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Download as PDF; Printable version; In other projects ... The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to ...
Terminology: The definition given here is the one used by the first edition of EGA and the Stacks Project. In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above.
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.
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Let ,,, be as in the above example. But this time let denote the scheme obtained by gluing , along the isomorphism given by . [3] So, geometrically, is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin.
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We also construct a sheaf on , called the “structure sheaf” as in the affine case, which makes it into a scheme.As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following.