K-theory and G-theory of derived algebraic stacks

@article{Khan2022KtheoryAG,
  title={K-theory and G-theory of derived algebraic stacks},
  author={Adeel A. Khan},
  journal={Japanese Journal of Mathematics},
  year={2022}
}
  • Adeel A. Khan
  • Published 13 December 2020
  • Mathematics
  • Japanese Journal of Mathematics
These are some notes on the basic properties of algebraic K-theory and G-theory of derived algebraic spaces and stacks, and the theory of fundamental classes in this setting. 
4 Citations
The Log Product Formula in quantum $K$-theory
We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. The proof requires
Generalized cohomology theories for algebraic stacks
We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck’s six operations. Objects in this
Foliations and stable maps
This paper is part of an ongoing series of works on the study of foliations on algebraic varieties via derived algebraic geometry. We focus here on the specific case of globally defined vector fields
Virtual excess intersection theory
We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement, and new proof, of R. Thomason's formula.

References

SHOWING 1-10 OF 123 REFERENCES
Algebraic K-theory and descent for blow-ups
We prove that algebraic K-theory satisfies ‘pro-descent’ for abstract blow-up squares of noetherian schemes. As an application we derive Weibel’s conjecture on the vanishing of negative K-groups.
Algebraic G-theory in motivic homotopy categories
We prove that algebraic G-theory in is representable in unstable and stable motivic homotopy categories; in the stable category we identify it with the Borel-Moore theory associated to algebraic
Algebraic Groups and Compact Generation of their Derived Categories of Representations
Let k be a field. We characterize the group schemes G over k, not necessarily affine, such that D-qc (B(k)G) is compactly generated. We also describe the algebraic stacks that have finite
𝐾-theory and 0-cycles on schemes
We prove Bloch’s formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in
The resolution property for schemes and stacks
Abstract We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a
Notes on G-theory of Deligne-Mumford stacks
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general
On Some Finiteness Questions for Algebraic Stacks
We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of
Algebraic K‐theory, A1‐homotopy and Riemann–Roch theorems
In this paper, we show that the combination of the constructions done in SGA 6 and the A1‐homotopy theory naturally leads to results on higher algebraic K‐theory. This applies to the operations on
On the algebraic K‐theory of higher categories
We prove that Waldhausen K ‐theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K ‐theory spaces admit canonical
Perfect complexes on algebraic stacks
We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks
...
...