K-theory of Hermitian Mackey functors, real traces, and assembly

Dotto, Emanuele; Ogle, Crichton

doi:10.2140/akt.2019.4.243

Cited by 8 publications

(12 citation statements)

References 32 publications

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“…Remark. The surgery results of this section can now be used to almost fully determine the L-groups of L b (ℤ) appearing in Corollary 1.2.24, or equivalently L g ( ) from [DO19]. Recall that in Corollary 1.2.24 we have determined the non-positive homotopy groups of L b (ℤ).…”

Section: Corollary Let Be An Invertible Module With Involution Over A...mentioning

confidence: 98%

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Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

Calmès¹,

Dotto²,

Harpaz³

et al. 2020

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We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring to the homotopy C 2 -orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of ℤ, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension is an equivalence in degrees ≥ +3. As an important tool, we establish the hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi. CONTENTS 10 1.1 L-theoretic preliminaries 10 1.2 Surgery for -quadratic structures 13 1.3 Surgery for -symmetric structures 24 2 L-theory of Dedekind rings 28 2.1 The localisation-dévissage sequence 28 2.2 Symmetric and quadratic L-groups of Dedekind rings 34 3 Grothendieck-Witt groups of Dedekind rings 43 3.1 The homotopy limit problem 43 3.2 Grothendieck-Witt groups of the integers 47 References 54

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Section: Corollary Let Be An Invertible Module With Involution Over A...mentioning

confidence: 98%

“…We denote the corresponding L-spectra by L u ( ), L t (ℤ) and L b (ℤ), respectively. The connective cover of L b (ℤ) is equivalent to the spectrum L g ( ) considered in[DO19].…”

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confidence: 99%

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

Calmès¹,

Dotto²,

Harpaz³

et al. 2020

Preprint

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“…Form rings and free modules over them were also studied by Dotto and Ogle in [8] under the name Hermitian Mackey functor. However, form parameter rings and form categories seem not to have been studied before though there is a related notion of stable infinity category with non-degenerate quadratic functor due to Lurie [16].…”

Section: Introductionmentioning

confidence: 99%

“…Our notation here deviates from[8]. Below, we will define a Burnside form ring R for any commutative ring R.…”

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confidence: 99%

Higher -Theory of Forms I. From Rings To exact categories

Schlichting¹

2019

J. Inst. Math. Jussieu

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We prove the analog for the $K$ -theory of forms of the $Q=+$ theorem in algebraic $K$ -theory. That is, we show that the $K$ -theory of forms defined in terms of an $S_{\bullet }$ -construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter.

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“…Theorem 1 also has consequences for other L-spectra, not associated to the genuine family of Poincaré structure Ϙ ≥m M considered so far. For instance, we can also treat the spectrum L b (Z) obtained from the Burnside Poincaré structure Ϙ b on Perf(Z), appearing in [CDH + 20c, Corollary 1.2.24] and [DO19]. This Burnside Poincaré structure is a non-abelian derived structure, and the results we state next apply more generally to any non-abelian derived structure on Perf(Z), for instance those associated to a form parameter in the sense of Bak, and more generally in the sense of Schlichting [Sch19], see [CDH + 20a, Definition 4.2.26] for a definition in the language of Poincaré categories.…”

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confidence: 99%

Remarks on chromatically localised hermitian K-theory

Land¹

2022

Preprint

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We describe chromatic localisations of genuine L-spectra of discrete rings and deduce that the purity property of K(1)-local K-theory of rings established by Bhatt-Clausen-Mathew also holds in Grothendieck-Witt theory. In addition, we collect some results on higher chromatic localisations of various L-theory spectra and their consequences for Grothendieck-Witt theory.The recent times have shown many advances in chromatically localised algebraic K-theory, notably its relation to étale K-theory, redshift, and purity phenomena [AKS20, AKQ20, BCM20, CM21, CMNN20, HW21, LMMT20, Mat21, Yua21]. In addition, in the series of papers [CDH + 20a, CDH + 20b, CDH + 20c, CDH + 22] we have shown that Grothendieck-Witt theory, also known as hermitian K-theory, sits in a fibre sequence with algebraic K-theory and L-theory. The purpose of this short note is to collect some results on chromatically localised L-theory and to use this to lift some of the purity and redshift results from algebraic K-theory to Grothendieck-Witt theory.

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K-theory of Hermitian Mackey functors, real traces, and assembly

Cited by 8 publications

References 32 publications

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

Higher -Theory of Forms I. From Rings To exact categories

Remarks on chromatically localised hermitian K-theory

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