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

Abstract

This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable ∞-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 ∞-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 ∞-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 ∞-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 ∞-categories, showing in particular that they form a bicomplete, closed symmetric monoidal ∞-category. We also study the process of tensoring and cotensoring a Poincaré ∞-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 ∞-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. CONTENTSCALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE 5 Monoidal structures and multiplicativity 78 5.1 Tensor products of hermitian ∞-categories 78 5.2 Construction of the symmetric monoidal structure 82 5.3 Day convolution of hermitian structures 88 5.4 Examples 95 6 The category of Poincaré categories 99 6.1 Limits and colimits 99 6.2 Internal functor categories 103 6.3 Cotensoring of hermitian categories 108 6.4 Tensoring of hermitian categories 113 6.5 Finite tensors and cotensors 116 6.6 Finite complexes and Verdier duality 121 7 Hyperbolic and metabolic Poincaré categories 123 7.1 Preliminaries: pairings and bifibrations 124 7.2 Bilinear and symmetric ∞-categories 129 7.3 The categorical Thom isomorphism 135 7.4 Genuine semi-additivity and spectral Mackey functors 146 7.5 Multiplicativity of Grothendieck-Witt and L-groups 152 References 156

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

“…Combined with our extension of Karoubi's fundamental theorem this implies that in the case of finite global dimension Karoubi's fundamental theorem holds in its classical form in sufficiently high degrees, allowing for many of the associated arguments to be picked up in this context. In a different direction, for such rings one can eventually deduce results about classical symmetric GW-groups from results on the corresponding homotopy coherent symmetric GW-groups, allowing one to exploit some of the useful properties of the latter, such as a dévissage property we prove in Paper [3] and the A 1 -invariance, which will be established in [16], for the benefit of the former. We exploit these ideas in Paper [3] to solve the homotopy limit problem for number rings, show that their Grothendieck-Witt groups are finitely generated, and produce an essentially complete calculation of the quadratic and symmetric Grothendieck-Witt groups (and their skew variants) of the integers, affirming, in particular, a conjecture of Berrick and Karoubi from [13].…”

“…The left hand map then sends [P] to the class of the associated hyperbolic form on P ⊕ Hom R (P, M), and is invariant under this C 2action. The sequence (1) can often be used to compute GW q 0 (R, M) from the two outer groups, and consequently obtain more complete information about quadratic forms. For example, in the case of the integers this sequence is split short exact and we have an isomorphism W q (Z) ∼ = Z given by taking the signature divided by 8 and an isomorphism K 0 (Z) C 2 ∼ = Z given by the dimension.…”

The Galois action on symplectic K-theory

“…We point out that Karoubi's notation L is a special case of the GW from[10][11][12]. The L-theory in the latter papers are more similar to Karoubi's W -groups.…”

“…For odd p and i > 0, the groups KSp i (Z; Z p ) are as in the following table, with the identifications given explicitly by the Betti-Hodge map: 7 The relationship between K -theory and Hermitian K -theory is more complicated when the prime 2 is not inverted, and is well understood only quite recently. See [10][11][12] 6 as well as [21,25,46].…”

“…In an earlier version of the paper, we explained in more detail how to deduce this isomorphism from Karoubi's main result. We decided this was unnecessary, not least in light of the thorough treatment in[10][11][12]24].…”