Related papers: D\'{e}vissage Hermitian Theory

A d\'evissage-type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $\mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $\mathscr{C}$ when a…

We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as…

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

We compute the $\mathbb{G}W$-spectrum (Karoubi--Grothendieck--Witt spectrum) of Grassmannians over divisorial schemes defined over fields of characteristic zero, and, as a corollary, determine their stabilized $\mathbb{L}$-theory spectrum.…

In this paper, we consider the Hermitian $K$-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the d\'{e}vissage theorem. This theorem is then used to compute the Hermitian $K$-theory of…

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…

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…

In this article we establish some formalism of Derived Witt-D\'evissage theory for resolving subcategories of abelian categories. Results directly apply to noetherian schemes.

In this article, we extend the theorem of heart\cite{Barwick_2015}, which implies Quillen's d\'evissage theorem by \cite{Efimov2025}, to generic small stable $\infty$-categories. To be precise, we establish a necessary and sufficient…

The (A)CGW categories of Campbell and Zakharevich show how finite sets and varieties behave like the objects of an exact category for the purpose of algebraic $K$-theory. These structures admit a well-behaved Q-construction akin to…

Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck-Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and…

For a reductive group scheme over a regular semi-local ring, we prove an equivarinat version of the Gersten conjecture. We draw some interesting consequences for the representation rings of such reductive group schemes. We also prove the…

We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian $K$-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be…

In this paper we define a notion of Witt group for sesquilinear forms in hermitian categories, which in turn provides a notion of Witt group for sesquilinear forms over rings with involution. We also study the extension of scalars for…

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $\lambda \in \ell$ such that $\lambda + \lambda^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action…

We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and…

Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…

In this article we deduce an analogue of Quillen's Local-Global Principle for the elementary subgroup of the general quadratic group and the hermitian group. We show that the unstable K_1-groups of the hermitian groups are nilpotent by…

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G=Q_p, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H=Z_p, the ring of…

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent…