Gabriel’s theorem and birational geometry

Calabrese, John; Pirisi, Roberto

doi:10.1090/proc/14990

Gabriel’s theorem and birational geometry
HTML articles powered by AMS MathViewer

by John Calabrese and Roberto Pirisi

Proc. Amer. Math. Soc. 149 (2021), 907-922

DOI: https://doi.org/10.1090/proc/14990

Published electronically: December 31, 2020

PDF | Request permission

Abstract:

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic away from a subset of a given dimension if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel’s reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field.

References

M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218, DOI 10.1007/BF01243916
M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
Martin Brandenburg, Rosenberg’s reconstruction theorem, Expo. Math. 36 (2018), no. 1, 98–117. MR 3780029, DOI 10.1016/j.exmath.2017.08.005
John Calabrese and Michael Groechenig, Moduli problems in abelian categories and the reconstruction theorem, Algebr. Geom. 2 (2015), no. 1, 1–18. MR 3322195, DOI 10.14231/AG-2015-001
Colin Diemer, Birational geometry and derived categories, Superschool on derived categories and D-branes, Springer Proc. Math. Stat., vol. 240, Springer, Cham, 2018, pp. 77–92. MR 3848732, DOI 10.1007/978-3-319-91626-2_{7}
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
Carl Faith, Algebra. I. Rings, modules, and categories, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 190, Springer-Verlag, Berlin-New York, 1981. Corrected reprint. MR 623254
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
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
Sven Meinhardt and Holger Partsch, Quotient categories, stability conditions, and birational geometry, Geom. Dedicata 173 (2014), 365–392. MR 3275309, DOI 10.1007/s10711-013-9947-x
Dennis Presotto and Michel Van den Bergh, Noncommutative versions of some classical birational transformations, J. Noncommut. Geom. 10 (2016), no. 1, 221–244. MR 3500820, DOI 10.4171/JNCG/232
Alexander Rosenberg, Spectra of ‘spaces’ represented by abelian categories, (2004), MPIM Preprints, 2004-115, http://www.mpim-bonn.mpg.de/preblob/2543.
Alexander L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347919, DOI 10.1007/978-94-015-8430-2
S. Paul Smith, Integral non-commutative spaces, J. Algebra 246 (2001), no. 2, 793–810. MR 1872125, DOI 10.1006/jabr.2001.8957
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2017.
Michel Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. MR 1846352, DOI 10.1090/memo/0734

AMS Website Down

Proceedings of the American Mathematical Society

Gabriel’s theorem and birational geometry
HTML articles powered by AMS MathViewer

Abstract: