Exercise 7, § 1

Commutative Algebra, I, 40

Let $U,\ V,\ W$ be three commutative groups and $f : U \to V,\ g : V \to W$ homomorphisms. (a) Consider the diagram

\begin{array}{ccccccccc} 0 & \longrightarrow & U & \xrightarrow{\alpha} & U \times V & \xrightarrow{\beta} & V & \longrightarrow & 0 \\ & & f \downarrow & & h \downarrow & & -g \downarrow & \\ 0 & \longrightarrow & V & \xrightarrow{\gamma} & W \times V & \xrightarrow{\delta} & W & \longrightarrow & 0 \end{array}

where $\alpha(u) = (u, f(u)),\ \beta(u, v) = v - f(u),\ \gamma(v) = (g(v), v),\ \delta(w, v) = w - g(v),\ h(u, v) = (g(f(u)), v)$ . Show that this diagram is commutative and that its rows are exact. (b) Deduce from (a) and Proposition 2 of no. 4 an exact sequence

0 \to \operatorname{Ker}(f) \to \operatorname{Ker}(g_{of}) \to \operatorname{Ker}(g) \to \operatorname{Coker}(f) \to \operatorname{Coker}(g_{of}) \to \operatorname{Coker}(g) \to 0.

Give a direct definition of this exact sequence.

Solution

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