Derived categories and quasi-isomorphisms

A chain isomorphism has a strict inverse in every degree. A homotopy equivalence has an inverse whose composites are homotopic to the identities. A quasi-isomorphism only requires isomorphisms on cohomology. The first implies the second and the second implies the third, but neither converse holds in general. In particular, an acyclic complex can represent a nonzero object of the homotopy category when its exact sequences do not split, yet it becomes isomorphic to zero after quasi-isomorphisms are inverted.

Let $K(R)$ denote the homotopy category of cochain complexes of left modules, so its morphisms are cochain maps modulo homotopy. The derived category $D(R)$ is obtained from it by inverting quasi-isomorphisms. One can use a set-theoretic universe convention to manage the size of module categories. The construction does not declare every cohomology isomorphism between graded objects to be a chain map: a quasi-isomorphism is first an actual map of complexes, with its compatibility equation already satisfied.

A derived morphism can be represented by a roof with a quasi-isomorphism pointing toward its source and an ordinary homotopy-class map pointing toward its target. The inverse of the left arrow exists in the localized category, so the roof describes their composite there. Different roofs can describe the same morphism after suitable refinements. Roofs are a presentation of morphisms, not unique coordinate expressions; composing them generally requires a common replacement rather than naive concatenation of unrelated chain maps.

The universal property says that a functor from the homotopy category which sends every quasi-isomorphism to an isomorphism factors through the localization, uniquely up to natural isomorphism. This tells us what the construction accomplishes independently of a chosen roof calculus. A one-sided exact functor need not already invert quasi-isomorphisms when applied termwise. Its derived version first chooses replacements on which that failure disappears, then descends to the localized setting.

For modules over a commutative ring, resolve one tensor factor projectively in nonpositive cochain degrees and tensor with the other. The resulting derived tensor product has cohomology in degree $-i$ equal to the homological group $\operatorname{Tor}_i$. Its degree-zero cohomology is the ordinary tensor product, but retaining only that group would discard the very data the derived operation adds. Derived Hom similarly packages Ext in nonnegative cohomological degrees for modules in degree zero.

Bounded-above complexes of projectives are sufficient replacements for ordinary module resolutions and for bounded-above derived constructions. Arbitrary unbounded complexes require appropriate K-projective, K-flat, or K-injective replacements; being projective or injective in each degree alone is insufficient. Also, equal graded cohomology groups do not generally force two complexes to be isomorphic in the derived category: extension information between degrees can remain. Localization preserves more structure than the list of cohomology modules, even while it forgets acyclic summands.