Exercise 1.1.3 of An Introduction to Homological Algebra, Charles A. Weibel states
'… show that … every chain complex of vector spaces is isomorphic to a complex of this fom'
An isomorphism of chain complexes has not been defined yet. My hunch was that it would be a morphism that induced isomorphisms on each homology module, but then 2 paragraphs later his notion is defined as a quasi-isomorphism.
So what am I supposed to show for exercise 1.1.3? I.e what is isomorphism vs quasi isomorphism of chain complexes?
I assume what I have provided is enough context for people more familiar with homological algebra but I can state th whole question if needed.
Best Answer
Isomorphism always means a morphism that has an inverse morphism.
There are a number of ways to show that a morphism of chain complexes is an isomorphism if and only if its map in each degree is an isomorphism.