# Homological algebra

The homological algebra is a branch of mathematics that has its origins in the algebraic topology has. The methods used there can be substantially generalized and also used in other mathematical areas. The publication of the now classic work Homological Algebra by Henri Cartan and Samuel Eilenberg in 1956 can be regarded as the beginning of homological algebra. In the following year, Alexander Grothendieck generalized these ideas for Abelian categories .

## Origins in algebraic topology

In certain of algebraic topology are topological spaces first so-called chain complexes or Kokettenkomplexe and then formed therefrom homology or cohomology in functorial associated manner. Chain complexes are consequences ${\ displaystyle {\ mathcal {C}}}$

${\ displaystyle \ ldots {\ xrightarrow {d_ {n + 2}}} C_ {n + 1} {\ xrightarrow {d_ {n + 1}}} C_ {n} {\ xrightarrow {d_ {n}}} C_ {n-1} {\ xrightarrow {d_ {n-1}}} \ ldots}$

of groups , modules , vector spaces or other structures and morphisms between them, so that always applies, that is, that the image of the core of lies. Therefore, one can form the factor groups called the -th homology group. A typical example are simplicial complexes , the homology groups derived from them are then called simplicial homology groups. If all arrows are turned around in the above considerations, the cohomology groups are obtained in an analogous manner. The general procedure can therefore be summarized as follows: ${\ displaystyle C_ {n}}$ ${\ displaystyle d_ {n}}$${\ displaystyle d_ {n} \ circ d_ {n + 1} = 0}$${\ displaystyle d_ {n + 1}}$${\ displaystyle d_ {n}}$ ${\ displaystyle H_ {n} ({\ mathcal {C}}) = \ mathrm {ker} (d_ {n}) / \ mathrm {im} (d_ {n + 1})}$${\ displaystyle n}$

Topological space (Ko) chain complex (Ko) homology groups.${\ displaystyle \ longrightarrow}$${\ displaystyle \ longrightarrow}$

In a first step, one abstracts from the topological spaces and proceeds directly from chain complexes. This means that (co) homology theories can also be built up for other mathematical structures. For example, the Hochschild homology results from a chain complex that is assigned to an algebra over a field . This approach easily leads to the investigation of exact sequences and their behavior among functors. Large parts of the theory can be carried out in any Abelian category . For many applications, however, the category of modules above a ring is sufficient, in which the basic ideas can be developed. In this context, reference is also made to Mitchell's embedding theorem .

Manin and Sergei Gelfand see the origin of homological algebra in Hilbert's investigation of syzygies .

## Hom functor and tensor functor

The application of the Hom functor to sequences is of particular importance . Be

${\ displaystyle 0 \ rightarrow A_ {1} {\ xrightarrow {\ alpha}} A_ {2} {\ xrightarrow {\ beta}} A_ {3} \ rightarrow 0}$

a short exact sequence, for example in the category of modules over a ring. Exactly means that the core and image of the morphisms involved are the same at every point , in particular the exactness at is equivalent to the injectivity of , the exactness at is equivalent to the surjectivity of . Short stands for the length 3 of the sequence, the terminal zero objects are not counted. Note that even shorter sequences are trivial: An exact sequence of length 2 only says that and are isomorphic, an exact sequence of length 1 is only possible for. If you apply the Hom functor to it, with a further module, or a further object from the category under consideration, you get an exact sequence ${\ displaystyle A_ {1}}$${\ displaystyle \ alpha}$${\ displaystyle A_ {3}}$${\ displaystyle \ beta}$${\ displaystyle A_ {1}}$${\ displaystyle A_ {2}}$${\ displaystyle A_ {1} = 0}$${\ displaystyle \ mathrm {Hom} (B, -)}$${\ displaystyle B}$

${\ displaystyle 0 \ rightarrow \ mathrm {Hom} (B, A_ {1}) {\ xrightarrow {\ alpha ^ {*}}} \ mathrm {Hom} (B, A_ {2}) {\ xrightarrow {\ beta ^ {*}}} \ mathrm {Hom} (B, A_ {3})}$,

where is defined by and analogously . In general, this sequence cannot be exactly extended with the zero object, i.e. it is generally not surjective. This leads on the one hand to the concept of the projective module , because precisely for projective modules all such sequences can be continued exactly with the zero object, on the other hand to the concept of the ext functor , which in the general case replaces the zero object on the right side of the sequence occurs. ${\ displaystyle \ alpha ^ {*} = \ mathrm {Hom} (B, \ alpha)}$${\ displaystyle \ alpha ^ {*} (f): = \ alpha \ circ f}$${\ displaystyle \ beta ^ {*}}$${\ displaystyle \ beta ^ {*}}$${\ displaystyle B}$

If one replaces the Hom functor by the tensor product with a module , one finds similar relationships. If you apply the functor to the above short exact sequence, you get the exact sequence ${\ displaystyle B}$${\ displaystyle (B \ otimes -)}$

${\ displaystyle B \ otimes A_ {1} {\ xrightarrow {\ alpha ^ {*}}} B \ otimes A_ {2} {\ xrightarrow {\ beta ^ {*}}} B \ otimes A_ {3} \ rightarrow 0}$,

where is now defined as , and analogously . This sequence cannot generally be continued exactly through 0 on the left side, that is, it is generally not injective. This leads, on the one hand, to the concept of the flat module , because precisely for flat modules , all such sequences can be continued exactly with the zero object, and on the other hand to the concept of the gate functor , which in the case of an exact continuation of the above sequence replaces the zero object on the left side of the Sequence occurs. ${\ displaystyle \ alpha ^ {*}}$${\ displaystyle \ mathrm {id} _ {B} \ otimes \ alpha}$${\ displaystyle \ beta ^ {*}}$${\ displaystyle \ alpha ^ {*}}$${\ displaystyle B}$

If one looks at the common features of the constructions presented by the functors and presented constructions, one obtains the concept of the derived functor , Ext and Tor can be understood as derivatives of these two functors. ${\ displaystyle \ mathrm {Hom} (B, -)}$${\ displaystyle (B \ otimes -)}$

## Sequences of homology groups

Another important topic of homological algebra are certain exact sequences from (co) homology groups that support their calculation, which will be briefly touched upon here. A homomorphism between two chain complexes and is understood to be a sequence of homomorphisms such that ${\ displaystyle {\ mathcal {C}} = ((C_ {n}) _ {n}, (d_ {n}) _ {n})}$${\ displaystyle {\ mathcal {C ^ {'}}} = ((C_ {n} ^ {'}) _ {n}, (d_ {n} ^ {'}) _ {n})}$${\ displaystyle (\ varphi _ {n}) _ {n}}$${\ displaystyle \ varphi _ {n}: C_ {n} \ rightarrow C_ {n} ^ {'}}$

${\ displaystyle {\ begin {array} {ccccccc} \ ldots {\ xrightarrow {d_ {n + 2}}} & C_ {n + 1} & {\ xrightarrow {d_ {n + 1}}} & C_ {n} & {\ xrightarrow {d_ {n}}} & C_ {n-1} & {\ xrightarrow {d_ {n-1}}} \ ldots \\\ ldots & \ downarrow _ {\ varphi _ {n + 1}} && \ downarrow _ {\ varphi _ {n}} && \ downarrow _ {\ varphi _ {n-1}} & \ ldots \\\ ldots {\ xrightarrow {d_ {n + 2} ^ {'}}} & C_ { n + 1} ^ {'} & {\ xrightarrow {d_ {n + 1} ^ {'}}} & C_ {n} ^ {'} & {\ xrightarrow {d_ {n} ^ {'}}} & C_ { n-1} ^ {'} & {\ xrightarrow {d_ {n-1} ^ {'}}} \ ldots \ end {array}}}$

is a commutative diagram . Cores and images of such homomorphisms are the chain complexes of the cores and images of the . With this one can speak of exact sequences of chain complexes and moves in a category that does not consist of modules over a ring. The homomorphism between the chain complexes induces homomorphisms by ${\ displaystyle \ varphi _ {n}}$${\ displaystyle \ varphi = (\ varphi _ {n}) _ {n}}$${\ displaystyle H_ {n} (\ varphi): H_ {n} ({\ mathcal {C}}) \ rightarrow H_ {n} ({\ mathcal {C}} ^ {'})}$

${\ displaystyle H_ {n} (\ varphi) (x + \ mathrm {im} (d_ {n + 1})): = \ varphi _ {n} (x) + \ mathrm {im} (d_ {n + 1 } ^ {'})}$ For ${\ displaystyle x \ in \ mathrm {ker} d_ {n}}$

sits down and is convinced of the well-definedness . A typical and fundamental result of homological algebra says:

If a short exact sequence of chain complexes, then the snake lemma gives homomorphisms , so that ${\ displaystyle 0 \ rightarrow {\ mathcal {C}} \, {\ xrightarrow {\ varphi}} \, {\ mathcal {C}} ^ {'} \, {\ xrightarrow {\ psi}} \, {\ mathcal {C}} ^ {''} \ rightarrow 0}$${\ displaystyle \ omega _ {n}: H_ {n} ({\ mathcal {C}} ^ {''}) \ rightarrow H_ {n-1} ({\ mathcal {C}})}$

${\ displaystyle \ ldots \ rightarrow H_ {n} ({\ mathcal {C}}) {\ xrightarrow {H_ {n} (\ varphi)}} H_ {n} ({\ mathcal {C}} ^ {'} ) {\ xrightarrow {H_ {n} (\ psi)}} H_ {n} ({\ mathcal {C}} ^ {''}) {\ xrightarrow {\ omega _ {n}}} H_ {n-1 } ({\ mathcal {C}}) \ rightarrow \ ldots}$

is an exact sequence.

If some of the homology groups that occur are 0, then isomorphisms can be constructed between others and thus statements about homology groups can be made. The above sentence is sometimes called the main clause about chain complexes and speaks of so-called long exact sequences . Similar sequences can be constructed for derivatives of additive functors . Further generalizations lead to the so-called spectral sequences .

## literature

• Henri Cartan, Samuel Eilenberg: Homological algebra. With an appendix by David A. Buchsbaum . Reprint of the original published in 1956. Princeton University Press, Princeton (1999) ISBN 0-691-04991-2
• David Eisenbud : Commutative Algebra. With a View Toward Algebraic Geometry. Springer-Verlag, 1999, ISBN 0-387-94269-6 (covers homological algebra from Chapter 16).
• John McCleary: A User's Guide to Spectral Sequences. Cambridge University Press, 2000, ISBN 0521567599 .
• Peter Hilton and Urs Stammbach: A course in homological algebra. 2nd edition, Springer-Verlag, Graduate Texts in Mathematics, 1997, ISBN 0-387-94823-6 .
• Saunders Mac Lane : Homology , Springer Basic Teachings of Mathematical Sciences Volume 114 (1967)
• Joseph J. Rotman: An Introduction to Homological Algebra. 2nd edition, Springer-Verlag, New York 2009, ISBN 978-0-387-24527-0 .
• Tilman Bauer: Homological Algebra and Group Homology . Lecture notes winter semester 2004/05, University of Münster, revised version from June 18, 2008. Accessed on September 3, 2014.