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\begin{matrix} A_1 & \longrightarrow & A_2 & \longrightarrow & A_3 & \longrightarrow & A_4 & \longrightarrow & A_5 \\\ \downarrow h_1 & \ &\ \downarrow h_2 & \ & \ & \ &\ \ \downarrow h_4 & \ &\ \ \downarrow h_5 \\B_1 & \longrightarrow & B_2 & \longrightarrow & B_3 & \longrightarrow & B_4 & \longrightarrow & B_5 \end{matrix}

I am looking for an example of a commutative diagram with exact rows and vertical maps $h_1, h_2, h_4, h_5$ isomorphisms for which there does not exist a map $h_3 : A_3\rightarrow B_3$ making the diagram commute.(Exercise 2.34 in An Introduction to Homological Algebra by Joseph J. Rotman)

I don't know how to start to construct such a diagram.

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1 Answer 1

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How about this:

$$\require{AMScd} \begin{CD} 0 @>>> {\mathbb{Z}_2} @>>> {\mathbb{Z}_4} @>>> {\mathbb{Z}_2} @>>> 0 \\ @VVV @VVV @. @VVV @VVV \\ 0 @>>> {\mathbb{Z}_2} @>>> {\mathbb{Z}_2\oplus\mathbb{Z}_2} @>>> {\mathbb{Z}_2} @>>> 0 \\ \end{CD} $$

I haven't specified what any of the maps are, since you can probably guess. Let me know if you'd like me to be more explicit.

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