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isomorphism of varieties
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(Definition)
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Definition 1 Let $V_1$ and $V_2$ be algebraic varieties. We say that $V_1$ and $V_2$ are isomorphic, and write $V_1 \cong V_2$ if there are regular maps $$\phi \colon V_1 \to V_2, \quad \psi\colon V_2 \to V_1$$ such that the compositions $\psi \circ \phi$ and
$\phi \circ \psi$ are the identity maps on $V_1$ and $V_2$ respectively.
Definition 2 Let $V_1$ and $V_2$ be varieties defined over a field $K$ We say that $V_1/K$ and $V_2/K$ are isomorphic over $K$ if $V_1$ and $V_2$ are isomorphic as in Definition 1 and the regular maps $\phi$ and $\psi$ can be defined over $K$
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"isomorphism of varieties" is owned by alozano.
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Cross-references: field, identity maps, compositions, regular maps, isomorphic, varieties, algebraic
There is 1 reference to this entry.
This is version 1 of isomorphism of varieties, born on 2005-03-01.
Object id is 6838, canonical name is IsomorphismOfVarieties.
Accessed 1513 times total.
Classification:
| AMS MSC: | 14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms) |
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Pending Errata and Addenda
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