Isomorphism of Chain Complex, definition/terminology

Question

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.

Just an isomorphism in the category of chain complexes; so a chain map that's an isomorphism in each dimension. — Angina Seng, Nov 22 '17 at 19:40
Ok so they are using isomorphism and quasi isomorphism the same? — Prince M, Nov 24 '17 at 01:41

score 3 · Answer 1 · answered Nov 22 '17 at 19:45

3

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.

answered Nov 22 '17 at 19:45

Can you give an example then of a quasi isomorphism that is not an isomorphism – Prince M Apr 10 '21 at 07:46
@PrinceM Think about the complex $\dots \to 0 \to X \overset{id}{\to} X \to 0\to \dots $. Its homology is zero at every degree, so that the zero map is a quasi-automorphism. But, of course, this is not an isomorphism. – Antonio Lorenzin Dec 14 '22 at 11:21

Isomorphism of Chain Complex, definition/terminology

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