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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
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack 𝔛 is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description
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
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
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
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, the cohomology of a stack is a collection of cohomological theories associated with algebraic stacks, extending familiar constructions
Cohomology_of_a_stack
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
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
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
Topics referred to by the same term
Bratton Stack (mathematics), a sheaf that takes values in categories rather than sets Algebraic stack, a special kind of stack commonly used in algebraic geometry
Stack
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
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
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)
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
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
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
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
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
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
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
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
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
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
Property of a mathematical space
sub-varieties of the given algebraic set (the length of such a chain is the number of "⊊"). Each variety can be considered as an algebraic stack, and its dimension
Dimension
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
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
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
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
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
mathematics, especially algebraic geometry and algebraic topology, a higher stack is a higher category generalization of a stack (a category-valued sheaf)
Higher_stack
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)
French mathematician (1928–2014)
λ-ring AB5 category Abelian category Accessible category Algebraic geometry Algebraic stack Approximation property – Mathematical concept Barsotti–Tate
Alexander_Grothendieck
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
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
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
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
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
an algebraic stack with coefficients in, say, the étale sheaf ℚl. To understand the problem that motivates the notion, consider the classifying stack B𝔾m
Smooth_topology
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
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
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)
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
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
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
Series of language models developed by DeepSeek
5 7B. Further pretrain with 500B tokens (6% DeepSeekMath Corpus, 4% AlgebraicStack, 10% arXiv, 20% GitHub code, 10% Common Crawl). This produced Base.
DeepSeek_(AI)
Moduli space in the 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
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
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
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
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
Algebraic geometry category satisfying lifting conditions
groupoids; e.g., an algebraic stack (since all morphisms are cartesian then.) If p : F S → C {\displaystyle p:F_{S}\to C} is the stack associated to a scheme
Prestack
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
Belgian mathematician (born 1944)
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
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)
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
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
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
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
nuclear C*-algebra is a C*-algebra A such that for every C*-algebra B the injective and projective C*-cross norms coincides on the algebraic tensor product
Nuclear_C*-algebra
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)
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
(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
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
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
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
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)
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
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
[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
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
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
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
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
Concept in algebraic geometry
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally
Normal_scheme
Theory of cohomology for commutative rings
In commutative algebra and algebraic geometry, André–Quillen cohomology is a cohomology theory for commutative rings and their modules, constructed using
André–Quillen_cohomology
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
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
theory) Toën, Bertrand (2002), "Stacks and Non-abelian cohomology" (PDF), Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian
Simplicial_presheaf
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
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
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
Object in algebraic geometry
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
Stacky_curve
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
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
Type of integrable system
moduli stack of Hitchin pairs, instead of the moduli space. Let g {\displaystyle {\mathfrak {g}}} be the Lie algebra of the reductive algebraic group G
Hitchin_system
for numerical and scientific computing. Packt Publishing Ltd. "Xtensor-stack/Xtensor". GitHub. 13 February 2022. scipy on GitHub armadillo on GitHub
Comparison of linear algebra libraries
Comparison_of_linear_algebra_libraries
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
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
essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by Joe Buhler
Essential_dimension
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
Generalizations of codimension-1 subvarieties of algebraic varieties
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common
Divisor_(algebraic_geometry)
On the cohomology ring of the moduli stack of principal bundles
{Bun} _{G}(X),\mathbb {Q} _{l})} of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The original
Atiyah–Bott_formula
travel, tourism, insurance
ALGEBRAIC STACK
ALGEBRAIC STACK
Surname or Lastname
English
English : habitational name from Stockhow in Cumbria, first attested in 1581 as Stackay.
Boy/Male
Gujarati, Indian, Jain, Marathi
Lotus Stack
Surname or Lastname
English
English : topographic name for someone who lived near a heap of some kind, from Middle English reke ‘stack’, ‘heap’.German : from Radeke, a pet form of a Germanic personal name formed with rÄd ‘advice’, ‘counsel’.Altered spelling of German Reeck.
Surname or Lastname
German (of Slavic origin)
German (of Slavic origin) : from a pet form of the personal name Pavel or Paweł, respectively the Czech and Polish forms of Paul, or from a Sorbian cognate.German (of Slavic origin) : nickname for a small man, from Slavic palac ‘thumb’.Irish : MacLysaght ascribes the origin of this surname in Ireland to the arrival there in the 15th century of a Lombard family of bankers named de Palatio.English : from Old French palis, paleis ‘palisade’, ‘fence’, hence a topographic name for someone who lived by a palisade or a metonymic occupational name for a maker of fences.English : possibly a metonymic occupational name for someone who worked at a palace (bishop’s, archbishop’s, or royal), from Old French, Middle English palais, paleis.English : metonymic occupational name for a worker at a straw stack, from Old French paille ‘straw’ + Middle English hous ‘house’.Greek : ornamental name or nickname from Albanian pallë ‘sword’.Catalan (Pallà s) : variant spelling of Pallars, a regional name from the Catalan district of Pallars, in the Pyrenees.
Girl/Female
Tamil
Lotus stack
Surname or Lastname
English (mainly West Midlands)
English (mainly West Midlands) : probably a habitational name from a place so named in North Yorkshire.
Surname or Lastname
English
English : nickname for a large, well-built man, from Middle English stack ‘haystack’ (from Old Norse stakkr). The surname is now less common in England than in Ireland (especially County Kerry), where it was first taken in the 13th century; it has been Gaelicized Stac.German : variant of Staack.Americanized form of Polish or Czech Stach.
Surname or Lastname
Norwegian
Norwegian : habitational name from any of several farmsteads, so named from Old Norse hlað ‘pile or stack’ (for example, of wood or stones) or ‘pavement’.North German : short form of Ladwig, a variant of Ludwig.English : topographic name for someone living by a road, path, or watercourse, Middle English lade, lode (Old English (ge)lÄd).
Surname or Lastname
English
English : variant or patronymic form of Stack.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Lotus Stack
Surname or Lastname
English
English : topographic name for someone who lived at a place where wood was stacked, from Old English wudu ‘wood’ + fīn ‘pile’.
Girl/Female
Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Lotus Stack; Intelligent; Princess
ALGEBRAIC STACK
ALGEBRAIC STACK
ALGEBRAIC STACK
ALGEBRAIC STACK
ALGEBRAIC STACK
ALGEBRAIC STACK
ALGEBRAIC STACK
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