General Homology Theory
To a large extent the present chapter is arrow-theoretic. There is a substantial body of linear algebra which can be formalized very systematically, and constitutes what Steenrod called abstract nonsense, but which provides a well-oiled machinery applicab
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XX
General Homology Theory
To a large extent the present chapter is arrow-theoretic . There is a substantial body of linear algebra which can be formalized very systematically, and constitutes what Steenrod called abstract nonsense , but which provides a well-oiled machinery applicable to many doma ins . References will be given along the way . Most of what we shall do applies to abelian categories , which were mentioned in Chapter III, end of §3. However, in first reading, I recommend that readers disregard any allusions to general abelian categories and assume that we are dealing with an abelian category of modules over a ring , or other specific abelian categories such as complexes of module s over a ring .
§1.
COMPLEXES
Let A be a ring. By an open complex of A-modules, one means a sequence of modules and homomorphisms {(E i , di)},
wher e i range s over all integers and d, maps Ei into
e:
1,
and such that
for all i. One frequently considers a finite sequence of homomorphisms, say E 1 .......
. . ........
E'
761 S. Lang, Algebra © Springer Science+Business Media LLC 2002
762
GENERAL HOMOLOGY THEORY
XX, §1
such that the composite of two successive ones is 0, and one can make this sequence into a complex by inserting 0 at each end :
Such a complex is called a finite or bounded complex. Remark. namely,
Complexes can be indexed with a descending sequence of integers,
--+
Ei + l
du
I
~
Ei--+ di E i-
I --+
When that notation is used systematically, then one uses upper indices for complexes which are indexed with an ascending sequence of integers:
In this book, I shall deal mostly with ascending indices . As stated in the introduction of this chapter, instead of modules over a ring, we could have taken objects in an arbitrary abelian category. The homomorphisms di are often called differentials, because some of the first complexes which arose in practice were in analysis, with differential operators and differential forms. Cf. the examples below . We denote a complex as above by (E, d). If the complex is exact , it is often useful to insert the kernels and cokernels of the differentials in a diagram as follows , letting M, = Ker di = 1m e>. ---->
Ei- 2 - - - - > Ei- 1 - - - - > Ei - - - - > Ei+ 1 - - - - >
\ / \ /\ / /\ / \ /\ o Mi -
M i+ 1
Mi
1
0
0
0
Thus by definition, we obtain a family of short exact sequences
If the complex is not exact, then of course we have to insert both the image of di - 1 and the kernel of di• The factor
will be studied in the next section. It is called the homology of the complex, and measures the deviation from exactness.
COMPLEXES
XX, §1
763
Let M be a module. By a resolution of M we mean an exact sequence
Thus a resolution is an exact complex who se furthest term on the right before Th e resolution is indexed as sho wn. We usuall y write EM for the part of complex form ed onl y of the E;'s, thus :
ois M.
stopping at Eo. We then write E for the complex obtained by sticking 0 on the right :
E is :
->
En
->
En-
1 -> .. . ->
Eo -> O.
If
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