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In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.
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Thomas Andrew Bridgeland is a Professor of Mathematics at the University of Sheffield. He was a senior research fellow in 2011–2013 at All Souls College, Oxford and, since 2013, remains as a Quondam Fellow. He is most well-known for defining Bridgeland stability conditions on triangulated categories.
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Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada.
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In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.
In algebraic geometry, the moduli stack of formal group laws is a stack classifying formal group laws and isomorphisms between them. It is denoted by . It is a "geometric “object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology.
In algebraic geometry, a toric stack is a stacky generalization of a toric variety. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking GIT quotients with that of taking quotient stacks. Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack.
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In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack , the Chow group of X is the same as the G-equivariant Chow group of Y.
Bertrand Toën is a mathematician who works as a director of research at the Centre national de la recherche scientifique (CNRS) at the Paul Sabatier University, Toulouse, France. He received his PhD in 1999 from the Paul Sabatier University, where he was supervised by Carlos Simpson and Joseph Tapia.
In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is
In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al.
Gabriele Vezzosi is an Italian mathematician, born in Florence (Italy). His main interest is algebraic geometry.
In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms.
References
- Toën, Bertrand (2014), Derived Algebraic Geometry, arXiv: 1401.1044
- Toën, Bertrand (2006), Higher and derived stacks: a global overview, arXiv: math/0604504
- Lurie, Jacob. "Derived Algebraic Geometry". hdl:1721.1/30144.
- Mathew, Akhil; Meier, Lennart (2013). "Affineness and chromatic homotopy theory". Journal of Topology. 8 (2): 476–528. arXiv: 1311.0514 . doi:10.1112/jtopol/jtv005. S2CID 119713516.
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