Suppose we have a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ and a projective resolution $P$ for $A$ and a projective resolution for $C$, Q. Is there a way we can construct a projective resolution for $B$ of the form $P \bigoplus Q$ in a way that we can have a short exact sequence of complexes $ 0 \rightarrow P \rightarrow P \bigoplus Q \rightarrow Q \rightarrow 0$ ? By this i mean can we construct specific maps for the resolution in $B$ in order to be a resolution and a chain transformation? Im just looking for an yes or no answer, i have tried to do this on my own but i havent quite managed to construct the specific maps myself, but would like to try some more if the answer is true. Thanks in advance.
2026-03-27 23:38:53.1774654733
Constructing a resolution from other projective resolutions and a short exact sequence
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Yes, this is called the Horseshoe Lemma.