exact sequence

DEFINITION 11.1   Let $A$ be a ring. Then a sequence

$$M_1 \overset{f_1}{\to} M_2 \overset{f_2}{\to} M_3$$

is exact if condition

$$\operatorname{Image}(f_1)=\operatorname{Ker}(f_2)$$

holds.

We also use the notion of exact sequences for sheaves on schemes. It is also defined likewise. This could be summarized in the theory of abelian categories. We postpone the precise argument to a near future.