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Generalization of algebraic spaces or schemes
specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves M g ,
Algebraic_stack
Generalisation of a sheaf; a fibered category that admits effective descent
There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks. Edidin (2003) and Fantechi (2001)
Stack_(mathematics)
Type of object in algebraic geometry
In algebraic geometry, a Deligne–Mumford stack is a stack that behaves, in many respects, like an algebraic variety or an orbifold, while still allowing
Deligne–Mumford_stack
Geometric space whose points represent algebro-geometric objects of some fixed kind
mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric
Moduli_space
Mathematics notation where operators follow operands
effects and implications depending on the actual implementation involving a stack. The description "Polish" refers to the nationality of logician Jan Łukasiewicz
Reverse_Polish_notation
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack X {\displaystyle {\mathfrak {X}}} is a generalization of a quasi-coherent sheaf on
Sheaf_on_an_algebraic_stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety
Quotient_stack
Branch of mathematics
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts
Derived_algebraic_geometry
Geometric space
moduli object may be constructed as a scheme, an algebraic space, or more naturally as an algebraic stack. In many cases there is both a coarse moduli space
Moduli_of_algebraic_curves
Type of functor
In algebraic geometry, given algebraic stacks p : X → C , q : Y → C {\displaystyle p:X\to C,\,q:Y\to C} over a base category C, a morphism f : X → Y {\displaystyle
Morphism_of_algebraic_stacks
Algebraic stack in mathematics
or M e l l {\displaystyle {\mathcal {M}}_{\mathrm {ell} }} , is an algebraic stack over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb {Z} )} classifying
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite
Behrend's_trace_formula
that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by Grothendieck
Quasi-separated_morphism
Open-source textbook on algebraic geometry
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry
Stacks_Project
Concept in algebraic geometry
In algebraic geometry, the quotient space of an algebraic stack F, denoted by |F|, is a topological space which as a set is the set of all integral substacks
Quotient space of an algebraic stack
Quotient_space_of_an_algebraic_stack
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
stack. More generally, a group algebraic-space, an algebraic-space analog of a group scheme, is a group-stack. Over a field k, a vector bundle stack V
Group_stack
Concept in differential geometry
differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable
Differentiable_stack
the moduli stack of principal bundles over X, denoted by Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} , is an algebraic stack given by: for
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
mathematics, especially algebraic geometry and algebraic topology, a higher stack is a higher category generalization of a stack (a category-valued sheaf)
Higher_stack
Topics referred to by the same term
values in categories rather than sets Algebraic stack, a special kind of stack commonly used in algebraic geometry Stacks Project, an open source collaborative
Stack
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space. All algebraic spaces are assumed
Keel–Mori_theorem
Generalization of a scheme
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory
Algebraic_space
Concept in algebraic geometry
In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles (or locally free sheaves) of rank n over
Moduli stack of vector bundles
Moduli_stack_of_vector_bundles
Generalization of algebraic variety
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety in several ways, such as taking
Scheme_(mathematics)
Topics referred to by the same term
topological spaces Quotient space (linear algebra), in case of vector spaces Quotient space of an algebraic stack Quotient metric space Quotient object This
Quotient_space
In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group
Cohomology_of_a_stack
Property of a mathematical space
algebraic set (the length of such a chain is the number of " ⊊ {\displaystyle \subsetneq } "). Each variety can be considered as an algebraic stack,
Dimension
Construct in mathematics
theory of non-abelian bundle gerbes. Twisted sheaf Azumaya algebra Twisted K-theory Algebraic stack Bundle gerbe String group Basic bundle theory and K-cohomology
Gerbe
Branch of algebra that studies commutative rings
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings
Commutative_algebra
functors as either algebraic spaces or as algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves
Artin's_criterion
French mathematician (1928–2014)
λ-ring AB5 category Abelian category Accessible category Algebraic geometry Algebraic stack Approximation property – Mathematical concept Barsotti–Tate
Alexander_Grothendieck
Jacobian Moduli of algebraic curves Hurwitz's theorem on automorphisms of a curve Clifford's theorem on special divisors Gonality of an algebraic curve Weil reciprocity
List of algebraic geometry topics
List_of_algebraic_geometry_topics
American mathematician (born 1937)
foundations came with algebraic spaces (more general than schemes but more restricted than Matsusaka's Q-varieties or Mumford's stacks). Algebraic spaces allow
David_Mumford
Binary tree representing a mathematical expression
two trees are merged and a pointer to the final tree remains on the stack. Algebraic expression trees represent expressions that contain numbers, variables
Binary_expression_tree
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Algebraic geometry analog of a principal bundle in algebraic topology
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski
Torsor_(algebraic_geometry)
In algebraic geometry, a derived stack is, roughly, a stack together with a sheaf of commutative ring spectra. It generalizes a derived scheme. Derived
Derived_stack
called an algebraic groupoid, to convey the idea it is a generalization of algebraic groups and their actions. For example, suppose an algebraic group G
Groupoid_object
Concept in algebraic geometry
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 X = [ Y /
Chow_group_of_a_stack
Concept in algebraic geometry
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by
Morphism_of_schemes
Mathematical concept
{\mathfrak {X}}} is quasi-compact and quasi-separated. In fact, for the algebraic stack B G a {\displaystyle B\mathbb {G} _{a}} , there are no compact objects
Compact_object_(mathematics)
definition of a Hopf algebroidpg301-302 is its a commutative algebraic representation of an algebraic stack which can be presented as affine schemes. More generally
Hopf_algebroid
Chinese artificial intelligence company
5 7B. Further pretrain with 500B tokens (6% DeepSeekMath Corpus, 4% AlgebraicStack, 10% arXiv, 20% GitHub code, 10% Common Crawl). This produced Base.
DeepSeek
Belgian mathematician
work came to be seen as an introduction to one form of the theory of algebraic stacks, and recently has been applied to questions arising from string theory
Pierre_Deligne
Generalization of vector bundles
Quasi-coherent sheaf on an algebraic stack Mumford 1999, Ch. III, § 1, Theorem-Definition 3. Stacks Project, Tag 01LA. Stacks Project, Tag 01BU. Serre 1955
Coherent_sheaf
Algebraic geometry category satisfying lifting conditions
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying
Prestack
solution. Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms
List_of_incomplete_proofs
In computer science, algebraic semantics is a formal approach to programming language theory that uses algebraic methods for defining, specifying, and
Algebraic semantics (computer science)
Algebraic_semantics_(computer_science)
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
Toric_stack
Concept in mathematics
and right shtukas are essentially the same. By varying U, we get an algebraic stack Shtukar of shtukas of rank r, a "universal" shtuka over Shtukar×X and
Drinfeld_module
Wayback Machine There, the cotangent stack on an algebraic stack X is defined as the relative Spec of the symmetric algebra of the tangent sheaf on X. (Note:
Cotangent_sheaf
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 M
Moduli stack of formal group laws
Moduli_stack_of_formal_group_laws
German mathematician
Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and
Kai_Behrend
Mathematical set with some added structure
to describe moduli of algebraic curves. A further generalization are the algebraic stacks, also called Artin stacks. DM stacks are limited to quotients
Space_(mathematics)
than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf Q l {\displaystyle \mathbb
Smooth_topology
Branch of mathematics
non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients)
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
(noncommutative) associative algebras". Stack Exchange. May 9, 2012. "How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3
Free product of associative algebras
Free_product_of_associative_algebras
Moduli space in the Grothendieck category of schemes
require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin). Work of Grothendieck and David Mumford
Moduli_scheme
Set of mathematical conjectures proposed by Douglas Ravenel
Jack, Hall; David, Rydh (2016-06-27). "The telescope conjecture for algebraic stacks". Journal of Topology. 10 (3): 776–794. arXiv:1606.08413. doi:10.1112/topo
Ravenel's_conjectures
American mathematician (born 1934)
theorem in local algebra as well as the "Existence theorem". This work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved
Michael_Artin
Dutch mathematician
His research interests include arithmetic geometry and algebraic geometry. He maintains the Stacks Project. De Jong was born in Bruges, Belgium on 30 January
Aise_Johan_de_Jong
Performing order of mathematical operations
replaced by the use of algebraic fractions. These are most explicitly and unambiguously written "vertically" with the numerator stacked above the denominator
Order_of_operations
Ways in which keystrokes are interpreted
calculators with algebraic entry system with parentheses (AESP) support the entry of parentheses. An input scheme known as algebraic operating system
Calculator_input_methods
in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory
Hodge_bundle
mathematics, especially in differential and algebraic geometries, an inertia stack of a groupoid X is a stack that parametrizes automorphism groups on X
Inertia_stack
Mathematical model for data types
It has a mathematical foundation in universal algebra. Formally, an ADT is analogous to an algebraic structure in mathematics, consisting of a domain
Abstract_data_type
In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category Coh G ( X ) {\displaystyle \operatorname {Coh}
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
theory) Toën, Bertrand (2002), "Stacks and Non-abelian cohomology" (PDF), Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian
Simplicial_presheaf
Seminal math text
definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of
Pursuing_Stacks
In algebraic geometry, Sumihiro's theorem, introduced by (Sumihiro 1974), states that a normal algebraic variety with an action of a torus can be covered
Sumihiro's_theorem
1960–67 foundational treatise on algebraic geometry by Alexander Grothendieck
French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published
Éléments de géométrie algébrique
Éléments_de_géométrie_algébrique
Mathematical concept that extends the intuitive idea of gluing in topology
was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract
Descent_(mathematics)
Mapping theorem in topology
compact supports. The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields. Fixed-point theorems Lefschetz zeta function Holomorphic
Lefschetz_fixed-point_theorem
articles Free/GNU Stacks Project English A mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed
List_of_online_encyclopedias
Restriction of scalars
{\displaystyle T\to S} of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties
Weil_restriction
1. Khan 2023, Definition 3.2.3. Khan, Adeel A. (2023), Lectures on Algebraic Stacks (PDF) https://ncatlab.org/nlab/show/simplicial+diagram https://ncatlab
Simplicial_diagram
Expresses the number of points of a variety over a finite field
Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology. One application of the Grothendieck
Grothendieck_trace_formula
Theorem about cohomology rings
MR 0051508. Behrend, Kai A. (2003). "Derived l-adic categories for algebraic stacks". Memoirs of the American Mathematical Society. 163 (774). doi:10.1090/memo/0774
Borel's_theorem
Subset (often algebraic set) that is not the union of subsets of the same nature
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic
Irreducible_component
Generalization of category
003. ISBN 978-1-139-54233-3. Khan, Adeel A. (2023). "Lectures on algebraic stacks". arXiv:2310.12456 [math.AG]. Kelly, G. M.; Street, Ross (1974). "Review
2-category
[arXiv:2204.00295] Khan, Adeel A. (2022). "A modern introduction to algebraic stacks". "Kerodon". Mazel-Gee, Aaron (2015). "A user's guide to co/cartesian
Cartesian_fibration
Theorem in geometry
algebraic geometry: Introduction to intersection theory in algebraic geometry https://mathoverflow.net/questions/25218/why-is-riemann-roch-for-stacks-so-hard
Riemann–Roch-type_theorem
html Khan 2023, Remark 4.2.4. Khan, Adeel A. (2023), Lectures on Algebraic Stacks (PDF) https://ncatlab.org/nlab/show/action+groupoid https://mathoverflow
Action_groupoid
Basic concepts of algebra
relationships in science and mathematics are expressed as algebraic equations. In mathematics, a basic algebraic operation is a mathematical operation similar to
Elementary_algebra
Type of Abelian category (in category theory in mathematics)
Given an (affine or projective) algebraic variety V {\displaystyle V} (or more generally: any scheme or algebraic stack), the category Qcoh ( V ) {\displaystyle
Grothendieck_category
American mathematician and professor
geometry of Deligne-Mumford stacks. In: Abramovich, D; Bertram, A; Katzarkov, L; Pandharipande, R; Thaddeus, M. Algebraic Geometry: Seattle 2005. Providence
Andrew_Kresch
approach to derived algebraic geometry" and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine. Just as affine algebraic geometry is equivalent
Derived_scheme
Topological space in which closed subsets satisfy the descending chain condition
sets must eventually be constant. A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain
Noetherian_topological_space
Compiler optimization technique
operations – replace several operations with one equivalent. Algebraic laws – use algebraic laws to simplify or reorder instructions. Special-case instructions
Peephole_optimization
Concept in mathematical category theory
(PDF). hdl:1773/20977 – via nlab. Khan, Adeel A. (2023), Lectures on Algebraic Stacks (PDF), arXiv:2310.12456 Vistoli, Angelo (September 2, 2008). "Notes
Category_of_elements
Theorem relating to algebraic topology
Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory
Landweber exact functor theorem
Landweber_exact_functor_theorem
Algebraic invariant of topological spaces
mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. Historically
Elliptic_cohomology
Construct in algebraic geometry
ringed spaces, schemes, and algebraic spaces into the theory. Suppose that X {\displaystyle X} and Y {\displaystyle Y} are algebraic varieties and that f :
Cotangent_complex
Computer algebra system
Coordinating Facility. Like most computer algebra systems, Maxima supports a variety of ways of reorganizing symbolic algebraic expressions, such as polynomial
Maxima_(software)
System which describes the computational effects of computer programs
language with algebraic effect handlers as a main feature. Eff is a statically typed functional programming language centered around algebraic effect handlers
Effect_system
Conjectures connecting number theory and geometry
structure of Galois groups in algebraic number theory to automorphic forms and, more generally, the representation theory of algebraic groups over local fields
Langlands_program
In algebraic geometry and complex geometry, the Harder–Narasimhan stratification is any of a stratification of the moduli stack of principal G-bundles
Harder–Narasimhan stratification
Harder–Narasimhan_stratification
In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the
Functor represented by a scheme
Functor_represented_by_a_scheme
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