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[parent] isomorphism of varieties (Definition)
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
$\displaystyle \phi \colon V_1 \to V_2, \quad \psi\colon V_2 \to V_1$
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$.



"isomorphism of varieties" is owned by alozano.
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See Also: the $j$-invariant classifies elliptic curves up to isomorphism


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Cross-references: field, identity maps, compositions, regular maps, isomorphic, varieties, algebraic
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This is version 1 of isomorphism of varieties, born on 2005-03-01.
Object id is 6838, canonical name is IsomorphismOfVarieties.
Accessed 1056 times total.

Classification:
AMS MSC: 14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms)
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