Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userK-theory and G-theory of derived algebraic stacks
222
0
0.0
(
0
)
Authors
Adeel A. Khan
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We introduce a notion of Milnor square of stable $infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the K-theory of algebraic stacks with affine diagonal and nice stabilizers. This yields a generalization of Weibels conjecture on the vanishing of negative K-groups for this class of stacks.
The family of Thom spectra $y(n)$ interpolate between the sphere spectrum and the mod two Eilenberg-MacLane spectrum. Computations of Mahowald, Ravenel, and Shick and the authors show that the $E_1$ ring spectrum $y(n)$ has chromatic complexity $n$. We show that topological periodic cyclic homology of $y(n)$ has chromatic complexity $n+1$. This gives evidence that topological periodic cyclic homology shifts chromatic height at all chromatic heights, supporting a variant of the Ausoni--Rognes red-shift conjecture. We also show that relative algebraic K-theory, topological cyclic homology, and topological negative cyclic homology of $y(n)$ at least preserve chromatic complexity.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $mathbb{C}$ and passing to spaces of $mathbb{C}$-valued points, we obtain new proofs of the topologic
In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial backgrounds are discussed.
We show that K_{2i}(Z[x,y]/(xy),(x,y)) is free abelian of rank 1 and that K_{2i+1}(Z[x,y]/(xy),(x,y)) is finite of order (i!)^2. We also compute K_{2i+1}(Z[x,y]/(xy),(x,y)) in low degrees.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
