Cohomological dimension

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In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory.

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Cohomological dimension of a group

As most cohomological invariants, the cohomological dimension involves a choice of a "ring of coefficients" R, with a prominent special case given by R = Z, the ring of integers. Let G be a discrete group, R a non-zero ring with a unit, and RG the group ring. The group G has cohomological dimension less than or equal to n, denoted cdR(G)  n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules P0, ..., Pn and RG-module homomorphisms dk: PkPk  1 (k = 1, ..., n) and d0: P0R, such that the image of dk coincides with the kernel of dk  1 for k = 1, ..., n and the kernel of dn is trivial.

Equivalently, the cohomological dimension is less than or equal to n if for an arbitrary RG-module M, the cohomology of G with coefficients in M vanishes in degrees k > n, that is, Hk(G,M) = 0 whenever k > n. The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups Hk(G,M){p}. [1]

The smallest n such that the cohomological dimension of G is less than or equal to n is the cohomological dimension of G (with coefficients R), which is denoted .

A free resolution of can be obtained from a free action of the group G on a contractible topological space X. In particular, if X is a contractible CW complex of dimension n with a free action of a discrete group G that permutes the cells, then .

Examples

In the first group of examples, let the ring R of coefficients be .

Now consider the case of a general ring R.

Cohomological dimension of a field

The p-cohomological dimension of a field K is the p-cohomological dimension of the Galois group of a separable closure of K. [4] The cohomological dimension of K is the supremum of the p-cohomological dimension over all primes p. [5]

Examples

See also

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References

  1. Gille & Szamuely (2006) p.136
  2. Baumslag, Gilbert (2012). Topics in Combinatorial Group Theory. Springer Basel AG. p. 16.
  3. Gruenberg, Karl W. (1975). "Review of Homology in group theory by Urs Stammbach". Bulletin of the American Mathematical Society . 81: 851–854. doi: 10.1090/S0002-9904-1975-13858-4 .
  4. Shatz (1972) p.94
  5. Gille & Szamuely (2006) p.138
  6. Gille & Szamuely (2006) p.139
  7. 1 2 Gille & Szamuely (2006) p.140