The Local Structure of Algebraic K-Theory

Dundas, Bjørn Ian; Goodwillie, Thomas G.; McCarthy, Randy

doi:10.1007/978-1-4471-4393-2

Cited by 74 publications

(128 citation statements)

References 219 publications

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“…This follows by the standard arguments proving the "fundamental cofibration sequence" for fixed points of topological Hochschild homology, as in [7,VI.1.3.8]. For a published account see [3, 5.2.5], but remove the intricacies which are present in the commutative situation where non-cyclic group actions are allowed.…”

Section: Lemma 33 the Homotopy Fiber Of The Restriction Mapmentioning

confidence: 94%

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Integral excision for $K$-theory

Dundas

Kittang²

2013

Homology, Homotopy and Applications

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If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace K(A) → T C(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision.The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and -more relevantly for our current application -the T-Tate spectrum of topological Hochschild homology, where T is the circle group.

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Section: Lemma 33 the Homotopy Fiber Of The Restriction Mapmentioning

confidence: 94%

“…Topological cyclic homology T C(A) of a connective S-algebra A is most effectively defined integrally, as in [7], by a cartesian square…”

Section: Relations Between T C and Homotopy T-fixed Pointsmentioning

confidence: 99%

Integral excision for $K$-theory

Dundas

Kittang²

2013

Homology, Homotopy and Applications

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“…Let

F : T H H {(A)}^{C_{p^{m}}} \to T H H {(A)}^{C_{p^{m - 1}}}

be the map forgetting part of the invariance, and let

T F false(A; p false) = {prefixholim}_{F, m} T H H {(A)}^{C_{p^{m}}}

be the sequential homotopy limit over the

F

‐maps. The

R

‐maps induce a self‐map of

T F (A; p)

, also denoted

R

, and the topological cyclic homology functor

T C (A; p)

can be defined as the homotopy equalizer of

i d

and

R :

The (integral) topological cyclic homology of

A

, denoted

T C (A)

, is defined as the homotopy pullback of two maps

0true \prod_{p prime} T C {false(A; p false)}_{p} ⟶ \prod_{p prime} \underset{F, m}{holim} T H H {false(A false)}_{p}^{h C_{p^{m}}} ⟵ T H H {false(A false)}^{h double-struckT},

see [, Definition 6.4.3.1]. The left‐hand map is defined in terms of

π : T C (A; p

…”

Section: Cubical Descentmentioning

confidence: 99%

“…The case

F = T C

remains. For this we appeal to [, Theorem 7.0.0.2] (where

K (B)

in the lower left‐hand corner of the displayed square should be replaced with

K (A)

), to see that for each

s ⩾ 1

the square

is homotopy Cartesian. This uses that

π_{0} X^{s + 1} false(T false)

is constant as a functor of

T

, by our assumptions on

R

A

B

and

ι

.…”

Section: Cubical Descentmentioning

confidence: 99%

“…Proof The assertion that

f^{n}

is left cofinal, that is, that the left fiber

f_{n} / false[q false]

has contractible nerve for each object

[q]

Δ^{< n}

, is proved in [, § 6]. Note that the argument offered at this point in [, A8.1.1] is flawed. The left fiber

f / [q]

is the increasing union of the left fibers

f_{n} / false[q false]

, hence its nerve is also contractible.…”

Section: Cosimplicial Descentmentioning

confidence: 99%

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Cubical and cosimplicial descent

Dundas

Rognes

2018

J. London Math. Soc.

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The Riemann–Roch strategy

Connes

Consani

2019

Advances in Noncommutative Geometry

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We show that by working over the absolute base S (the categorical version of the sphere spectrum) instead of S[±1] improves our previous Riemann-Roch formula for Spec Z. The formula equates the (integer-valued) Euler characteristic of an Arakelov divisor with the sum of the degree of the divisor (using logarithms with base 2) and the number 1, thus confirming the understanding of the ring Z as a ring of polynomials in one variable over the absolute base S, namely S[X], 1 + 1 = X + X 2 .

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The Local Structure of Algebraic K-Theory

Cited by 74 publications

References 219 publications

Integral excision for $K$-theory

Integral excision for $K$-theory

Cubical and cosimplicial descent

The Riemann–Roch strategy

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