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An algebraic stack or Artin stack is a stack in groupoids X over the fppf site such that the diagonal map of X is representable and there exists a smooth surjection from (the stack associated to) a scheme to X. A morphism Y X of stacks is representable if, for every morphism S X from (the stack associated to) a scheme to X, the fiber product Y ...
A word identifying a person or a group of people in relation to a particular place, usually derived from the name of the place (which may be any kind of place, formal or informal, of any size or scale, from a town or city to a region, province, country, or continent) and used to describe all residents or natives of that place, regardless of any ...
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Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas , where is the stack associated to some scheme . If the atlas U → X {\displaystyle {\mathcal {U}}\to {\mathcal {X}}} is moreover étale, then X {\displaystyle {\mathcal {X}}} is said to be a Deligne–Mumford stack .
A morphism f : Y → X is called quasi-compact, if for some (equivalently: every) open affine cover of X by some U i = Spec B i, the preimages f −1 (U i) are quasi-compact. quasi-finite The morphism f : Y → X has finite fibers if the fiber over each point is a finite set.
Stacks of coins. Any of these stacks can be split into two stacks of different sizes: once the leftmost stack of three has been split, it can be split no further. Grundy's game is a two-player mathematical game of strategy. The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two ...
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