Real topological Hochschild homology

Dotto, Emanuele; Moi, Kristian; Patchkoria, Irakli; Reeh, Sune Precht

doi:10.4171/jems/1007

Cited by 13 publications

(63 citation statements)

References 30 publications

(48 reference statements)

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“…Remark. Real topological Hochschild homology and its (genuine) real cyclotomic structure have been defined more generally for ring spectra with anti-involution in [DMPR21] and [Høg16]. The new contribution of Theorem A is deducing that THR(A) admits the structure of a C 2 -E ∞ -algebra in real cyclotomic spectra; to our knowledge, this genuine equivariant multiplicative structure on THR(A) has not been discussed in previous work.…”

Section: Resultsmentioning

confidence: 97%

“…Dotto [Dot12,Dot16] and Dotto-Ogle [DO19] then studied various trace maps out of real algebraic K-theory, while Høgenhaven [Høg16] defined real topological cyclic homology and made some initial computations. Dotto-Moi-Patchkoria-Reeh [DMPR21] pushed the study of real topological Hochschild homology further and made some seminal computations, which Dotto-Moi-Patchkoria [DMP19,DMP21] leveraged to study real topological restriction homology and real topological cyclic homology.…”

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confidence: 99%

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On the equivalence of two theories of real cyclotomic spectra

Quigley¹,

Shah²

2021

Preprint

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We give a new formula for real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for topological cyclic homology to one involving genuine C 2 -spectra. To accomplish this, we give a new definition of the ∞-category of real cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction. We then define an ∞-categorical version of Høgenhaven's O(2)-orthogonal cyclotomic spectra, construct a forgetful functor relating the two theories, and show that this functor restricts to an equivalence between full subcategories of appropriately bounded-below objects. As an application, we compute the real topological cyclic homology of perfect Fp-algebras for all primes p. Contents 1. Introduction 1.1. Motivation 1.2. Main results 1.3. Conventions, notation, and prerequisites 1.4. Acknowledgments 2. Two theories of real p-cyclotomic spectra 2.1. Borel real p-cyclotomic spectra 2.2. Genuine real p-cyclotomic spectra 2.3. C 2 -symmetric monoidal structures 3. Comparison of the theories 3.1. The dihedral Tate orbit lemma 3.2. The comparison at finite level 3.3. Exchanging a lax equalizer for an equalizer 3.4. Proof of the main theorem 3.5. Variant: O(2)-spectra at the prime p 4. Extension to integral theories 4.1. Borel real cyclotomic spectra 4.2. F-genuine real cyclotomic spectra 4.3. Integral comparison 5. Real cyclotomic structure on THR of C 2 -E ∞ -algebras 6. TCR of the constant mod p Mackey functor 6.1. Computation for p odd 6.2. Computation for p = 2 6.3. Extension to perfect F p -algebras Appendix A. On the geometric fixed points of real topological cyclic homology

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Section: Resultsmentioning

confidence: 97%

mentioning

confidence: 99%

On the equivalence of two theories of real cyclotomic spectra

Quigley¹,

Shah²

2021

Preprint

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“…In this paper, we will give an explicit construction of B σ and prove Theorem 1.1. Our proof is similar to that in [Dotto12], but we fill in elementary details.…”

Section: Introductionmentioning

confidence: 84%

“…The idea for such a functor goes back at least to [BF85], where B σ is defined on certain monoids related to simplicial Hermitian rings and used to study the extended Hermitian algebraic K-theory. Theorem 1.1, which explains its key property, was first proved in [Dotto12] and [Stiennon13], with two different methods, and was generalized in [Moi13] to the simplicial case with more discussions of homology. The functor B σ is now being used in many computations concerning real spaces and spectra.…”

Section: Introductionmentioning

confidence: 99%

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Investigation of novel embedded piezoelectric ultrasonic transducers on crack and corrosion monitoring of steel bar

Liu

Chen

et al. 2020

Construction and Building Materials

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Hermitian K-theory for stable $$\infty $$-categories I: Foundations

Calmès

Dotto

Harpaz

et al. 2022

Sel. Math. New Ser.

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This paper is the first in a series in which we offer a new framework for hermitian $${\text {K}}$$ K -theory in the realm of stable $$\infty $$ ∞ -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré $$\infty $$ ∞ -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived $$\infty $$ ∞ -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on $$\infty $$ ∞ -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré $$\infty $$ ∞ -categories, showing in particular that they form a bicomplete, closed symmetric monoidal $$\infty $$ ∞ -category. We also study the process of tensoring and cotensoring a Poincaré $$\infty $$ ∞ -category over a finite simplicial complex, a construction featuring prominently in the definition of the $${\text {L}}$$ L - and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré $$\infty $$ ∞ -category using generators and relations. We extract its basic properties, relating it in particular to the 0th $${\text {L}}$$ L - and algebraic $${\text {K}}$$ K -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

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Real topological Hochschild homology

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Cited by 13 publications

References 30 publications

On the equivalence of two theories of real cyclotomic spectra

On the equivalence of two theories of real cyclotomic spectra

Investigation of novel embedded piezoelectric ultrasonic transducers on crack and corrosion monitoring of steel bar

Hermitian K-theory for stable $$\infty $$-categories I: Foundations

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