This chapter will place our previous developments in a more general setting. Fitrs, we have already noted that modules may be replaced by objects in an abelian category; our first three sections develop this technique and show how those ideas of homological algebra which do not involve tensor products can be carried over to any abelian category. Second, the relative and the absolute Ext functors can be treated to-gether, as cases of the general theory of “proper” exact sequences developed here in §§4–7. The next sections describe the process of forming “derived” functors: Hom R leads to the functors Ext R n , ⊗ R to the Tor n R , and any additive functor T to a sequence of “satellite” functors. Finally, an application of these ideas to the category of complexes yields a generalized KÜNNETH formula in which the usual exact sequence is replaced by a spectral sequence.
KeywordsAbelian Group Exact Sequence Spectral Sequence Short Exact Sequence Natural Transformation
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