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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 spaces or schemes
general theory of stacks, the notion of algebraic stacks was defined by Michael Artin. One of the motivating examples of an algebraic stack is to consider
Algebraic_stack
Type of object in algebraic geometry
space of algebraic curves Quotient stack Stacky curve "Deligne–Mumford stacks", The Stacks project, The Stacks Project Authors Deligne, Pierre; Mumford
Deligne–Mumford_stack
Topics referred to by the same term
Channing "Stacks" Lorenzo (born 1997), an American professional wrestler Parnell "Stacks" Edwards, a key associate in the Lufthansa heist Robert Stack Pierce
Stack
Dutch mathematician
include arithmetic geometry and algebraic geometry. He maintains the Stacks Project. De Jong was born in Bruges, Belgium on 30 January 1966. He attended
Aise_Johan_de_Jong
Cloud computing software
services. OpenStack began in 2010 as a joint project of Rackspace Hosting and NASA. As of 2012[update], it was managed by the OpenStack Foundation, a
OpenStack
(004V)—The Stacks project". "Lemma 5.8.3 (004W)—The Stacks project". "Section 5.9 (0050): Noetherian topological spaces—The Stacks project". Hart, Klaas
Hyperconnected_space
spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05. "Section 17.8 (01B1): Modules locally generated by sections—The Stacks project". stacks
Closed_immersion
American professional wrestler (born 1997)
to WWE where he performs on the NXT brand under the ring name Channing "Stacks" Lorenzo and is a member of the BirthRight stable. He is a former two-time
Channing_"Stacks"_Lorenzo
Term in algebraic geometry
1. Liu (2002), Lemma 3.3.17. Stacks Project, Tag 02YJ. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; Stacks Project, Tag 02LQ. Grothendieck, EGA IV
Proper_morphism
Concept in algebraic geometry
MR 0463157 p. 91 The Stacks Project Authors. "Tag 033H: Normal schemes". The Stacks project. Retrieved 2026-09-29. The Stacks Project Authors. "Section 29
Normal_scheme
Generalisation of a sheaf; a fibered category that admits effective descent
Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced
Stack_(mathematics)
Search engine based on Apache Lucene
alongside Logstash, Kibana, and Beats as part of the Elastic Stack (formerly the ELK Stack). Shay Banon created the precursor to Elasticsearch, called
Elasticsearch
Generalizations of codimension-1 subvarieties of algebraic varieties
cone, above. Dieudonné (1985), section VI.6. Stacks Project, Tag 00PF. Stacks Project, Tag 02MC. Stacks Project, Tag 02MD. Kollár (2013), Notation 1.2. Hartshorne
Divisor_(algebraic_geometry)
Construction in algebraic geometry
Stacks Project, Tag 020D. Grothendieck, EGA I, Théorème 3.2.6; Hartshorne (1977), Theorem II.3.3. Hartshorne (1977), section II.3. Stacks Project, Tag
Fiber_product_of_schemes
Algebraic geometry scheme
Birational Geometry of Algebraic Varieties, Cambridge University Press, ISBN 0-521-63277-3, MR 1658959 The Stacks Project Authors, The Stacks Project
Gorenstein_scheme
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, §13
Coherent_sheaf
Mathematics text
"online resource for homotopy-coherent mathematics" inspired by the Stacks Project. Higher Topos Theory covers two related topics: ∞-categories and ∞-topoi
Higher_Topos_Theory
Module over a ring
Torsion-free modules, The University of Chicago Press, Chicago-London, ISBN 978-0-226-51074-3, MR 0344237 The Stacks Project Authors, The Stacks Project
Torsion-free_module
algébrique 0I, 1.7.1. The Stacks Project authors (2017). Stacks Project, Tag 01B4. The Stacks Project authors (2017). Stacks Project, Tag 01AS. Eisenbud, David
Support_of_a_module
In fact, this is equivalent to condition (2) above. Stacks 01XF. Serre (1957). Stacks 01XF. Stacks 01XE, Lemma 29.3.2. Hartshorne, Robin (1977), Algebraic
Serre's_theorem_on_affineness
Concept in algebraic geometry
Stacks project" Stacks Project, Tag 01AM Stacks Project, Tag 01PS Stacks Project, Tag 01QE Stacks Project, Tag 01VU Stacks Project, Tag 02NP Stacks Project
Ample_line_bundle
See the cited Stacks Project page for more details. "Section 28.9 (02IR): Regular schemes". stacks.math.columbia.edu. The Stacks project. Retrieved 2022-02-18
Regular_scheme
Type of functor
stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism. § 8.6 of F. Meyer, Notes on algebraic stacks Stacks
Morphism_of_algebraic_stacks
Construction in homological algebra
"History of homological algebra", History of topology (PDF), Amsterdam: North-Holland, pp. 797–836, MR 1721123 The Stacks Project Authors, The Stacks Project
Tor_functor
Cloud computing software and services company
which OpenStack components to run. Project Murano, an OpenStack application catalog project developed by Mirantis and other members of the OpenStack community
Mirantis
Concept in algebraic geometry
Stacks project". stacks.math.columbia.edu. Retrieved 2020-02-23. Vakil. "FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 35 AND 36" (PDF). Stacks Project,
Coherent_sheaf_cohomology
Theorem in algebraic geometry
5; Stacks Project, Tag 0A9X. Hartshorne (1977), Theorem III.7.11; Stacks Project, Tag 0BQZ. Hartshorne (1966), Corollary VII.3.4(c); Stacks Project, Tag
Serre_duality
Category in mathematics
(1994), Definition 10.2.6. Neeman (2001), Remark D.1.5. Stacks Project, Tag 09IS, Stacks Project, Tag 09M1. Neeman (2001), Lemma D.1.3. Neeman (1996), Theorems
Triangulated_category
Concept in algebraic geometry
28.5.7 (0BA8)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. "Lemma 28.5.8 (01P0)—The Stacks project". stacks.math.columbia.edu
Noetherian_scheme
Geometric space whose points represent algebro-geometric objects of some fixed kind
(01NE)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-09-12. "algebraic geometry - What does projective space classify?". Mathematics Stack Exchange
Moduli_space
Type of Abelian category (in category theory in mathematics)
arXiv:1411.0331, doi:10.1007/s00029-016-0263-9, MR 3624905 Stacks Project, Tag 077P; Stacks Project, Tag 0781. Faith, Carl (1973). Algebra: Rings, Modules
Grothendieck_category
Bitcoin smart contract platform
In October 2020 the community rebranded Blockstack to Stacks. With the introduction of the Stacks Blockchain 2.0 on 14 January 2021, Hiro PBC revoked their
Stacks_blockchain
Cloud-based online integrated development environment for web development
common frameworks, and a cloud-based development container for full-stack projects. AI code completion Angular Assemblyline4 Astro Astro Starlight Documentation
CodeSandbox
In algebra, integer associated to a module
2013-03-02. Retrieved 2020-05-22. Alt URL "Lemma 10.51.2 (02LZ)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-22. Fulton, William, 1939-
Length_of_a_module
vol. 123, Gauthier-Villars, MR 0054995 The Stacks Project Authors. "Stacks Project — Tag 0323". stacks.math.columbia.edu. Retrieved 2018-08-13. v t
Cohen_structure_theorem
Generalization of algebraic variety
viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified. Grothendieck originally introduced stacks as a tool
Scheme_(mathematics)
Topological space in which closed subsets satisfy the descending chain condition
Mathematics Stack Exchange. "Lemma 5.9.3 (04Z8)—The Stacks project". stacks.math.columbia.edu. "Lemma 5.9.4 (0053)—The Stacks project". stacks.math.columbia
Noetherian_topological_space
Construct in mathematics
target—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-27. "Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project". stacks.math
Gerbe
In mathematics, a mapping between categories
modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20. "Section 48.3 (0A9D): Right adjoint of pushforward—The Stacks project". stacks.math
Direct_image_functor
Singularities of algebraic varieties
Geometry (2 vols.), Berlin: Springer-Verlag, doi:10.1007/978-3-642-18808-4, ISBN 3-540-22533-1, MR 2095471 The Stacks Project Authors, The Stacks Project
Normal_crossing_singularity
Mathematics notation where operators follow operands
of just dropping values off the stack on overflow as with fixed-sized stacks. It also changed the behaviour of the stack to no longer duplicate the top
Reverse_Polish_notation
Concept in commutative algebra
G-rings—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. "Section 15.46 (07P6): The singular locus—The Stacks project". stacks.math.columbia
Excellent_ring
Federal cloud computing platform
research projects. One application Open Sourced by NASA and developed by the Nebula project, Nova, became one of the two founding projects of the OpenStack project
Nebula_(computing_platform)
1090/CONM/388. ISBN 9780821837191. The Stacks Project authors. "30.24 Grothendieck's existence theorem, I". The Stacks Project authors. "75.42 Grothendieck's
Grothendieck existence theorem
Grothendieck_existence_theorem
Subset (often algebraic set) that is not the union of subsets of the same nature
(004U): Irreducible components—The Stacks project". "Section 5.8 (004U): Irreducible components—The Stacks project". This article incorporates material
Irreducible_component
Software company based in Canada
over by Google in 2014. Gridcentric supported the OpenStack project and was used by several OpenStack based cloud computing companies including Piston Cloud
Gridcentric
Correspondence between subfields and subgroups
theory at PlanetMath. The Stacks Project authors. "Theorem 9.21.7 (Fundamental theorem of Galois theory)". The Stacks Project authors. "Theorem 9.22.4
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
In algebra, completion w.r.t. powers of an ideal
Quasi-unmixed ring "Stacks Project — Tag 05GG". stacks.math.columbia.edu. Retrieved 2025-05-14. "Stacks Project — Tag 0316". stacks.math.columbia.edu.
Completion_of_a_ring
Data structure used in computer programs
computer instruction sets provide special instructions for manipulating stacks. A call stack is used for several related purposes, but the main reason for having
Call_stack
Stacks Project - Nagata compactification - See Lemma 38.33.8 first, then backtrack Stacks Project - Derived lower shriek via compactifications Stacks
Nagata's compactification theorem
Nagata's_compactification_theorem
Mathematical construction
1017/CBO9780511661761. ISBN 9780521207843. stalk in nLab The Stacks Project authors. "6.11 Stalks". The Stacks Project authors. "6.27 Skyscraper sheaves and stalks"
Stalk_(sheaf)
Construction in homological algebra
needed] Koszul–Tate complex Syzygy (mathematics) The Stacks Project, section 0601 The Stacks Project, section 0601, Lemma 15.28.12 Indeed, by linearity
Koszul_complex
Mathematical concept
"Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project". Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics
Gluing_schemes
Programming paradigm that relies on a stack machine model
Stack-oriented programming is a programming paradigm that relies on one or more stacks to manipulate data and/or pass parameters. Programming constructs
Stack-oriented_programming
Mathematical group
of number fields". Mathematics Stack Exchange. Retrieved 2020-11-11. "9.22 Infinite Galois theory". The Stacks project. Milne. "Field Theory" (PDF). p
Galois_group
ISBN 978-0-387-90244-9, MR 0463157 The Stacks Project authors. "28 Properties of Schemes/28.10 Dimension". The Stacks Project authors. "29.29 Morphisms of given
Dimension_of_a_scheme
Set of software subsystems or components needed to create a complete platform
platforms like npm. The increasing popularity of these stacks reflects a broader shift toward full-stack JavaScript development in modern web engineering.
Solution_stack
Open-source cloud computing software
integration and support of Windows Server 2008 R2 Hyper-V to the OpenStack project. Citrix Systems purchased Cloud.com on July 12, 2011, for approximately
Apache_CloudStack
irreducible scheme.[citation needed] The Stacks project, Tag 02D4 The Stacks project, Tag 01OK The Stacks project, Tag 01ON The Stacks project, Tag 01J2
Irreducible_ring
Property of functions which is weaker than continuity
MR 0163911. "Stacks Project — Lemma 37.30.4". Stacks.math.columbia.edu. Retrieved 2025-04-27. "Stacks Project — Lemma 37.30.5". Stacks.math.columbia
Semi-continuity
Function that preserves distinctness
"Section 7.3 (00V5): Injective and surjective maps of presheaves". The Stacks project. Retrieved 2019-12-07. Farlow, S. J. "Section 4.2 Injections, Surjections
Injective_function
Construct in algebraic geometry
(08UX): Deformations of ringed spaces and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02. "Section 91.23 (08V3):
Cotangent_complex
Mathematical concept that extends the intuitive idea of gluing in topology
flat descent Monadic descent Descent data for quasi-coherent sheaves, Stacks Project SGA 1, Ch VIII – this is the main reference Siegfried Bosch; Werner
Descent_(mathematics)
Result in category theory
The 2-Yoneda lemma—The Stacks project". stacks.math.columbia.edu. "Lemma 4.41.2 (2-Yoneda lemma)—The Stacks project". stacks.math.columbia.edu. "2.2
2-Yoneda_lemma
Properties of mathematical functions
"Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2019-12-07. "Earliest Known Uses of
Bijection, injection and surjection
Bijection,_injection_and_surjection
Chinese rapper (born 1998)
2024, Skai Isyourgod released his second album, Stacks from All Sides (八方来财). Songs such as "Stacks from All Sides", "Karma Code" (因果), and "Blueprint
Skai_Isyourgod
South Korean co-ed music group
AllDay Project (Korean: 올데이 프로젝트, stylized in all caps or abbreviated to ADP) is a South Korean co-ed group formed and managed by The Black Label. The
AllDay_Project
Direct summand of a free module (mathematics)
(help) "Section 10.95 (05A4): Descending properties of modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-11-03. Exercises 4.11 and 4.12
Projective_module
Mathematical concept
{A/B}})\to 0} Section 12.10 The Stacks Project "109.76 The category of modules modulo torsion modules". The Stacks Project. Gabriel, Pierre, Des categories
Quotient of an abelian category
Quotient_of_an_abelian_category
proven in 1910 by Ernst Steinitz. Lemma 9.19.1 (Primitive element), The Stacks project. Accessed on line July 19, 2023. Steinitz, Ernst (1910). "Algebraische
Steinitz's theorem (field theory)
Steinitz's_theorem_(field_theory)
Railroad cars carrying two layers of intermodal containers
required to securely mate container stacks. China had started[when?] to experiment using reduced-size containers to be stacked on to normal containers to allow
Double-stack_rail_transport
Open-source analytics and visualization platform
backend and infrastructure telemetry in the LGTM stack for full-stack observability. The Faro project was announced by Grafana Labs in 2022 and is released
Grafana
MR 0238860. "Lemma 37.11.7 (02H6): Infinitesimal lifting criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-07. Formally smooth with smooth
Formally_smooth_map
cloud system was built on the OpenStack open source cloud framework, as well as many other open source software projects. Nebula was founded as Fourth Paradigm
Nebula_(company)
Category theory concept
[math.CT]. "Section 4.32 (02XG): Categories over categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-16. Lurie, Jacob (2008-07-31)
Overcategory
ix+269. MR 0255390. https://stacks.math.columbia.edu/tag/04ZC Topological definition of (local) constructibility https://stacks.math.columbia.edu/tag/054H
Constructible_set_(topology)
discrete valuation ring. EGA II, proposition 7.2.3 and théorème 7.3.8. Stacks Project, tags 01KA, 01KY, and 0BX4. Grothendieck, Alexandre; Jean Dieudonné
Valuative_criterion
Haskell development tool
Stack is a tool to build projects and manage their dependencies for the programming language Haskell. It uses the Cabal library but with a curated version
Stack_(Haskell)
Concept in algebraic geometry
28.6 (0663)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-01. "Lemma 15.28.13 (0913)—The Stacks project". stacks.math.columbia.edu
Local_cohomology
Scheme parametrizing flags in the fibers of a vector bundle
arXiv:1510.07852. doi:10.1142/S0129167X1750077X. The Stacks Project Authors. "Grassmannians". The Stacks Project. Retrieved 18 July 2026. Berthelot, Pierre; Grothendieck
Flag_bundle
Faithfully flat morphism of schemes
descent theory" (PDF). arXiv:math/0412512. Bibcode:2004math.....12512V. Stacks Project, "The fpqc Topology." http://stacks.math.columbia.edu/tag/03NV
Fpqc_morphism
Wash., 1972). Lecture Notes in Math. Vol. 341. Berlin: Springer. pp. 266–292. MR 0347943. A spectral sequence of Quillen at the Stacks Project v t e
Quillen_spectral_sequence
Differential form in commutative algebra
an enhancement of the de Rham complex for the ring of Witt vectors. "Stacks Project". Retrieved 2022-11-21. Hartshorne (1977, p. 172) Laurent-Gengoux, C
Kähler_differential
Non-profit educational organization
challenges), “Ziplines” (front-end projects), and “Basejumps” (full-stack projects). Completing the front-end and full-stack projects awarded the student with respective
FreeCodeCamp
Manmade calcium sulfate hydrate by-product
applications. As a result, there are about a billion tons of phosphogypsum stacked in 25 stacks in Florida (22 are in central Florida) and about 30 million additional
Phosphogypsum
Integrated development environment from Microsoft
preview build was released shortly thereafter. On November 18, 2015, the project "Visual Studio Code – Open Source" (also known as "Code – OSS"), on which
Visual_Studio_Code
Type of morphism in algebraic geometry
Théorème 8.11.1. "Lemma 02LS". The Stacks Project. Retrieved 31 January 2022. "Definition 29.15.1". The Stacks Project. Retrieved 15 August 2023. EGA IV3
Quasi-finite_morphism
Ruled surface over the projective line
20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-23. Cox, David A.; Little
Hirzebruch_surface
Branch of mathematics
generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer
Algebraic_geometry
archive.org/web/20121108104319/http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf A book worked out by many authors. Glossary of arithmetic and Diophantine
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Analogs of homology groups for algebraic varieties
Fulton. Intersection Theory, section 1.2 and Appendix A.3. Stacks Project, https://stacks.math.columbia.edu/tag/0BE9 Fulton, Intersection Theory, section
Chow_group
Type of sheaf of modules
divisor. First Chern class Birkhoff–Grothendieck theorem EGA 0I, 5.4. Stacks Project, tag 01CR, [1]. Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments
Invertible_sheaf
United States historic place
later rebuilt). The main entrance to the Gardner (Main) Stacks is through Doe, but the stacks are also connected to nearby Moffitt Library by means of
Doe_Memorial_Library
Algebraic stack in mathematics
American Mathematical Society, ISBN 978-1470427986 moduli+stack+of+elliptic+curves at the nLab "The moduli stack of elliptic curves", Stacks project
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Mathematical concept
18 (0FNB): Double complexes and associated total complexes—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-07-08. Weibel, Charles A. (1994)
Double_complex
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
ISBN 978-1-4008-7123-0 "Section 97.9 (0CZX): The Hilbert functor—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-06-17. "3264 and all that" (PDF)
Hilbert_scheme
Free general-purpose operating system
(/ˈdɛbiən/) is a free, general-purpose operating system developed by the Debian Project, a worldwide volunteer association founded by Ian Murdock on August 16
Debian
alternative definition of pseudo-coherent complex". MathOverflow. "15.74 Perfect complexes". The Stacks project. "perfect module". ncatlab.org. v t e
Perfect_complex
functor. "Section 5.17 (005M): Characterizing proper maps—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-25. Schnürer, Olaf M.; Soergel
Direct image with compact support
Direct_image_with_compact_support
travel, tourism, insurance
STACKS PROJECT
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travel, tourism, insurance