Definition:Coherent Sequence of Inverse System of Groups

Definition

Let $\sequence {G_n}_{n \mathop \in \N}$ be a inverse system of groups with group homomorphisms $\sequence {\theta_n}_{n \mathop \in \N_{>0} }$.

Let $\prod_{n \mathop \in \N} G_n$ be the Cartesian product of $\sequence {G_n}_{n \mathop \in \N}$.

Then $\sequence {x_n}_{n \mathop \in \N} \in \prod_{n \mathop \in \N} G_n$ is a coherent sequence (of groups) if and only if:

$\forall n \in \N : \map {\theta_{n + 1} } {x_{n + 1} } = x_n$

Also see

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• Definition:P-adically Coherent Sequence: A special case with $G_n = \Z / p^n \Z$ and $\map {\theta_{n+1} } {x \bmod p^{n+1} } = x \bmod p^n$

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $10$: Completion