I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows:
Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to M'\to M\to M''\to 0$$ a complex of finitely generated flat $A$-modules. Show that if there exists an ideal $I\subset \mathfrak m$ such that $C^\bullet\otimes_AA/I$ is exact, then $C^\bullet$ is exact.
The notation $C^\bullet$ is new for me so does it mean only that $C^\bullet$ is a name of an exact sequence, and $C^\bullet\otimes_AA/I$ is the sequence $0\to M'\otimes_AA/I\to M\otimes_AA/I\to M''\otimes_AA/I\to 0$? I am unable to solve the problem.
I don't know if you solved the problem, but this follows easily from the following remark: if $0\to M'\otimes_AA/I\to M\otimes_AA/I$ is exact, then $0\to M'\to M$ is also exact, and moreover the image of $M'$ is a direct summand of $M$ (the argument is the same as the one given in this answer).