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.

Cohomological dimension of a group[edit]

As most (co)homological 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: Pk\toPk − 1 (k = 1, …, n) and d0: P0\toR, 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 coeffients 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 n = cdR(G).

A free resolution of Z 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 cdZ(G) ≤ n.

Examples[edit]

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

  • A free group has cohomological dimension one. As shown by John Stallings (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups.
  • The fundamental group of a compact, connected, orientable Riemann surface other than the sphere has cohomological dimension two.
  • More generally, the fundamental group of a compact, connected, orientable aspherical manifold of dimension n has cohomological dimension n. In particular, the fundamental group of a closed orientable hyperbolic n-manifold has cohomological dimension n.
  • Nontrivial finite groups have infinite cohomological dimension over Z. More generally, the same is true for groups with nontrivial torsion.

Now let us consider the case of a general ring R.

  • A group G has cohomological dimension 0 if and only if its group ring RG is semisimple. Thus a finite group has cohomological dimension 0 if and only if its order (or, equivalently, the orders of its elements) is invertible in R.
  • Generalizing the Stallings–Swan theorem for R = Z, Dunwoody proved that a group has cohomological dimension at most one over an arbitrary ring R if and only if it is the fundamental group of a connected graph of finite groups whose orders are invertible in R.

Cohomological dimension of a field[edit]

The p-cohomological dimension of a field K is the p-cohomological dimension of the Galois group of a separable closure of K.[2] The cohomological dimension of K is the supremum of the p-cohomological dimension over all primes p.[3]

Examples[edit]

  • Every field of non-zero characteristic has cohomological dimension at most 1.[4]
  • Every finite field has absolute Galois group isomorphic to \mathbf{\hat Z} and so has cohomological dimension 1.[5]
  • The field of formal Laurent series k((t)) over an algebraically closed field k of non-zero characteristic also has absolute Galois group isomorphic to \mathbf{\hat Z} and so cohomological dimension 1.[5]

See also[edit]

References[edit]

  1. ^ Gille & Szamuely (2006) p.136
  2. ^ Shatz (1972) p.94
  3. ^ Gille & Szamuely (2006) p.138
  4. ^ Gille & Szamuely (2006) p.139
  5. ^ a b Gille & Szamuely (2006) p.140