In Lang's algebra book page 164, he provided the following theorem:
Theorem Given $(A_n), (B_n), (C_n)$ be inverse systems of abelian groups, assume $(A_n)$ satisfies Mittag-Leffler condition. If the sequence $0 \xrightarrow{} (A_n) \xrightarrow{} (B_n) \xrightarrow{g} (C_n) \xrightarrow{} 0$ is exact, then so is the sequence $0 \xrightarrow{} \varprojlim A_n \xrightarrow{} \varprojlim B_n \xrightarrow{g} \varprojlim C_n \xrightarrow{} 0$
It's obvious that the only point is to prove the surjectivity on the right. Let $(c_n)$ be an element of the inverse limit, Lang states that each inverse image $g^{-1}(c_n)$ is a coset of $A_n$, so in bijection with $A_n$. Also, the Mittag-Leffler condition on $(A_n)$ implies the Mittag-Leffler condition on $(g^{-1}(c_n))$. I actually don't understand what he means by "$g^{-1}(c_n)$ is a coset of $A_n$, so in bijection with $A_n$"? Because there are possibilities that $g^{-1}(c_n)$ would lie outside the image of $A_n$ in $B_n$.
Does anyone know what he really means by a coset of $(A_n)$? Any helps would be appreciated! Thanks
Best Answer
It means that there is some $b_n\in B_n$ such that $g^{-1}(c_n)=b_n+A_n$. This follows from the exactness of the sequence $0\to A_n\to B_n\to C_n\to 0$.