## A unified Kummer-Artin-Schreier sequence.(English)Zbl 0608.12026

For $$b$$ in a commutative ring $$R_0$$, let $$G\{b\}$$ be the affine group scheme defined by $$\{(x,y)\mid x(1-by)=1\}$$. This is smooth with connected fibers of dimension one, and it turns out that $$H^1(R,G\{b\})$$ is a relative Picard group familiar from $$K$$-theory for any $$R_0$$-algebra $$R$$. Now let $$p$$ be a prime, and let $$\zeta$$ be a $$p$$-th root of unity. Then over $$R_0 = \mathbb{Z}[\zeta]$$ there is an exact sequence $1\longrightarrow \mathbb{Z}/p\mathbb{Z}\longrightarrow G\{1-\zeta \}\longrightarrow G\{(1-\zeta)^p\}\longrightarrow 1.$ The cohomology for this sequence reduces to the Kummer sequence when $$1/p\in R$$, and it reduces to the Artin-Schreier sequence when $$p=0$$ in $$R$$. For $$p=2$$ (and arbitrary $$R)$$ it yields the description of étale quadratic algebras previously derived by adhoc methods.

### MSC:

 12G05 Galois cohomology 14L15 Group schemes
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### References:

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