On the symbol length in Galois cohomology.

Online seminar Quadratic Forms, Linear Algebraic Groups and Beyond, 8:30-9:30am

https://uottawa-ca.zoom.us/j/99397061432?pwd=LzNBS1UrSVNJQ1ZiU28yWHAyNlV...

Fix a prime p and let F be a field with characteristic not p. Let G_F be the absolute Galois group of F and let \mu_{p^s} be the G_F-module of roots of unity of order
 dividing p^s in a fixed algebraic closure of F.  Let \alpha \in H^n(F,\mu_{p^s}^{\otimes n}) be a symbol (i.e \alpha=a_1 U ... U a_n where a_i \in H^1(F, \mu_{p^s})) with effective exponent dividing p^{s-1} (that is p^{s-1} \alpha=0 in H^n(G_F,\mu_p^{\otimes n})). In this talk I will explain how to write \alpha as a sum of symbols coming from H^n(F,\mu_{p^{s-1}}^{\otimes n}), that is of symbols of the form p \gamma for \gamma \in H^n(F,\mu_{p^s}^{\otimes n}). If n >3 and p \neq 2, we assume that F is prime-to-p closed and of characteristic zero.