... --> 0 --> 0 --> R --> 0 --> ...
with R located in degree n-1, and let D^n be the chain complex
... --> 0 --> R === R --> 0 --> ...
with R located in degrees n and n-1. There is a chain map S^{n-1} -->
D^n that is given by the identity map R = R in degree n-1. We use the
general theory from last time to show the following result.
Theorem. The category Ch(R) has a cofibrantly generated model structure, where the weak equivalences are the chain maps f : X --> Y such that the induced map f_* : H_n(X) --> H_n(Y) is an isomorphism, for all integers n, and where
I = { S^{n-1} --> D^n | n integer }
J = { 0 ----> D^n | n integer }
are sets of generating cofibrations and generating trivial
cofibrations, respectively.Today we proved conditions (i), (iii), and (iv) (b) from the theorem on the note of Lecture 7. We began the proof of condition (ii). We showed that the fibrations are surjections. We also showed that if A is a cofibrant object, then the R-modules A_n are projective, for all integers n. We showed that, conversely, if A is a complex such that the R-modules A_n are projective, for all integers n, and if A_n is zero, for n << 0, then A is a cofibrant object.
We plan to complete the proof next time and to give some examples of calculations of Ext-groups.