In Martin Krieger's book "Doing Mathematics: Convention Subject, Calculation, Analogy" (2003) I find the following statement (apparently, a quote from somone else) : "Homological algebra starts from the regrettable fact that not all modules are projective". And then he adds "Those regrettable facts, the breaks or failures of exactness, are measured by the nonzero homology group".
Could anyone shed some light on those sentences? Any insights into the origins and history of homological algebra?
Thanks in advance.