Group cohomology

Group cohomology is a technical mathematical tool that originally served to study groups , but later also found applications in topology and number theory in particular . The group cohomology of Galois groups is also known as Galois cohomology and plays an important role in number theory. In the topology, group cohomology plays an important role as the cohomology of Eilenberg-MacLane spaces .

Definition as a derived functor

definition

It is a group . The functor from the category of the modules into the category of the Abelian groups , which assigns the subgroup of the invariant elements to a module , is exact left . Its n th right derivative is the n th cohomology group of with coefficients in a module . ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle A^{G}}$ ${\displaystyle G}$ ${\displaystyle H^{n}(G,A)}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle A}$

Relationship to Ext

The group cohomology can also be defined using the Ext functor :

${\displaystyle \mathrm {H} ^{n}(G,A)=\mathrm {Ext} _{\mathbb {Z} [G]}^{n}(\mathbb {Z} ,A);}$

It is the group ring of and with the trivial provided operation. ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle G}$

Definition of coquette

From the description with the aid of the Ext functor it can be seen that the group cohomology can be calculated with the aid of a projective resolution of the trivial module once selected . It can be specified as explicit: ${\displaystyle G}$ ${\displaystyle (\mathbb {Z} [G^{n}],d_{n})}$

${\displaystyle d_{n}(\sigma _{1},\ldots ,\sigma _{n})=\sum _{i=1}^{n-1}(-1)^{i}(\sigma _{1},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n});}$

is there

${\displaystyle (\sigma _{1},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n}):=(\sigma _{1},\ldots ,\sigma _{i-1},\sigma _{i+1},\ldots ,\sigma _{n}),}$

d. H. Index is omitted. ${\displaystyle i}$

The group cohomology is then the cohomology of the complex with ${\displaystyle (C^{n},d^{n})}$

${\displaystyle C^{n}=\{f\colon G^{n+1}\to A\mid f(\sigma \sigma _{1},\ldots ,\sigma \sigma _{n+1})=\sigma \cdot f(\sigma _{1},\ldots ,\sigma _{n+1})\}}$

and

${\displaystyle (d^{n-1}f)(\sigma _{1},\ldots ,\sigma _{n+1})=\sum _{i=1}^{n+1}(-1)^{i}f(\sigma _{1},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n+1}).}$

The elements of this complex are called homogeneous coquettes .

Inhomogeneous coquettes

The condition of the -invariance of the coquettes allows the number of copies to be reduced by one: the group homology can also be defined via the complex of inhomogeneous coquettes : ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle ({\tilde {C}}^{n},{\tilde {d}}^{n})}$

${\displaystyle {\tilde {C}}^{n}=\{f\colon G^{n}\to A\}}$

and

${\displaystyle ({\tilde {d}}^{n-1}f)(\sigma _{1},\ldots ,\sigma _{n})=\sigma _{1}\cdot f(\sigma _{2},\ldots ,\sigma _{n})+{}}$

${\displaystyle {}+\sum _{i=1}^{n-1}(-1)^{i}f(\sigma _{1},\ldots ,\sigma _{i}\sigma _{i+1},\ldots ,\sigma _{n})+(-1)^{n}f(\sigma _{1},\ldots ,\sigma _{n-1}).}$

For example is

${\displaystyle \mathrm {H} ^{1}(G,A)=\{c\colon G\to A\mid c(\sigma \tau )=c(\sigma )+\sigma c(\tau )\}/\{c_{a}(\tau )=\tau a-a\mid a\in A\}.}$

The inhomogeneous 1-coccycles

${\displaystyle c\colon G\to A,\quad c(\sigma \tau )=c(\sigma )+\sigma c(\tau )}$

are called entangled homomorphisms .

Definition of classifying spaces

The group cohomology can be defined equivalently as the cohomology of the Eilenberg-MacLane space , i.e. the classifying space of the group provided with the discrete topology : ${\displaystyle K(G,1)}$

${\displaystyle H^{*}(G,A)=H^{*}(K(G,1),A)}$.

This definition is often more useful than other definitions for practical calculations.

See also

• Group homology

Web links

• Jon F. Carlson : The mod-2 cohomology of 2-groups. Department of Mathematics, University of Georgia , May 31, 2001, archived from the original July 3, 2007 ; accessed on September 4, 2017 . 

literature

• David J. Benson: Representations and cohomology (=  Cambridge studies in advanced mathematics . Volume 31 ). 2nd Edition. II: Cohomology of groups and modules . Cambridge University Press, Cambridge 1998, ISBN 0-521-63652-3 .
• Kenneth S. Brown: Cohomology of groups (= Graduate Texts in Mathematics 87). Corrected 2nd printing. Springer, New York a. a. 1994, ISBN 0-387-90688-6 .
• George Janelidze, Bodo Pareigis, Walter Tholen (Eds.): Galois Theory, Hopf Algebras, and Semiabelian Categories (= Fields Institute Communications 43). American Mathematical Society, Providence RI 2004, ISBN 0-8218-3290-5 .
• Jürgen Neukirch : class field theory . BI university scripts, 713 / 713a *. Bibliographisches Institut, Mannheim / Vienna / Zurich 1969, ISBN 978-3-642-17324-0 , x + 308 pp.