The prime divisors of the period and index of a Brauer class

Abstract

We show that in locally-ringed connected topoi the primes dividing the period and index of a Brauer class coincide. The result applies in particular to Brauer classes on connected schemes, algebraic stacks, topological spaces and to the projective representation theory of profinite groups.

The topological period–index problem over 8‐complexes, I

“…Therefore, the lifting problem shown by the following diagram has a unique solution f 5 since H 4 (X; Z) = 0, and for the same reason, the composition κ 5 · f 5 is λ 1 R n ∈ H 7 (X; Z), for some λ 1 ∈ Z/ 3 (n)n, according to Lemma 6.4. On the other hand, it follows from Lemma 6.1 and (6.3) that in H 7 (BP (n, mn); Z) we have nR n = 0, whereas R n is of degree 3n in H 7 (BP (n, mn) [5]; Z). Therefore, we have 0 = nR n ∈ κ 5 ∈ H 7 (BP (n, mn) [5]; Z).…”

“…On the other hand, it follows from Lemma 6.1 and (6.3) that in H 7 (BP (n, mn); Z) we have nR n = 0, whereas R n is of degree 3n in H 7 (BP (n, mn) [5]; Z). Therefore, we have 0 = nR n ∈ κ 5 ∈ H 7 (BP (n, mn) [5]; Z).…”

“…Proof. It follows from (4.11) that H 7 (BP (n, mn) [5]; Z) ∼ = H 7 (K(Z/n, 2) × K(Z, 4); Z) ∼ = Z/ 3 (n)n × Z/n × Z/2. Therefore, H 7 (BP (n, mn); Z) ∼ = H 7 (BP (n, mn) [6]; Z) ∼ = Z/ 3 (n)n × Z/n × Z/2/(κ 5 ), and the result follows.…”

“…Remark 6.3. Since we have the homotopy equivalence BP (n, mn) [4] = BP (n, mn) [5] K(Z/n, 2) × K(Z, 4), the induced homomorphism H * (K(Z/n, 2) × K(Z, 4); Z) → H * (BP (n, mn); Z)…”

“…Remark In its full generality, Brauer group can be defined for any locally ringed topos. See , for example.…”