map of modules induces chain map

Last updated January 27, 2022

Let $f:M\to N$ be a map of $A$-modules.

1. Given a resolution for $N$ (i.e. an exact sequence $Q_\ast\to N\to 0$) and a projective complex over $M$ (see chain complex) (i.e. a chain complex $P_\ast\to M\to 0$ where the $P_\ast$ are projective modules), then there exists a collection of maps $f_i:P_i\to Q_i$ which determine a chain map. Moreover, this map is unique up to chain homotopy.