The resolution property holds away from codimension three

Mathur, Siddharth; Schröer, Stefan

doi:10.1090/tran/8709

The resolution property holds away from codimension three
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Trans. Amer. Math. Soc. 376 (2023), 1041-1063 Request permission

Abstract:

The purpose of this paper is to verify a conjecture of Gross under mild hypotheses: all reduced, separated, and excellent schemes have the resolution property away from a closed subset of codimension $\geq 3$. Our technique uses formal-local descent and the existence of affine flat neighborhoods to reduce the problem to constructing certain modules over commutative rings. Once in the category of modules we exhibit enough locally free sheaves directly, thereby establishing the resolution property for a specific class of algebraic spaces. A crucial step is showing it suffices to resolve a single coherent sheaf.

References

Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI 10.2307/2372985
Olivier Benoist, Quasi-projectivity of normal varieties, Int. Math. Res. Not. IMRN 17 (2013), 3878–3885. MR 3096912, DOI 10.1093/imrn/rns163
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
Dan Edidin, Brendan Hassett, Andrew Kresch, and Angelo Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), no. 4, 761–777. MR 1844577, DOI 10.1353/ajm.2001.0024
Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no. 4, 553–585 (French, with English and French summaries). MR 2044495, DOI 10.24033/bsmf.2455
P. Gross, Vector bundles as generators on schemes and stacks, http://d-nb.info/1007550295/34, 2010. Dissertation.
Philipp Gross, The resolution property of algebraic surfaces, Compos. Math. 148 (2012), no. 1, 209–226. MR 2881314, DOI 10.1112/S0010437X11005628
Philipp Gross, Tensor generators on schemes and stacks, Algebr. Geom. 4 (2017), no. 4, 501–522. MR 3683505, DOI 10.14231/2017-026
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
Jack Hall and David Rydh, Coherent Tannaka duality and algebraicity of Hom-stacks, Algebra Number Theory 13 (2019), no. 7, 1633–1675. MR 4009673, DOI 10.2140/ant.2019.13.1633
J. Hall and D. Rydh, Mayer-Vietoris squares in algebraic geometry, arXiv preprint arXiv:1606.08517, 2016.
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
G. Horrocks, Birationally ruled surfaces without embeddings in regular schemes, Bull. London Math. Soc. 3 (1971), 57–60. MR 284437, DOI 10.1112/blms/3.1.57
B. Conrad (https://mathoverflow.net/users/3927/bcnrd). Smooth linear algebraic groups over the dual numbers. MathOverflow. URL: https://mathoverflow.net/q/22078 (version: 2012-08-25).
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076, DOI 10.1007/3-540-27950-4
János Kollár, Non-quasi-projective moduli spaces, Ann. of Math. (2) 164 (2006), no. 3, 1077–1096. MR 2259254, DOI 10.4007/annals.2006.164.1077
János Kollár, Coherence of local and global hulls, Methods Appl. Anal. 24 (2017), no. 1, 63–70. MR 3694300, DOI 10.4310/MAA.2017.v24.n1.a5
Andrew Kresch and Angelo Vistoli, On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc. 36 (2004), no. 2, 188–192. MR 2026412, DOI 10.1112/S0024609303002728
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927, DOI 10.1007/978-3-540-24899-6
Mireille Martin, Sur une caractérisation des ouverts affines d’un schéma affine, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A38–A40 (French). MR 286805
Siddharth Mathur, Experiments on the Brauer map in high codimension, Algebra Number Theory 16 (2022), no. 3, 747–775. MR 4449398, DOI 10.2140/ant.2022.16.747
Siddharth Mathur, The resolution property via Azumaya algebras, J. Reine Angew. Math. 774 (2021), 93–126. MR 4250478, DOI 10.1515/crelle-2021-0002
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
Masayoshi Nagata, Existence theorems for nonprojective complete algebraic varieties, Illinois J. Math. 2 (1958), 490–498. MR 97406
Sam Payne, Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom. 18 (2009), no. 1, 1–36. MR 2448277, DOI 10.1090/S1056-3911-08-00485-2
Markus Perling and Stefan Schröer, Vector bundles on proper toric 3-folds and certain other schemes, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4787–4815. MR 3632550, DOI 10.1090/tran/6813
Stefan Schröer, On non-projective normal surfaces, Manuscripta Math. 100 (1999), no. 3, 317–321. MR 1726231, DOI 10.1007/s002290050203
Stefan Schröer and Gabriele Vezzosi, Existence of vector bundles and global resolutions for singular surfaces, Compos. Math. 140 (2004), no. 3, 717–728. MR 2041778, DOI 10.1112/S0010437X0300071X
C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225–274. MR 466154, DOI 10.1016/0001-8708(77)90041-X
The Stacks Project Authors, stacks project, http://stacks.math.columbia.edu, 2017.
R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16–34. MR 893468, DOI 10.1016/0001-8708(87)90016-8
Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. MR 2108211, DOI 10.1515/crll.2004.2004.577.1