194:, and complexes thereof. These more general geometric objects can be useful when one wants to work with categories that have good limit behavior. There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine
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The definition of
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of the Hopf algebra, exchanging the multiplication and comultiplication.
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to the abelian group of group scheme homomorphisms from the base change
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Groupes Algébriques (Bruxelles, 1962)
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for commutative group schemes. It was introduced by
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267:{\displaystyle {\widehat {\mathbf {G} }}_{m}}
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