# What is known about the cohomological dimension of algebraic number fields?

What is the cohomological dimension of algebraic number fields like $$\Bbb{Q}$$, $$\Bbb{Q}[i]$$, $$\Bbb{Q}[\sqrt{3}]$$ or similar? I'm interested in computing the cohomological dimension of $$\Bbb{A}^1_k$$ with $$k$$ a finite extension of $$\Bbb{Q}$$.

By definition, an algebraic number field is a finite extension of the field of rational numbers $$\Bbb Q$$. An algebraic number field $$K$$ is called totally imaginary if it has no embeddings into $$\Bbb R$$. For example, the field $$\Bbb Q(i)$$ is totally imaginary, while $$\Bbb Q$$ and $$\Bbb Q(\sqrt{3})$$ are not.

We fix an algebraic closure $$\overline K$$ of $$K$$, and we write $$\Gamma_K={\rm Gal}(\overline K/K)$$ for the absolute Galois group of $$K$$. Let $$p$$ be a prime number. By definition, the $$p$$-cohomological dimension $${\rm cd}_p(K)$$ is the supremum of the degrees of nonzero cohomology over all finite $$\Gamma_K$$-modules $$M$$ whose cardinality is a power of $$p$$. The cohomological dimension $${\rm cd}(K)$$ is the supremum of $${\rm cd}_p(K)$$ over all prime numbers $$p$$.

Theorem 1 (well-known) The cohomological dimension $${\rm cd}(K)$$ of an algebraic number field $$K$$ is $$2$$ if $$K$$ is totally imaginary, and $$\infty$$ otherwise.

Following a suggestion of David Loeffler, I state and prove two other theorems, from which Theorem 1 follows.

Theorem 2. The 2-cohomological dimension $${\rm cd}_2(K)$$ of an algebraic number field $$K$$ is $$2$$ if $$K$$ is totally imaginary, and $$\infty$$ otherwise.

Proof. Assume that $$K$$ is not totally imaginary. Let $$\Omega_{\Bbb R}$$ denote the set of real embeddings of $$K$$; then $$\Omega_{\Bbb R}$$ is nonempty. We take $$M=\mu_2$$. For any odd natural number $$n\ge 3$$ we have $$H^{n}(K,\mu_2)\cong\bigoplus_{v\in\Omega_{\Bbb R}} H^n(K_v,\mu_2)\cong \bigoplus_{v\in\Omega_{\Bbb R}} H^1(\Gamma_{\Bbb R},\mu_2) \cong\bigoplus_{v\in\Omega_{\Bbb R}}\Bbb Z/2\Bbb Z\neq 0.$$ (For the first isomorphism, see J. S. Milne, Arithmetic Duality Theorems, Theorem I.4.10(c). For the second isomorphism, see Atiyah and Wall, Cohomology of Groups, IV.8, Theorem 5, in: Cassels and Fröhlich (eds.), Algebraic Number Theory.) Thus $${\rm cd}_2(K)=\infty$$, as required.

Now assume that $$K$$ is totally imaginary. Then for any $$n\ge 3$$ and any finite $$\Gamma_K$$-module $$M$$ (in particular, for any $$\Gamma_K$$-module $$M$$ whose cardinality is a power of 2) we have $$H^n(K,M)\cong\bigoplus_{v\in\Omega_{\Bbb R}}H^n(K_v,M)=0,$$ because $$\Omega_{\Bbb R}=\varnothing$$. (For the isomorphism, again see Milne, Theorem I.4.10(c).) Thus $${\rm cd}_2(K)\le 2$$.

Let $$\Omega_f$$ denote the set of finite places of $$K$$. We take $$M=\mu_2$$. We have $$H^2(K, \mu_2)\cong\left\{(a_v)\in\bigoplus_{v\in\Omega_f}\Bbb Z/2\Bbb Z\ \mid\ \sum_{v\in \Omega_f} a_v=0\right\},$$ and hence, $$H^2(K,\mu_2)\ne 0$$. Thus $${\rm cd}_2(K)=2$$, as required.

Theorem 3. For any odd prime $$p$$, the $$p$$-cohomological dimension $${\rm cd}_p(K)$$ of any algebraic number field $$K$$ equals $$2$$.

Proof. For any $$n\ge 3$$ and any finite $$\Gamma_K$$-module $$M$$ whose cardinality is a power of $$p$$, we have $$H^n(K,M)\cong\bigoplus_{v\in\Omega_{\Bbb R}}H^n(K_v,M)\cong \bigoplus_{v\in\Omega_{\Bbb R}} H^1(\Gamma_{\Bbb R},M)=0,$$ where $$H^1(\Gamma_{\Bbb R},M)=0$$ because the group $$\Gamma_{\Bbb R}$$ is of order 2 while $$M$$ is of odd cardinality (this is an easy exercise; see also Atiyah and Wall, IV.6, Corollary 1 of Proposition 8). Thus $${\rm cd}_p(K)\le 2$$.

Let $$\Omega_f$$ denote the set of finite places of $$K$$. We take $$M=\mu_p$$. We have $$H^2(K, \mu_p)\cong\left\{(a_v)\in\bigoplus_{v\in\Omega_f}\Bbb Z/p\Bbb Z\ \mid\ \sum_{v\in \Omega_f} a_v=0\right\},$$ and hence, $$H^2(K,\mu_p)\ne 0$$. Thus $${\rm cd}_p(K)=2$$, as required.

• Maybe it's worth remarking that the infinite cohomological dimension is a local problem at the prime 2 only. For $K$ any number field and $p$ an odd prime, the $p$-cohomological dimension (i.e. the sup of the degrees of nonzero cohomology among all finite $Gal(\overline{K} / K)$-modules whose cardinality is a power of $p$) is always 2. Jan 19, 2020 at 8:37
• @DavidLoeffler: Thank you! I will edit my answer. Jan 19, 2020 at 10:20