ $(i \in I)$ be any family of $A$-modules, and let $M$ be their direct sum. Prove that $M$ is flat $\Leftrightarrow$ each $M_i$ is flat.

Amit Awasthi · Accepted Answer

\begin{lemma}\label{up:dirsum}
  Let $M = \bigoplus_{i \in I} M_i$ be the direct sum and $j_i\colon M_i 	o M$ be the canonical inclusion maps. $M$ satisfies the following universal property:
  \par If $\{f_i \colon M_i 	o P\}_{i \in I}$ is a family of $A$-module homomorphisms, then there exists a unique $A$-module homomorphism $f\colon M 	o P$ such that $f \circ j_i = f_i$ for all $i$.
\end{lemma}
\begin{proof}[Proof of Lemma $ef{up:dirsum}$]
  Define $f \colon M 	o P$ to be the map $(m_i)_{i \in I} \mapsto \sum_{i \in I} f_i(m_i)$; this is an $A$-module homomorphism since each $f_i$ is. $f \circ j_i = f_i$ by definition, and $f$ is unique since the $j_i(m_i)$ for $m_i \in M_i$ generate $M$, hence the $f_i$ uniquely determine $f$ by its behavior on generators.
\end{proof}
\begin{lemma}\label{l2.4}
  Tensor products commute with direct sums, i.e., for any $A$-module $N$, $N \otimes_A \left(\bigoplus_{i \in I} M_iight) \cong \bigoplus_{i \in I}(N \otimes_A M_i)$.
\end{lemma}
\begin{proof}[Proof of Lemma $ef{l2.4}$]
  Denoting $M = \bigoplus_{i \in I}M_i$, define the map
  \begin{equation*}
    g \colon N 	imes M 	o \bigoplus_{i \in I}(N \otimes M_i) \qquad (n,(m_i)_{i\in I}) \mapsto (n \otimes m_i)_{i \in I}.
  \end{equation*}
  This is $A$-bilinear, for
  \begin{align*}
    g(ax+y,(m_i)_{i\in I}) &= ((ax+y) \otimes m_i)_{i \in I} = a(x \otimes m_i)_{i \in I} + (y \otimes m_i)_{i \in I}\
    &= ag(x,(m_i)_{i\in I}) + g(y,(m_i)_{i\in I})\
    g(n,a(x_i)_{i\in I}+(y_i)_{i\in I}) &= (n \otimes ax_i+y_i)_{i\in I} = a(n \otimes x_i)_{i \in I} + (n \otimes y_i)_{i \in I}\
    &= ag(n,(x_i)_{i\in I}) = g(n,(y_i)_{i\in I})
  \end{align*}
  It then suffices by Prop.~2.12 to show that $\bigoplus_{i \in I}(N \otimes M_i)$ satisfies the universal property for $N \otimes M$, i.e., that for any $A$-bilinear map $f\colon N 	imes M 	o P$, there exists a unique homomorphism $f'\colon \bigoplus_{i \in I}(N \otimes M_i) 	o P$ making the diagram below commute:
  \begin{equation*}
  \begin{array}{ccc}
    N 	imes M & \xrightarrow{g} & \bigoplus_{i \in I}(N \otimes M_i) \
    \downarrow{f} & & \downarrow[dashed]{f'} \
    & P & 
  \end{array}
  \end{equation*}
  Now we know that for any $m_i \in M_i$, commutativity requires that for each direct summand $N \otimes M_i$, $n \otimes m_i \mapsto f(n,j_i(m_i))$, where $j_i \colon M_i 	o M$ is the canonical inclusion. But by Lemma ef{up:dirsum}, this induces a unique $A$-module homomorphism $f' \colon \bigoplus_{i \in I}(N \otimes M_i) 	o P$, and so we are done.
\end{proof}
\begin{proof}[Main Proof]
  Let $N' 	o N$ be injective. By Prop.~$2.19$, it suffices to show $f \otimes 1_M\colon N' \otimes M 	o N \otimes M$ is injective if and only if $f\colon N' \otimes M_i 	o N \otimes M_i$ is injective for each $i$. But by Lemma ef{l2.4}, we see that we have the commutative diagram
  \begin{equation*}
  \begin{array}{ccc}
    N' \otimes M & \xrightarrow{f \otimes 1_M} & N \otimes M \
    \downarrow{\cong} & & \downarrow{\cong} \
    \bigoplus_{i \in I}(N' \otimes M_i) & \xrightarrow{g} & \bigoplus_{i \in I}(N \otimes M_i)
  \end{array}
  \end{equation*}
  where $g$ is the map induced by Lemma ef{up:dirsum} by the maps
  \begin{equation*}
  N' \otimes M_i \xrightarrow{1_{N'} \otimes j_i} N' \otimes M \xrightarrow{f \otimes 1_M} N \otimes M \xrightarrow{\sim} \bigoplus_{i \in I}(N \otimes M_i)
  \end{equation*}
  Note that by uniqueness in Lemma ef{up:dirsum}, $g = \bigoplus_{i \in I} (f \otimes 1_{M_i})$ as well, hence by the diagram above, $f \otimes 1_M$ is injective if and only if $g$ is injective, which holds if and only if $f \otimes 1_{M_i}$ is injective for all $i \in I$.
\end{proof}

Exercise 2.4

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Exercise 2.4

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