Taiwanese Journal of Mathematics

On the Grothendieck Groups of Toric Stacks

Zheng Hua

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In this short note, we give an elementary proof for the fact that the Grothendieck group of complete toric Deligne-Mumford stack is torsion free.

Article information

Taiwanese J. Math., Volume 21, Number 3 (2017), 665-670.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 16E20: Grothendieck groups, $K$-theory, etc. [See also 18F30, 19Axx, 19D50] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Grothendieck group $K$-theory toric stacks Stanley-Reisner ring


Hua, Zheng. On the Grothendieck Groups of Toric Stacks. Taiwanese J. Math. 21 (2017), no. 3, 665--670. doi:10.11650/tjm/7347. https://projecteuclid.org/euclid.twjm/1498874612

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  • V. V. Batyrev Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–-535.
  • V. V. Batyrev and L. A. Borisov Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), no. 1, 183–-203.
  • L. A. Borisov, L. Chen and G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193–215.
  • L. A. Borisov and R. P. Horja, On the $K$-theory of smooth toric DM stacks, in Snowbird Lectures on String Geometry, 21–42, Contemp. Math. 401, Amer. Math. Soc., Providence, RI, 2006.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.
  • D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.
  • R. Goldin, M. Harada and T. S. Holm, Torsion in the full orbifold $K$-theory of abelian symplectic quotients, Geom. Dedicata 157 (2012), no. 1, 187–204.
  • Y. Kawamata,Derived categories of toric varieties, Michigan Math. J. 54 (2006), no. 3, 517–-535.