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presentation—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29. "Section 34.7 (021L): The fppf topology—The Stacks project". stacks.mathSerre duality (3,295 words) [view diff] exact match in snippet view article find links to article
5; Stacks Project, Tag 0A9X. Hartshorne (1977), Theorem III.7.11; Stacks Project, Tag 0BQZ. Hartshorne (1966), Corollary VII.3.4(c); Stacks Project, TagCoherent sheaf (6,934 words) [view diff] exact match in snippet view article find links to article
stack Mumford 1999, Ch. III, § 1, Theorem-Definition 3. Stacks Project, Tag 01LA. Stacks Project, Tag 01BU. Serre 1955, §13 Grothendieck & Dieudonné 1960Fiber product of schemes (1,178 words) [view diff] exact match in snippet view article find links to article
Stacks Project, Tag 020D. Grothendieck, EGA I, Théorème 3.2.6; Hartshorne (1977), Theorem II.3.3. Hartshorne (1977), section II.3. Stacks Project, TagHyperconnected space (1,807 words) [view diff] case mismatch in snippet view article find links to article
(004V)—The Stacks project". "Lemma 5.8.3 (004W)—The Stacks project". "Section 5.9 (0050): Noetherian topological spaces—The Stacks project". Hart, KlaasAmple line bundle (6,874 words) [view diff] exact match in snippet view article find links to article
Stacks project" Stacks Project, Tag 01AM Stacks Project, Tag 01PS Stacks Project, Tag 01QE Stacks Project, Tag 01VU Stacks Project, Tag 02NP Stacks ProjectDivisor (algebraic geometry) (6,612 words) [view diff] exact match in snippet view article
cone, above. Dieudonné (1985), section VI.6. Stacks Project, Tag 00PF. Stacks Project, Tag 02MC. Stacks Project, Tag 02MD. Kollár (2013), Notation 1.2. HartshorneNoetherian scheme (1,308 words) [view diff] case mismatch in snippet view article find links to article
"Lemma 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.columbiaIrreducible ring (564 words) [view diff] case mismatch in snippet view article find links to article
irreducible scheme.[citation needed] The Stacks project, Tag 02D4 The Stacks project, Tag 01OK The Stacks project, Tag 01ON The Stacks project, Tag 01J2Support of a module (904 words) [view diff] exact match in snippet view article find links to article
algébrique 0I, 1.7.1. The Stacks Project authors (2017). Stacks Project, Tag 01B4. The Stacks Project authors (2017). Stacks Project, Tag 01AS. Eisenbud, DavidTorsion-free module (597 words) [view diff] exact match in snippet view article find links to article
Torsion-free modules, The University of Chicago Press, Chicago-London, ISBN 978-0-226-51074-3, MR 0344237 The Stacks Project Authors, The Stacks ProjectTor functor (2,068 words) [view diff] exact match in snippet view article find links to article
"History of homological algebra", History of topology (PDF), Amsterdam: North-Holland, pp. 797–836, MR 1721123 The Stacks Project Authors, The Stacks ProjectCoherent sheaf cohomology (4,664 words) [view diff] exact match in snippet view article find links to article
exists. (Hartshorne 1977, (III.1.1A) and section III.2.) Stacks Project, Tag 01X8. Stacks Project, Tag 01XE. (Hartshorne 1977, Theorem III.5.1.) (HartshorneAnnihilator (ring theory) (2,160 words) [view diff] case mismatch in snippet view article
"Lemma 10.39.5 (00L2)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-13. "Lemma 10.39.9 (00L3)—The Stacks project". stacks.math.columbiaGorenstein scheme (679 words) [view diff] exact match in snippet view article find links to article
Birational Geometry of Algebraic Varieties, Cambridge University Press, ISBN 0-521-63277-3, MR 1658959 The Stacks Project Authors, The Stacks ProjectGrothendieck category (2,491 words) [view diff] exact match in snippet view article find links to article
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, ModulesCohen structure theorem (371 words) [view diff] exact match in snippet view article find links to article
local ring. All this material is developed carefully in the Stacks Project "Stacks Project — Tag 0323". stacks.math.columbia.edu. Retrieved 2018-08-13Irreducible component (951 words) [view diff] case mismatch in snippet view article find links to article
(004U): Irreducible components—The Stacks project". "Section 5.8 (004U): Irreducible components—The Stacks project". This article incorporates materialLength of a module (2,157 words) [view diff] case mismatch in snippet view article find links to article
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-Direct image functor (969 words) [view diff] case mismatch in snippet view article find links to article
modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20. "Section 48.3 (0A9D): Right adjoint of pushforward—The Stacks project". stacksGrothendieck existence theorem (325 words) [view diff] exact match in snippet view article find links to article
1090/CONM/388. ISBN 9780821837191. The Stacks Project authors. "30.24 Grothendieck's existence theorem, I". The Stacks Project authors. "75.42 Grothendieck'sScheme (mathematics) (7,139 words) [view diff] exact match in snippet view article
David Mumford, Can one explain schemes to biologists? The Stacks Project Authors, The Stacks Project https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/Completion of a ring (1,582 words) [view diff] exact match in snippet view article find links to article
topology Quasi-unmixed ring "Stacks Project — Tag 05GG". stacks.math.columbia.edu. Retrieved 2025-05-14. "Stacks Project — Tag 0316". stacks.math.columbiaExcellent ring (1,465 words) [view diff] case mismatch in snippet view article find links to article
G-rings—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. "Section 15.46 (07P6): The singular locus—The Stacks project". stacks.mathDimension of a scheme (1,254 words) [view diff] exact match in snippet view article find links to article
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 givenQuotient of an abelian category (1,640 words) [view diff] exact match in snippet view article find links to article
{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 categoriesStalk (sheaf) (1,588 words) [view diff] exact match in snippet view article
1017/CBO9780511661761. ISBN 9780521207843. stalk in nLab The Stacks Project authors. "6.11 Stalks". The Stacks Project authors. "6.27 Skyscraper sheaves and stalks"Triangulated category (5,798 words) [view diff] exact match in snippet view article find links to article
(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), TheoremsGerbe (3,462 words) [view diff] case mismatch in snippet view article find links to article
target—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-27. "Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project". stacksDescent (mathematics) (927 words) [view diff] exact match in snippet view article
descent Faithfully flat descent Descent data for quasi-coherent sheaves, Stacks Project SGA 1, Ch VIII – this is the main reference Siegfried Bosch; WernerUniversity of Florida Digital Collections (759 words) [view diff] exact match in snippet view article find links to article
Judaica Digital Collections Natural Sciences Digital Collection Open Stacks Project Performing Arts Digital Collection Psychological Study of the Arts DigitalFundamental theorem of Galois theory (3,001 words) [view diff] exact match in snippet view article find links to article
theory at PlanetMath. The Stacks Project authors. "Theorem 9.21.7 (Fundamental theorem of Galois theory)". The Stacks Project authors. "Theorem 9.22.4Finite morphism (855 words) [view diff] exact match in snippet view article find links to article
Shafarevich 2013, p. 62, Def. 1.2. Hartshorne 1977, Section II.3. Stacks Project, Tag 01WG. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4. GrothendieckOvercategory (756 words) [view diff] case mismatch in snippet view article find links to article
09375 [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)Injective function (2,558 words) [view diff] case mismatch in snippet view article find links to article
"Section 7.3 (00V5): Injective and surjective maps of presheaves". The Stacks project. Retrieved 2019-12-07. Farlow, S. J. "Section 4.2 Injections, SurjectionsConstructible topology (260 words) [view diff] case mismatch in snippet view article find links to article
authors prefer the term quasicompact here. "Lemma 5.23.8 (0905)—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20. "Reconciling two differentValuative criterion (277 words) [view diff] exact match in snippet view article find links to article
discrete valuation ring. EGA II, proposition 7.2.3 and théorème 7.3.8. Stacks Project, tags 01KA, 01KY, and 0BX4. Grothendieck, Alexandre; Jean DieudonnéDecentralized application (1,971 words) [view diff] case mismatch in snippet view article find links to article
processing and exposed the scaling limitations of public blockchains. Stacks project – a platform for developing decentralized applications. Freelance –Perfect complex (562 words) [view diff] case mismatch in snippet view article find links to article
alternative definition of pseudo-coherent complex". MathOverflow. "15.74 Perfect complexes". The Stacks project. "perfect module". ncatlab.org. v t eGodement resolution (577 words) [view diff] exact match in snippet view article find links to article
doi:10.1017/CBO9781139644136, ISBN 978-0-521-55987-4, MR 1269324 The Stacks Project authors. "20.30 Godement resolution". Goresky, Mark. "Introduction toApproximation property (ring theory) (206 words) [view diff] exact match in snippet view article
1216/rmjm/1181071964. JSTOR 44238106. "Tag 07BW: Smoothing Ring Maps". The Stacks Project. Columbia University, Department of Mathematics. Retrieved 2018-02-19Koszul complex (5,545 words) [view diff] exact match in snippet view article find links to article
needed] Koszul–Tate complex Syzygy (mathematics) The Stacks Project, section 0601 The Stacks Project, section 0601, Lemma 15.28.12 Indeed, by linearityGalois group (3,232 words) [view diff] case mismatch in snippet view article find links to article
Exchange. Retrieved 2020-11-11. "9.22 Infinite Galois theory". The Stacks project. Milne. "Field Theory" (PDF). p. 98. Archived (PDF) from the originalCotangent complex (4,731 words) [view diff] case mismatch in snippet view article find links to article
(08UX): Deformations of ringed spaces and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02. "Section 91.23 (08V3):Direct image with compact support (435 words) [view diff] case mismatch in snippet view article find links to article
zero functor. "Section 5.17 (005M): Characterizing proper maps—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-25. Schnürer, Olaf M.;Invariant basis number (1,027 words) [view diff] exact match in snippet view article find links to article
generating number) must have invariant basis number. (Lam 1999, p. 3) "Stacks Project, Tag 0FJ7". stacks.math.columbia.edu. Retrieved 4 March 2023. (HungerfordKähler differential (4,377 words) [view diff] exact match in snippet view article find links to article
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, CAlgebraic space (1,594 words) [view diff] case mismatch in snippet view article find links to article
ISBN 978-3-540-05496-2, MR 0302647 Danilov, V.I. (2001) [1994], "Algebraic space", Encyclopedia of Mathematics, EMS Press Algebraic space in the stacks projectRegular local ring (1,922 words) [view diff] exact match in snippet view article find links to article
ISBN 978-1-4612-0525-8. Chap.5.G. Jean-Pierre Serre, Local algebra, Springer-Verlag, 2000, ISBN 3-540-66641-9. Chap.IV.D. Regular rings at The Stacks ProjectConnected space (3,874 words) [view diff] exact match in snippet view article find links to article
space Pixel connectivity "5.12 Quasi-compact spaces and maps". The Stacks Project. Columbia University. Archived from the original on 17 March 2025. RetrievedFormally smooth map (620 words) [view diff] case mismatch in snippet view article find links to article
MR 0238860. "Lemma 37.11.7 (02H6): Infinitesimal lifting criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-07. Formally smooth withProjective module (3,081 words) [view diff] case mismatch in snippet view article find links to article
(help) "Section 10.95 (05A4): Descending properties of modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-11-03. Exercises 4.11 andModuli space (4,050 words) [view diff] case mismatch in snippet view article find links to article
Curves: Classical and Tropical" (PDF). AMS. "Lemma 27.13.1 (01NE)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-09-12. "algebraic geometryℓ-adic sheaf (1,103 words) [view diff] exact match in snippet view article find links to article
(PMS-33). Princeton University Press. p. 163. ISBN 978-0-691-08238-7. Stacks Project, Tag 03UL. Scholze, Peter; Bhatt, Bhargav (2013-09-04). "The pro-étaleLocal cohomology (4,306 words) [view diff] case mismatch in snippet view article find links to article
"Lemma 15.28.6 (0663)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-01. "Lemma 15.28.13 (0913)—The Stacks project". stacks.math.columbiaJacobson radical (2,879 words) [view diff] case mismatch in snippet view article find links to article
the result. "Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-24. Isaacs 1994, p. 182Restricted power series (1,377 words) [view diff] exact match in snippet view article find links to article
, φ ( H ) = h {\displaystyle F=GH,\,\varphi (G)=g,\varphi (H)=h} . Stacks Project, Tag 0AKZ. Grothendieck & Dieudonné 1960, Ch. 0, § 7.5.1. Bourbaki 2006Sheaf of algebras (930 words) [view diff] exact match in snippet view article find links to article
I, Definition 9.1.1. harvnb error: no target: CITEREFEGA1971 (help) Stacks Project, Tag 01S5. EGA 1971, Ch. I, Proposition 9.1.5. harvnb error: no target:Hirzebruch surface (1,693 words) [view diff] case mismatch in snippet view article find links to article
27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-23. Cox, David A.; LittleNisnevich topology (1,767 words) [view diff] case mismatch in snippet view article find links to article
"Section 10.154 (0BSK): Henselization and strict henselization—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-01-25. "counterexamples -Algebraic geometry (7,498 words) [view diff] exact match in snippet view article find links to article
Wisconsin–Milwaukee. Archived from the original on 2021-11-22 – via YouTube. The Stacks Project, an open source textbook and reference work on algebraic stacks andBijection, injection and surjection (2,207 words) [view diff] case mismatch in snippet view article find links to article
"Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2019-12-07. "Earliest Known UsesCohen–Macaulay ring (3,123 words) [view diff] case mismatch in snippet view article find links to article
archived (PDF) from the original on 5 Mar 2020 "Lemma 31.4.4 (0BXG)—The Stacks project", stacks.math.columbia.edu, retrieved 2020-03-05 Wiegand, Roger (DecemberFpqc morphism (369 words) [view diff] exact match in snippet view article find links to article
descent theory" (PDF). arXiv:math/0412512. Bibcode:2004math.....12512V. Stacks Project, "The fpqc Topology." http://stacks.math.columbia.edu/tag/03NVFunctor represented by a scheme (1,260 words) [view diff] exact match in snippet view article find links to article
R\mapsto Y(R)} determines a morphism of schemes X →Y in a natural way. The Stacks Project, 01J5 The functor of points, Yoneda's lemmma, moduli spaces and universalDouble complex (568 words) [view diff] case mismatch in snippet view article find links to article
"Section 12.18 (0FNB): Double complexes and associated total complexes—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-07-08. Weibel, Charles ARinged topos (836 words) [view diff] case mismatch in snippet view article find links to article
ISSN 0010-3616. "Section 26.15 (01JF): A representability criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-28. The standard referenceGlossary of algebraic geometry (12,496 words) [view diff] exact match in snippet view article find links to article
arXiv:1410.1716 [math.AG]. Hartshorne 1977, Exercise II.3.11(d) The Stacks Project, Chapter 21, §4. Grothendieck & Dieudonné 1960, 4.2.1 Hartshorne 19772-Yoneda lemma (1,228 words) [view diff] case mismatch in snippet view article find links to article
"94.5 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. "2Hilbert scheme (3,409 words) [view diff] case mismatch in snippet view article find links to article
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"Derived category (4,514 words) [view diff] exact match in snippet view article find links to article
14. ISBN 978-3-642-85844-4. Weibel 1994, remark 10.4.5 and errata Stacks Project, tag 079P. Markarian, Nikita (2009). "The Atiyah class, Hochschild cohomologyInjective module (3,919 words) [view diff] case mismatch in snippet view article find links to article
submodule can be extended to the whole module. "Lemma 47.7.5 (08Z6)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-02-25. Eisenbud. IntroductionHochschild homology (3,399 words) [view diff] case mismatch in snippet view article find links to article
Geometry". arXiv:math/0506603. "Section 23.6 (09PF): Tate resolutions—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-31. Cartan, Henri; EilenbergFrancis Mechner (2,758 words) [view diff] case mismatch in snippet view article find links to article
queenspaideiaschool.org/ Queens Paideia School Web site. Mechner, F. (1976). The STACKS project. Archives of The Mechner Foundation at 200 Central Park South, New YorkChow group (4,204 words) [view diff] exact match in snippet view article find links to article
geometry) Fulton. Intersection Theory, section 1.2 and Appendix A.3. Stacks Project, https://stacks.math.columbia.edu/tag/0BE9 Fulton, Intersection TheoryReflection principle (3,584 words) [view diff] exact match in snippet view article find links to article
978-3-540-88866-6. "Section 3.8 (000F): Reflection principle". The Stacks Project. 2022. Retrieved 7 September 2022. T. Jech, 'Set Theory: The Third MillenniumIntegral element (5,304 words) [view diff] case mismatch in snippet view article find links to article
Exercise 35 "Section 32.14 (05JW): Universally closed morphisms—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-11. Stein. ComputationalModuli stack of elliptic curves (2,344 words) [view diff] case mismatch in snippet view article find links to article
American Mathematical Society, ISBN 978-1470427986 moduli+stack+of+elliptic+curves at the nLab "The moduli stack of elliptic curves", Stacks projectSheaf of modules (3,458 words) [view diff] case mismatch in snippet view article find links to article
"Section 30.2 (01X8): Čech cohomology of quasi-coherent sheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2023-12-07. Costa, Miró-Roig &Weyl algebra (4,164 words) [view diff] case mismatch in snippet view article find links to article
pp. 600–601. "Section 41.13 (039P): Étale and smooth morphisms—The Stacks project". stacks.math.columbia.edu. Retrieved 2024-09-29. "etale morphism ofList of humorous names in mathematics (458 words) [view diff] exact match in snippet view article find links to article
soimeme.org. Retrieved 20 May 2025. "Section 10.66 (0546): Weakly associated primes". The Stacks Project. Columbia University. Retrieved 20 May 2025.