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

Venue: | Invent. Math |

Citations: | 14 - 6 self |

### BibTeX

@ARTICLE{Cortiñas_theobstruction,

author = {Guillermo Cortiñas},

title = {The obstruction to excision in K-theory and cyclic homology},

journal = {Invent. Math},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

Abstract Let f: A → B be a ring homomorphism of not necessarily unital rings and I ⊳ 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) → 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. Algebraic K-theory does not satisfy excision. This means that if f: A → B is a ring homomorphism and I ⊳ A is an ideal carried isomorphically to an ideal of B, then the map of relative K-groups K∗(A: I) → K∗(B: I): = K∗(B: f(I)) is not an isomorphism in general. The obstruction is measured by birelative groups

### Citations

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(Show Context)
Citation Context ...owing observations. The homotopy exact sequence of a level fibration is an inverse system of exact sequences, whence pro-exact. Thus EnZ → Z is a homotopy pro-equivalence (notation is as in [16]). By =-=[1]-=-, III.3.4, this implies that it is also a homology equivalence, whence Z can be replaced by EnZ, in which case the assertion is clear. - Step two. Conclude from step one that if Z is simply connected ... |

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19 |
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(Show Context)
Citation Context ...to have a gap (cf. [9], page 591, line 1). For I ∩ J = 0 and ∗ ≤ 1 it is well-known that both birelative K- and cyclic homology groups vanish. The case ∗ = 2 for I ∩ J = 0 was proved independently in =-=[12]-=- and [14]. An application of 0.2 to the computation of the K-theory of particular rings –other that coordinate rings of curves– in terms of their cyclic homology was given in [9], thm 3.1; see also [6... |

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(Show Context)
Citation Context ...inted simplicial set. The rational plus construction of the general linear group is the Bousfield-Kan Q-completion K Q (A) := Q∞BGl(A) The space K Q (A) is the 0-th space of a nonconnective spectrum (=-=[10]-=-) K Q A := (K Q (Σ n A) ∼ → ΩK Q (Σ n+1 A))n≥0 where Σ is Karoubi’s suspension functor. The general framework of the previous section produces a character c τ : K Q (A) → τK Q (A) which at the level o... |

13 |
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(Show Context)
Citation Context .... The theorem is trivially true for ∗ ≤ 0, in which case both birelative K- and cyclic homology groups are zero. The particular case of the isomorphism of the theorem when ∗ = 1 was proved in [7]. In =-=[6]-=- the statement of the theorem was conjectured to hold when A and B are commutative unital Q-algebras and B is a finite integral extension of A, (KABI conjecture) and it was shown its validity 1 CONICE... |

11 |
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(Show Context)
Citation Context ...e isomorphism is integral. Proof. When I ∩ J = 0 this the particular case of 0.1 when f is surjective. The general case of the corollary follows from this and from Goodwillie’s rational isomorphisms (=-=[11]-=-) K∗(A : I ∩ J) ⊗ Q ∼ =K∗(A ⊗ Q : (I ∩ J) ⊗ Q) ∼= HC∗−1(A : I ∩ J) ⊗ Q ∼ = HC∗−1(A ⊗ Q : (I ∩ J) ⊗ Q) The result of the corollary above for A unital was announced in [17]; however the proof in loc. ci... |

3 |
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(Show Context)
Citation Context ... HP∗ for negative cyclic and periodic cyclic homology. There is a long exact sequence (1) HCn−1(A, B : I) → HNn(A, B : I) → HPn(A, B : I) → HCn−2(A, B : I) In view of Cuntz-Quillen’s excision theorem =-=[4]-=-, we have HP∗(A, B : I) = 0, and therefore HC∗−1(A, B : I) ∼ = HN∗(A, B : I). To prove 0.1 it suffices to show that the Jones-Goodwillie Chern character ch∗ : K∗(R) → HN∗(R) (R a Q-algebra) induces an... |

2 | Periodic cyclic homology as sheaf cohomology, K-theory 20
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(Show Context)
Citation Context ...homotopy fibrations, that when applied to the periodic cyclic spectrum is again the periodic cyclic spectrum (cf. [2], 4.0) and that when applied to the negative cyclic spectrum is nullhomotopic (cf. =-=[3]-=-, proof of thm. 5.1). Note 2.8 is not particularly helpful to prove excision for HP , since Wodzicki’s theorem only gives ∞-excision for HC. The proof in [4] uses a different method. An example when 2... |

2 |
Hodge decompostion of Loday symbols in K-theory and cyclic homology, K-theory 8
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(Show Context)
Citation Context ... (I ∩ J) ⊗ Q) ∼= HC∗−1(A : I ∩ J) ⊗ Q ∼ = HC∗−1(A ⊗ Q : (I ∩ J) ⊗ Q) The result of the corollary above for A unital was announced in [17]; however the proof in loc. cit. turned out to have a gap (cf. =-=[9]-=-, page 591, line 1). For I ∩ J = 0 and ∗ ≤ 1 it is well-known that both birelative K- and cyclic homology groups vanish. The case ∗ = 2 for I ∩ J = 0 was proved independently in [12] and [14]. An appl... |