# 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} }

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.

## 48 Citations

On the K -theory of Toric Stack Bundles

- Mathematics
- 2008

Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack…

On the Grothendieck groups of toric stacks

- Mathematics
- 2009

In this note, we prove that the Grothendieck group of a smooth complete toric Deligne-Mumford stack is torsion free. This statement holds when the generic point is stacky. We also construct an…

The Grothendieck and Picard groups of a complete toric DM stack

- Mathematics
- 2008

We compute the Grothendieck and Picard groups of a complete smooth toric DM stack by using a suitable category of graded modules over a polynomial ring.

ON THE CONJECTURE OF KING FOR SMOOTH TORIC DELIGNE-MUMFORD STACKS

- Mathematics
- 2008

We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that…

COMPUTATION OF THE GROTHENDIECK AND PICARD GROUPS OF A TORIC DM STACK BY USING A HOMOGENEOUS COORDINATE RING FOR

- MathematicsGlasgow Mathematical Journal
- 2010

Abstract We compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading…

Smooth toric DM stacks

- Mathematics
- 2007

We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In…

Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks

- Mathematics
- 2012

A note on toric Deligne-Mumford stacks

- Mathematics
- 2007

We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which…

Tate resolutions on toric varieties

- Mathematics
- 2021

. We develop an analogue of Eisenbud-Fløystad-Schreyer’s Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier-Mukai transform, works quite…

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