Related papers: Remarks on chromatically localised hermitian K-the…
We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case…
We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a K\"unneth formula in that setting.…
We show that the map from $K({\mathbb S})$ to its chromatic completion is a connective cover and identify the fiber in $K$-theoretic terms. We combine this with recent work of Land-Mathew-Meier-Tamme to prove a form of "Waldhausen's…
We establish new structures on Grothendieck-Witt rings, including a GW(k)-module structure on the unit group GW(k)^x and a presentation of \ul{GW}^x as an infinite Gm-loop sheaf. Even though our constructions are motivated by speculations…
Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient.…
We show that the results of the paper Symplectic Reduction and Riemann-Roch for Circle Actions of Duistermaat, Guillemin, Meinrenken and Wu can be expressed entirely in K-theory. We show that their quantization is simply a pushforward in…
We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our…
We study a family of subrings, indexed by the natural numbers, of the mod-p cohomology of a finite group G. These subrings are based on a family of v_n-periodic complex oriented cohomology theories and are constructed as rings of…
We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$. As an application the cohomological purity conjecture is reduced to the complete case.
Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it…
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,…
The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, i.e. an $N\times N$ matrix of the form $A=UTV$, with $U, V$ some…
Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the…
We study the $K$-theory and Swan theory of the group ring $R[G]$, when $G$ is a finite group and $R$ is any ring or ring spectrum. In this setting, the well-known assembly map for $K(R[G])$ has a companion called the coassembly map. We…
Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition theorems for thick subcategories of various triangulated categories including the derived categories of rings, Noetherian stable homotopy categories, stable module…
Chromatic redshift phenomena suggest that algebraic K-theory increases the height of a commutative ring spectrum by one. In many cases, the chromatic redshift is already detected by negative topological cyclic homology. This paper explores…
In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic…
We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal…
The K-ring of symmetric vector bundles over a scheme X, the so-called Grothendieck-Witt ring of X, can be endowed with the structure of a (special) $\lambda$-ring. The associated $\gamma$-filtration generalizes the fundamental filtration on…