# On the $K$-theory of smooth toric DM stacks

@article{Borisov2005OnT,
title={On the \$K\$-theory of smooth toric DM stacks},
author={Lev Borisov and Richard Paul Horja},
journal={arXiv: Algebraic Geometry},
year={2005}
}
• Published 14 March 2005
• Mathematics
• arXiv: Algebraic Geometry
We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced smooth toric DM stacks.
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## References

SHOWING 1-10 OF 25 REFERENCES
The orbifold Chow ring of toric Deligne-Mumford stacks
• Mathematics
• 2004
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a
Equivariant K-theory of smooth toric varieties
We characterize the smooth toric varieties for which the Merkurjev spectral sequence, connecting equivariant and ordinary K-theory, degenerates. We find under which conditions on the support of the
Log Crepant Birational Maps and Derived Categories
We extend the conjecture on the derived equivalence and K-equivalence to the logarithmic case and prove it in the toric case.
Orbifold Gromov-Witten Theory
• Mathematics
• 2001
In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.
Twisted Orbifold K-Theory
• Mathematics
• 2003
Abstract: We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a
Algebraic orbifold quantum products
• Mathematics
• 2001
The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a
Stringy K-theory and the Chern character
• Mathematics
• 2007
We construct two new G-equivariant rings: $\mathcal{K}(X,G)$, called the stringy K-theory of the G-variety X, and $\mathcal{H}(X,G)$, called the stringy cohomology of the G-variety X, for any smooth,
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Introduction to toric varieties
The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio