Algebraic Topology: Lecture 9

We completed the proof that the category Ch(R) of unbounded chain complexes of left modules over a ring R has a model structure as described in the summary of Lecture 8. The associated homotopy category

    D(R) := Ho(Ch(R))

is called the unbounded derived category of the category of left R-modules. We showed that the set of morphisms between two objects X and Y of D(R) may be described as follows. Let X and Y be two chain complexes. We define a new chain complex

       d                     d                 d                    d
   ...---> Hom_R(X,Y)_{n+1} ---> Hom_R(X,Y)_n ---> Hom_R(X,Y)_{n-1} ---> ....

The group in degree n consists of all tuples (f_s) of R-linear maps f_s : X_s --> Y_{n+s}, where s runs through the integers, and d(f_s) = (df_s), where df_s : X_s --> Y_{n-1+s} is defined by (df_s)(x) = d(f_s(x)) - (-1)^s f_s(dx). We note that Z_0(Hom_R(X,Y)) is equal to the group of chain maps from X to Y and H_0(Hom_R(X,Y)) is equal to the group of chain homotopy classes of chain maps from X to Y. Let Q(X) --> X be a cofibrant replacement of X. Then we showed that

    Hom_{D(R)}(X,Y) = H_0(Hom_R(Q(X),Y)).

If X is a chain complex, we write X[q] for the chain complex defined by X[q]_n = X_{q+n} and d[q](x) = (-1)^q d(x). Now let M and N be two left R-modules. We can view M and N as chain complexes concentrated in degree zero. Let Q(M) --> M be a cofibrant replacement of M, or equivalently, a resolution of M by projective left R-modules. Then one defines

    Ext_R^q(M,N) = H_{-q}(Hom_R(Q(M),N)).

It follows that for left R-modules M and N, we have

    Hom_{D(R)}(M,N[-q]) = Ext_R^q(M,N).

We showed by example that these groups can often be effectively evaluated.