Realization-obstruction exact sequences for Clifford system extensions

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Publication date: 19 January 2020

Abstract: For every action

p h i i n e x t H o m (G, e x t {A u t}_{k} (K))

of a group

G

on a commutative ring

K

we introduce two abelian monoids. The monoid

e x t {C l i f f}_{k} (p h i)

consists of equivalent classes of

G

-graded Clifford system extensions of type

p h i

of

K

-central algebras. The monoid

m a t h c a l C_{k} (p h i)

consists of equivariant classes of generalized collective characters of type

p h i

from

G

to the Picard groups of

K

-central algebras. Furthermore, for every such

p h i

there is an exact sequence of abelian monoids

0 o H^2(G,K^*_{phi}) o ext{Cliff}_k(phi) omathcal{C}_k{(phi)} o H^3(G,K^*_{phi}).

The rightmost homomorphism is often surjective, terminating the above sequence. When

p h i

is a Galois action, then the restriction-obstruction sequence of Brauer groups is an image of an exact sequence of sub-monoids of this sequence.

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