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of integers is an algebraic space that is not quasi-separated. This algebraic space is also an example of a group object in the category of algebraic spaces that is not a scheme; quasi-separated algebraic spaces that are group objects are always group schemes. There are similar examples given by
105:) are automatically quasi-separated. Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated.
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The quotient of an algebraic space by an infinite discrete group acting freely is often not quasi-separated. For example, if
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The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi-compact.
235:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie"
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is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme
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and morphisms between them are defined in a similar way, though some authors include the condition that
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is quasi-separated as part of the definition of an algebraic space or algebraic stack
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by an infinite subgroup, or the quotient of the complex numbers by a lattice.
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is an "infinite dimensional vector space with two origins" over a field
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over a field is quasi-separated over the field but not separated.
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glued together by identifying the nonzero points in each copy.
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is a locally
Noetherian scheme then any morphism from
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Any separated scheme or morphism is quasi-separated.
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to any scheme is quasi-separated, and in particular
197:then the quotient of the affine line by the group
94:. Quasi-separated morphisms were introduced by
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16:In algebraic geometry, a morphism of schemes
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204:taking the quotient of the group scheme
172:is not quasi-separated. More precisely
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240:Publications Mathématiques de l'IHÉS
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96:Grothendieck & Dieudonné (1964
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178:consists of two copies of Spec
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193:is a field of characteristic
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38:if the diagonal map from
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