Fundamental lemma of homological algebra

The fundamental lemma of homological algebra (also the main lemma of homological algebra ) is a technical lemma from the mathematical field of homological algebra , it guarantees the continuity of chain mappings between chain complexes.

The fundamental lemma shows that the definition of homology groups is independent of choices in certain constructions.

lemma

Let and two chain complexes . For an integer let ${\displaystyle (C_{*},d_{*})}$ ${\displaystyle (C_{*}^{\prime },d_{*}^{\prime })}$ ${\displaystyle r}$

${\displaystyle (f_{i}\colon C_{i}\to C_{i}^{\prime })_{i\leq r}}$

a family of homomorphisms with

${\displaystyle d_{i}^{\prime }f_{i}=f_{i-1}d_{i}}$for . ${\displaystyle i\leq r}$

We assume that all of projective modules and that from degree of homology of disappears, so for all . ${\displaystyle C_{i}}$ ${\displaystyle i>r}$ ${\displaystyle r}$ ${\displaystyle C^{\prime }}$ ${\displaystyle H_{i}(C^{\prime })=0}$ ${\displaystyle i\geq r}$

Then a chain homomorphism can be made ${\displaystyle (f_{i}\colon C_{i}\to C_{i}^{\prime })_{i\leq r}}$

${\displaystyle f\colon C\to C^{\prime }}$

continue with for and this continuation is unambiguous except for chain homotopy . For every two continuations the chain homotopy can be chosen so that for . ${\displaystyle f=f_{i}}$ ${\displaystyle i\leq r}$ ${\displaystyle h}$ ${\displaystyle h_{i}=0}$ ${\displaystyle i\leq r}$

Inferences

A direct consequence of the fundamental lemma is the following theorem:

For every two projective resolutions and one module there is an augmentation-preserving chain homotopy equivalence (which is unique except for chain homotopy) . ${\displaystyle F}$ ${\displaystyle F^{\prime }}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle f\colon F\to F^{\prime }}$

A typical application of this fact can be found in the definition of homology groups. For example, the group homology of a group with coefficients in an Abelian group (e.g. or ) is defined as the homology of the chain complex ${\displaystyle H_{*}(G,A)}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle A=\mathbb {Z} }$ ${\displaystyle A=\mathbb {R} }$

${\displaystyle F_{*}\otimes _{\mathbb {Z} G}\mathbb {Z} }$,

where denotes any projective resolution of the module (with the trivial effect). The above sentence shows the independence of the homology groups defined in this way from the choice of the projective resolution. For concrete calculations it is often very helpful that you can choose any projective resolution to determine the homology. ${\displaystyle F_{*}}$ ${\displaystyle \mathbb {Z} G}$ ${\displaystyle A}$ ${\displaystyle \mathbb {Z} G}$

Another application is the definition of the gate functor , which is also defined by means of projective resolutions and where the above sentence also implies the independence of the functor from the selected projective resolution.

In general, the above theorem can be used to prove the well-definedness of left - derivative functors .

Dual version

The fundamental allemma also has a dual version for coquette complexes .

Let and two Kokettenkomplexe. For an integer let ${\displaystyle (C^{*},d^{*})}$ ${\displaystyle ({C^{\prime }}^{*},{d^{\prime }}^{*})}$ ${\displaystyle r}$

${\displaystyle (f_{i}\colon C^{i}\to {C^{\prime }}^{i})_{i\leq r}}$

a family of homomorphisms with

${\displaystyle {d^{\prime }}^{i}f_{i}=f_{i+1}d^{i}}$for . ${\displaystyle i\leq r}$

We assume that all are with injective modules and that from degree the cohomology of vanishes, i.e. for all . ${\displaystyle C_{i}}$ ${\displaystyle i>r}$ ${\displaystyle r}$ ${\displaystyle C^{\prime }}$ ${\displaystyle H^{i}(C^{\prime })=0}$ ${\displaystyle i\geq r}$

Then a chain homomorphism can be made ${\displaystyle (f_{i}\colon C_{i}\to {C^{\prime }}^{i})_{i\leq r}}$

${\displaystyle f\colon C\to C^{\prime }}$

continue with for and this continuation is unambiguous except for chain homotopy . For every two continuations the chain homotopy can be chosen so that for . ${\displaystyle f=f_{i}}$ ${\displaystyle i\leq r}$ ${\displaystyle h}$ ${\displaystyle h_{i}=0}$ ${\displaystyle i\leq r}$

The dual version is used in the definition of cohomology groups, for example in group cohomology , or in the definition of the ext functor .

literature

• KS Brown: Cohomology of groups. Corrected reprint of the 1982 original. Graduate Texts in Mathematics, 87. Springer-Verlag, New York, 1994. ISBN 0-387-90688-6
• E. Ossa: topology. Vieweg studies: advanced course in mathematics, 42. Friedr. Vieweg & Sohn, Braunschweig, 1992. ISBN 3-528-07242-3

Web links

• C. Schweigert: Higher Algebra: Representation Theory and Homological Algebra (Chapter 6)
• T. Bauer: Homological Algebra and Group Cohomology (Chapter 3)
• JF Davis, P. Kirk: Lectures in Algebraic Topology (Chapter 2)

Individual evidence

1. ↑ Ossa, op.cit., Sentence 6.1.8
2. ↑ Lemma I.7.4 in Brown, op. Cit.
3. ↑ Theorem I.7.5 in Brown, op. Cit.
4. ↑ Ossa, op.cit., Chapter 6.1