Some exact sequences; negative K-theory

AbstractLet B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B = ℤp[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ≅ H ⋊ ℤp, this set S is Venjakob's canonical Ore set. In particular, our result implies thatis split exact for each n ≥ 0. We also prove the corresponding result for the localisation of ℤp[[G]][$\frac{1}{p}$] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.

Download Full-text

Algebraic K-Theory of ∞-Operads

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014008019jkt277 ◽

2014 ◽

Vol 14 (3) ◽

pp. 614-641 ◽

Cited By ~ 1

Author(s):

Thomas Nikolaus

Keyword(s):

Homotopy Groups ◽

Combinatorial Model ◽

K Theory ◽

Definition Of ◽

Exact Sequences

AbstractThe theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal setDsatisfying certain lifting conditions.In this paper we give a definition of K-groupsKn(D) for a dendroidal setD. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Using results from [Heu11b] and [BN12] we show that theK-theory groups ofDcan be realized as homotopy groups of a K-theory spectrum.

Download Full-text

The Cuntz-Pimsner extension and mapping cone exact sequences

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-115634 ◽

2019 ◽

Vol 125 (2) ◽

pp. 291-319 ◽

Cited By ~ 1

Author(s):

Francesca Arici ◽

Adam Rennie

Keyword(s):

Exact Sequence ◽

Finite Index ◽

Mapping Cone ◽

K Theory ◽

Exact Sequences

For Cuntz-Pimsner algebras of bi-Hilbertian bimodules with finite Jones-Watatani index satisfying some side conditions, we give an explicit isomorphism between the $K$-theory exact sequences of the mapping cone of the inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the Cuntz-Pimsner exact sequence. In the process we extend some results by the second author and collaborators from finite projective bimodules to certain finite index bimodules, and also clarify some aspects of Pimsner's `extension of scalars' construction.

Download Full-text

An elementary differential extension of odd K-theory

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is013002018jkt218 ◽

2013 ◽

Vol 12 (2) ◽

pp. 331-361 ◽

Cited By ~ 6

Author(s):

Thomas Tradler ◽

Scott O. Wilson ◽

Mahmoud Zeinalian

Keyword(s):

Abelian Group ◽

Equivalence Relation ◽

Unitary Group ◽

Differential Forms ◽

Equivalence Classes ◽

Smooth Maps ◽

Type Form ◽

K Theory ◽

Chern Simons ◽

Exact Sequences

AbstractThere is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Download Full-text