Flat modules and direct limit.

Question

Let $\{M_i, u_{ij}\}$ be a direct system of $R $-modules over the directed set $I $, where $R $ is a commutative ring with unit, and let $F $ a $M_{i}$-flat module for each $i\in I $. Make $M $ the direct limit of the $M_i $. Is $F $ $M $-flat?

Answer 1

With your definition, it is clear that if $F$ is not $M$-flat, you can find a finitely generated submodule $M'\subset M$ with $F\otimes M'\to F\otimes M$ not injective. Since these generators come from a $M_i$ for some $i$ (being a finite set), you may assume that $M'\subset M_i$. So, you will get a non-zero element $\sum f_i\otimes m_i\in F\otimes M'$ which goes to zero in $F\otimes M$ and since these can be interpreted as finitely many relations in $F\otimes M$, they too happen in some $M_i$. Thus, you will get $M'\subset M_i$ and $F\otimes M'\to F\otimes M_i$ not injective.

Answer 2

Let $M'$ be a finitely generated submodule of $M$ and $v:M'\rightarrow M$ the canonical inclusion. Then there are $y'_1,y'_2,...,y'_n\in M$ such that $M'=<y'_1,...,y'_n>$ and, since $M$ is the direct limit of $M_i$, if $u_i:M_i\rightarrow M$ is the canonical map, there are $l_1,...,l_n\in I$ such that $y'_i=u_{l_i}(y_{l_i})$ with $y_{l_i}\in M_{l_i}$. Since $I$ is a directed set, there is $k\in I$ such that $l_i\leq k$ and $y'_i=u_k(y_i)$ for all $i\in\{1,...,n\}$.

Take $z\in \ker(1_F\otimes v)$. There are $x_1,...,x_n\in F$ such that $\displaystyle z=\sum_{i=1}^{n} x_i\otimes u_k(y_i)$. Then $$\left(1_F\otimes u_k\right)\left(\sum x_i\otimes y_i\right)=0$$ in $F\otimes M$.

It is clear that $\left(F\otimes M_i,1_F\otimes u_{ij}\right)$ form a direct system of modules with direct limit has been $F\otimes M$ and $1_F\otimes u_i$ the canonical projections. From the last equality we have that there is $l\in I$ such that $k\leq l$ and $$\left(1_F\otimes u_{kl}\right)\left(\sum x_i\otimes y_i\right)=0$$ in $F\otimes M_l$.

Now, define, for each $r\in I$ such that $k\leq r$, $M'_r=<u_{rk}(y_1),...,u_{rk}(y_n)>$. Let $u'_{sr}$ the restriction of the map $u_{sr}$ to $M'_s$. It is clear that $(M'_r,u'_{sr})$ form a direct system of modules and so $(F\otimes M'_r,1_F\otimes u'_{rs})$ also. Let $\gamma$ be the canonical map $\displaystyle\gamma:\lim_{\longrightarrow}M'_r\rightarrow M$ and consider the canonical composition $$(1_F\otimes\gamma)\circ(1_F\otimes u'_l):F\otimes M'_l\longrightarrow F\otimes\left(\lim_{\longrightarrow}M'_r\right)\longrightarrow F\otimes M$$ Note that $$\left(1_F\otimes\gamma\right)\circ\left(1_F\otimes u'_l\right)\left(1\otimes u_{kl}\left(\sum x_i\otimes y_i\right)\right)=z$$ and, since $$\left(1\otimes u_{kl}\right)\left(\sum x_i\otimes y_i\right)=0$$ in $F\otimes M'_l$, we have $$z=\left(1_F\otimes\gamma\right)\circ\left(1_F\otimes u'_l\right)\left(1\otimes u_{kl}\left(\sum x_i\otimes y_i\right)\right)=0.$$ Therefore the map $1_F\otimes v$ is injective.