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irreducible component
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(Definition)
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Let $G \subset {\mathbb{C}}^N$ be an open set.
Definition 1 A locally analytic set (or an analytic variety) $V \subset G$ is said to be irreducible if whenever we have two locally analytic sets $V_1$ and $V_2$ such that $V = V_1 \cup V_2$ then either $V = V_1$ or $V = V_2$ Otherwise $V$ is said to be reducible. A maximal irreducible subvariety of $V$ is said to be an irreducible component of $V$ Sometimes irreducible components are called ircomps.
Note that if $V$ is an analytic variety in $G$ then a subvariety $W$ is an irreducible component of $V$ if and only if $W^*$ (the set of regular points of $W$ is a connected complex analytic manifold. This means that the irreducible components of $V$ are the closures of the connected
components of $V^*$
- 1
- Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.
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"irreducible component" is owned by jirka. [ full author list (2) ]
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See Also: analytic set
| Also defines: |
irreducible analytic variety, irreducible locally analytic set, irreducible analytic variety, reducible locally analytic set, reducible analytic variety |
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Cross-references: connected components, closures, complex analytic manifold, connected, regular points, subvariety, irreducible, analytic variety, locally analytic, open set
There is 1 reference to this entry.
This is version 2 of irreducible component, born on 2005-02-22, modified 2005-02-23.
Object id is 6806, canonical name is IrreducibleComponent2.
Accessed 5529 times total.
Classification:
| AMS MSC: | 32A60 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Zero sets of holomorphic functions) | | | 32C25 (Several complex variables and analytic spaces :: Analytic spaces :: Analytic subsets and submanifolds) |
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Pending Errata and Addenda
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