Algebraic Topology: Lecture 7
We discussed the following example of a model category. Let R be a
ring, and let A be the category whose objects are the left R-modules
and whose morphisms are the R-linear maps. More generally, A can be
any abelian category with enough projectives. Let C = C^+(A) be the
category whose objects are the chain complexes in A that are bounded
below and whose morphisms are the chain maps. Then C has a model
structure where the weak equivalences are the maps that induce
isomorphism on homology, where the fibrations are the maps that, in
each degree, are epimorphisms, and where the cofibrations are the maps
that, in each degree, are monomorphisms whose cokernels are projective
objects in A. We note that every object M. in C is fibrant and that an
object P. in C is cofibrant if and only if each P_q is a projective
object in A. The associated homotopy category Ho(C) is called the
bounded derived category of A and denoted D^+(A). We showed that the
set of morphisms in Ho(C) from an object M. to an object N. is given
as follows: We choose a weak equivalence p_M : P. --> M. from a cofibrant
object. Then set of maps in Ho(C) from M. to N. is canonically
isomorphic to the set of chain homotopy classes of chain maps from
P. to N. An object M in A determines an object of C that we also
denote by M, namely, the complex
.... --> 0 --> 0 --> M --> 0 --> 0 --> ...
where M is degree 0. An if M. is any object of C, we write M[q]. for
the object with M[q]_i = M_{q+i} and d = (-1)^q d. We finally showed
that if M and N are two objects of A, then
Hom_{D^+(A)}(M,N[-q]) = Ext_A^q(M,N).