Consider a $\mathbb{Z}$-graded chain complex $A^{\bullet}$, I know that a bounded below complex is one such that $A^i = 0$ for $i$ sufficiently small, and a bounded above complex is one such that $A^i = 0$ for sufficiently large $i$, every bounded above (or below) complex has a Cartan-Eilenberg resolution and every unbounded complex has a resolution too (according to Spaltenstein and other authors).
I wanted to ask:
- Does this apply to unbounded complexes that are not $\mathbb{Z}$-graded? Or finite complexes? Arbitrary chain complexes? An $\mathbb{N}$-graded complex for example? Do they too have resolutions?
I just wanted to know as I don't think I know the reason as to why boundedness is such a key thing, I know that the classical construction of resolutions for complexes works only for bounded complexes but I don't know why or where it fails for unbounded complexes.