Algebraic Topology: Lecture 10

We completed the proof that the category of bounded below chain complexes of left modules over a ring R has a model structure where the weak equivalences are the quasi-isomorphisms, where the fibrations are the epimorphisms, and where the cofibrations are the maps that have the LLP with respect to trivial fibrations. We showed in Lecture 7 that if M and N are two R-modules, then the set of maps in the homotopy category associated with the model category above from M considered as a chain complex M located in degree zero to N considered as a chain complex N[-q] located in degree q is canonically isomorphic to the q'th Ext-group of the R-modules M and N: 
      Hom_{Ho C^+(R-modules)}(M,N[-q]) = Ext_A^q(M,N).
We gave a number of examples of calculations of these groups. We recalled that a ring R is defined to be (left) regular if every (left) R-module has a finite resolution by projective (left) R-modules. This implies in particular that for every (left) R-module M, there must exists a non-negative integer m such that for all (left) R-modules N, Ext_R^q(M,N) is zero if q is greater than or equal to m. We showed that if R is a PID then every R-module has a projective resolution of length 1, and hence, a PID is a regular ring. On the other hand we showed that if R = Z[C_m] is the group ring of the cyclic group C_m = {1,t,...,t^{m-1}} of order m, then Z, considered as an R-module via the map e : Z[C_m] --> Z that takes t^i to 1, has a periodic resolution of period two given by 
       t-1     N     t-1     N     t-1     e
    ...---> R ---> R ---> R ---> R ---> R ---> Z ---> 0
where N = 1 + t + ... + t^{m-1} is the norm element. It follows that if M is an R-module then the group Ext_R^q(Z,M) is the q'th cohomology group of the cochain complex 
        t-1     N     t-1     N     t-1
    ...<--- M <--- M <--- M <--- M <--- M
If also M = Z we find that Ext_R^q(Z,Z) is Z, if q = 0, Z/mZ, if q is odd, and zero, if q > 0 is even. In particular the ring Z[C_m] is not regular since infinitely many of the Ext-groups are non-zero. Finally, we mentioned that, in general, if R = Z[G] is the group ring of a group G and if M is an R-module, the group 
     H^q(G,M) := Ext_{Z[G]}^q(Z,M)
is called the q'th group cohomology group of G with coefficients in M.