Journal of Commutative Algebra

Valuative and geometric characterizations of Cox sheaves

Benjamin Bechtold

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Abstract

We give an intrinsic characterization of Cox sheaves on Krull schemes in terms of their valuative algebraic properties. We also provide a geometric characterization of their graded relative spectra in terms of good quotients of graded schemes, extending the existing theory on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an irredundant characterization of Cox rings which, in turn, produces a normality criterion for certain graded rings.

Article information

Source
J. Commut. Algebra, Volume 10, Number 1 (2018), 1-43.

Dates
First available in Project Euclid: 18 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1526608933

Digital Object Identifier
doi:10.1216/JCA-2018-10-1-1

Mathematical Reviews number (MathSciNet)
MR3804845

Zentralblatt MATH identifier
06875412

Subjects
Primary: 13A02: Graded rings [See also 16W50] 13A18: Valuations and their generalizations [See also 12J20] 14A20: Generalizations (algebraic spaces, stacks)

Keywords
Cox rings Krull schemes graded schemes

Citation

Bechtold, Benjamin. Valuative and geometric characterizations of Cox sheaves. J. Commut. Algebra 10 (2018), no. 1, 1--43. doi:10.1216/JCA-2018-10-1-1. https://projecteuclid.org/euclid.jca/1526608933


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References

  • D.F. Anderson, Graded Krull domains, Comm. Algebra 7 (1979), 79–106.
  • I. Arzhantsev, U. Derenthal, J. Hausen and A. Laface, Cox rings, Cambridge University Press, Cambridge, 2014, arXiv:1003.4229v2.
  • I.V. Arzhantsev, On the factoriality of Cox rings, Mat. Z. 85 (2009), 643–651 (in Russian); Math. Notes 85 (2009), 623–629 (in English).
  • H. Bäker, Good quotients of Mori dream spaces, Proc. Amer. Math. Soc. 139 (2011), 3135–3139.
  • B. Bechtold, Factorially graded rings and Cox rings, J. Algebra 369 (2012), 351–359.
  • ––––, Cox sheaves on graded schemes, algebraic actions and $\mathbb{F}_1$-schemes, Ph.D. thesis, Eberhard Karls Universität Tübingen, in preparation.
  • F. Berchtold and J. Hausen, Homogeneous coordinates for algebraic varieties, J. Algebra 266 (2003), 636–670.
  • ––––, Cox rings and combinatorics, Trans. Amer. Math. Soc. 359 (2007), 1205–1252.
  • M. Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), 61–99.
  • A. Canonaco, The Beilinson complex and canonical rings of irregular surfaces, Mem. Amer. Math. Soc. 183 (2006).
  • A.-M. Castravet and J. Tevelev, $\overline{M}_{0,n}$ is not a Mori dream space, arXiv:1311.7673.
  • D.A. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom. 4 (1995), 17–50.
  • A. Deitmar, Congruence schemes, Inter. J. Math. 24 (2013).
  • ––––, F1-schemes and toric varieties, Beitr. Alg. Geom. 49 (2008), 517–525.
  • E.J. Elizondo, K. Kurano and K. Watanabe, The total coordinate ring of a normal projective variety, J. Algebra 276 (2004), 625–637.
  • R.M. Fossum, The divisor class of a Krull domain, Springer, Berlin, 1973.
  • R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.
  • J. Hausen, Cox rings and combinatorics, II, Moscow Math. J. 8 (2008), 711–757.
  • J. Hausen and H. Süß, The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), 977–1012.
  • Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  • Y. Kamoi, Noetherian rings graded by an Abelian group, Tokyo J. Math. 18 (1995), 31–48.
  • M.D. Larsen and P.J. McCarthy, Multiplicative theory of ideals, Academic Press, New York, 1971.
  • H. Lee and M. Orzech, Brauer groups, class groups and maximal orders for a Krull scheme, Canad. J. Math. 34 (1982), 996–1010.
  • M. Perling, Toric varieties as spectra of homogeneous prime ideals, Geom. Ded. 127 (2007), 121–129.
  • F. Rohrer, Coarsenings, injectives and Hom functors, Rev. Roum. Math. Pures Appl. 57 (2012), 275–287.
  • ––––, Graded integral closures, Beitr. Alg. Geom. 55 (2014), 347–364.
  • B. Sturmfels and M. Velasco, Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties, J. Commutative Alg. 2 (2010), 223–244.
  • M. Temkin, On local properties of non-Archimedean spaces, II, Israel J. Math. 140 (2004), 1–27.