Let $$0\rightarrow M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$$ be an exact sequence for R-modules $M_i$ with $R$-module homomorphisms $f_j:M_j\rightarrow M_{j+1}$, $j=0,1,2$.

There are four non-zero modules in this sequence. Given this, is there a natural way of constructing an exact sequence with only three non-zero modules? Perhaps using the universal property of direct sum, or for example can we always find a map $g:M_1\rightarrow M_2$ such that

$$0\rightarrow M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow 0$$

is an exact sequence (with $f_0:M_0\rightarrow M_1$ as before)?


1 Answer 1


Both of the following work

$$0\rightarrow M_0\rightarrow M_1\rightarrow {\rm Im}f_1\rightarrow 0$$

$$0\rightarrow {\rm Im}f_1\rightarrow M_2\rightarrow M_3\rightarrow 0$$ where inclusion is the map between ${\rm Im}f_1$ and $M_2$


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