Unbounded projective cofibrations

Proposition 20.2.1 (Unbounded projective cofibrations, [Lurie (2017), Propositions 7.1.2.8 and 7.1.2.11]). Let \(R\) be a commutative ring and let \(W\) be the class of quasi-isomorphisms in \(\Ch (R)\). There is a class \(I_{\proj }\) of projective cofibrations with the following properties:

(1)

The triple \((\Ch (R),W,I_{\proj })\) is a homotopy cocomplete \(\infty \)-category with weak equivalences and cofibrations.

(2)

The unit \(R[0]\) is cofibrant, and a finite tensor product of cofibrant complexes is again cofibrant.

(3)

If \(P\) is cofibrant, then \(P\otimes _R-\) is homotopy cocontinuous and preserves quasi-isomorphisms.