arXiv subjects

Kristian Moi

Publications and source records attributed to Kristian Moi.

7 recordsLinked to original sources

arXiv2021

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

This paper is the first in a series in which we offer a new framework for hermitian 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 this 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 L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0-th Grothendieck-Witt group of a Poincaré $\infty$-category using generators and relations. We extract its basic properties, relating it in particular to the 0-th L- and algebraic 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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arXiv2021

On the geometric fixed-points of real topological cyclic homology

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect $\mathbb{F}_p$-algebras, and $2$-torsion free rings with perfect modulo $2$ reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.

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arXiv2020

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.

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arXiv2020

Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

We define Grothendieck-Witt spectra in the setting of Poincar\'e $\infty$-categories and show that they fit into an extension with a K- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. Our set-up allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations: For example, the periodicity theorem holds for complex oriented $\mathrm{E}_1$-rings, and we show that the Grothendieck-Witt theory of parametrised spectra recovers Weiss and Williams' LA-theory. Our Grothendieck-Witt spectra are defined via a version of the hermitian Q-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version -- along with a concise proof -- of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism theoretic description of the aforementioned LA-spectra.

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arXiv2020

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

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-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 $R$ 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 $\mathbb{Z}$, 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 $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

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arXiv2017

Real topological Hochschild homology

This paper interprets Hesselholt and Madsen's real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for group-algebras. Using these structural results we determine the homotopy type of THR($\mathbb{F}_p$) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of $H\mathbb{F}_p$. We then calculate the homotopy type of THR($\mathbb{Z}$) away from the prime $2$, and the homotopy ring of the geometric fixed-points spectrum $Φ^{\mathbb{Z}/2}$THR($\mathbb{Z}$).

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arXiv2014

Homotopy theory of G-diagrams and equivariant excision

Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on homotopy (co)limits. A case of particular interest is when the indexing category is a cube. We use homotopy limits and colimits over such diagrams to produce loop and suspension spaces with respect to permutation representations of $G$. We go on to develop a theory of enriched equivariant homotopy functors and give an equivariant "linearity" condition in terms of cubical $G$-diagrams. In the case of $G$-topological spaces we prove that this condition is equivalent to Blumberg's notion of $G$-linearity. In particular we show that the Wirthmüller isomorphism theorem is a direct consequence of the equivariant linearity of the identity functor on $G$-spectra.

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