Excision in Algebraic K-Theory

Suslin, Andrei; Wodzicki, Mariusz

doi:10.2307/2946546

Cited by 81 publications

(122 citation statements)

References 26 publications

Supporting

Mentioning

116

Contrasting

Unclassified

Order By: Relevance

“…In particular this shows that (24) comes from a map of spectra which is functorially derived from K Q .…”

Section: The Chern Character Is the Tautological Charactermentioning

confidence: 91%

See 1 more Smart Citation

The obstruction to excision in K-theory and in cyclic homology

Cortiñas

2005

Invent. math.

View full text Add to dashboard Cite

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is measured by the birelative groups $K_*(A,B:I)$. We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are $\Q$-algebras we obtain an integral isomorphism.Comment: Final version to appear in Inventione

show abstract

“…In particular this shows that (24) comes from a map of spectra which is functorially derived from K Q .…”

Section: The Chern Character Is the Tautological Charactermentioning

confidence: 91%

“…, n}, and ∪ τ,n BT τ n ⊂ BGL is the union over all n and all finitely supported partial orders τ . The definition of each of the groups just mentioned for nonunital A can be found in [24]. If G ⊂ GL(A) is a subgroup, we write …”

Section: Gl(a) := Ker(gl(ã) → Gl(z))mentioning

confidence: 99%

The obstruction to excision in K-theory and in cyclic homology

Cortiñas

2005

Invent. math.

View full text Add to dashboard Cite

show abstract

“…It follows from the Suslin-Wodzicki Theorem [62,63] that the functor K is excisive on C * . Since maximal C * -tensor product is exact, the functor −⊗O ∞ preserves exactness in C * whence K(−⊗O ∞ ) is excisive.…”

Section: The Generalized Homology Theory K(−⊗o ∞ )mentioning

confidence: 99%

“…Thanks to the Karoubi conjecture, which is now a Theorem [62,63], we know that the nonconnective algebraic K-theory of a stable C * -algebra is isomorphic to its topological Ktheory. In fact, there is a canonical comparison map of spectra that induces the isomorphisms c n (A) : K n (A) → K top n (A) for all n ∈ Z when A is stable [30] (see also [55]).…”

Section: The Generalized Homology Theory K(−⊗o ∞ )mentioning

confidence: 99%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

View full text Add to dashboard Cite

Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C * -algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗O ∞ ∼ = A, deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing, which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we believe that it is worthwhile to have an algebraic formalism for T-duality at our disposal. One aspect of this theory is the Bunke-Schick topological T-duality, which has received a lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

show abstract

“…They were used by Goodwillie in [25, p. 392] and shown to be natural in [48, 5.11]. As observed on p. 85 of [48], naturality implies that if Q ⊆ A there is a natural quasi-isomorphism…”

Section: Corollary A5 the Direct Sum Of Matrices Makes X (A I ) + Imentioning

confidence: 99%