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Register a new userHermitian K-theory for stable $infty$-categories III: Grothendieck-Witt groups of rings
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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 Ranickis 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 Quillens localisation-devissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.
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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 Luries notion of a Poincare $infty$-category, which permits an abstract counterpart of unimodular forms called Poincare objects. We analyse the special cases of hyperbolic and metabolic Poincare objects, and establish a version of Ranickis algebraic Thom construction. For derived $infty$-categories of rings, we classify all Poincare 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 Poincare structures on $infty$-categories of parametrised spectra, recovering the visible signature of a Poincare duality space. We conduct a thorough investigation of the global structural properties of Poincare $infty$-categories, showing in particular that they form a bicomplete, closed symmetric monoidal $infty$-category. We also study the process of tensoring and cotensoring a Poincare $infty$-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 Poincare $infty$-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.
We define Grothendieck-Witt spectra in the setting of Poincare $infty$-categories and show that they fit into an extension with a L- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubis 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.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.
We prove a version of J.P. Mays theorem on the additivity of traces, in symmetric monoidal stable $infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $infty$-categories and construct a morphism of spectra $mathrm{THH}(mathbf C)to mathrm{End}(mathbf 1_mathbf C)$ for $mathbf C$ a stably symmetric monoidal rigid $infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $operatorname{Pic}(X)$ and pointed monoid of regular functions $Gamma(X, mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
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