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Date | 2024-03-22 |
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Speaker | Mikhail Borovoi |

Dept. | Tel Aviv University |

Room | 27-325 |

Time | 16:00-18:00 |

Let K be a number field (say, Q) and let G be a connected reductive group over K (say, SO(n)). One needs the first Galois cohomology H^1(K,G) for classification problems in algebraic geometry and linear algebra over K.

For a number field K admitting a real embedding (say, K=Q), we show that it is impossible to define a *group structure*, functorial in G, on the Galois cohomology pointed set H^1(K,G) for all connected reductive K-groups G. However, over an arbitrary number field K, we define an *operation of raising to power n* (which we denote by Diamond)

(x,n) mapsto x^{Diamond n}: H^1(K,G) times Z -> H^1(K,G).

We show that this new operation has nice functorial properties. When G is a torus (hence an abelian group), the pointed set H^1(K,G) has a natural abelian group structure, and our new operation coincides with the usual power operation

(x,n)mapsto x^n.

For a cohomology class x in H^1(K,G), we define the *period* (or order) per(x) to be the least integer n>0 such that the n-th power x^{Diamond n}=1, and we define the *index* ind(x) to be the greatest common divisor of the degrees [L:K] of finite extensions L/K splitting x. We show that per(x) divides ind(x), but they need not be equal. However, per(x) and ind(x) have the same set of prime factors.

All terms will be defined and examples will be given.

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