In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Alexander Grothendieck[1] to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. After Grothendieck developed the general theory of descent,[2] and Giraud the general theory of stacks,[3] the notion of algebraic stacks was defined by Michael Artin.[4]
^A'Campo, Norbert; Ji, Lizhen; Papadopoulos, Athanase (2016-03-07). "On Grothendieck's construction of Teichmüller space". arXiv:1603.02229 [math.GT].
^Grothendieck, Alexander; Raynaud, Michele (2004-01-04). "Revêtements étales et groupe fondamental (SGA 1). Expose VI: Catégories fibrées et descente". arXiv:math.AG/0206203.
^Giraud, Jean (1971). "II. Les champs". Cohomologie non abelienne. Grundlehren der mathematischen Wissenschaften. Vol. 179. pp. 64–105. doi:10.1007/978-3-662-62103-5. ISBN 978-3-540-05307-1.
^Artin, M. (1974). "Versal deformations and algebraic stacks". Inventiones Mathematicae. 27 (3): 165–189. Bibcode:1974InMat..27..165A. doi:10.1007/bf01390174. ISSN 0020-9910. S2CID 122887093.
specific to algebraicstacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves M g ,...
values in categories rather than sets Algebraicstack, a special kind of stack commonly used in algebraic geometry Stacks Project, an open source collaborative...
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ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings...
theory of non-abelian bundle gerbes. Twisted sheaf Azumaya algebra Twisted K-theory Algebraicstack Bundle gerbe String group Basic bundle theory and K-cohomology...
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two trees are merged and a pointer to the final tree remains on the stack. Algebraic expression trees represent expressions that contain numbers, variables...
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