Related papers: Higher Semiadditive Algebraic K-Theory and Redshif…

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…

This is an overview of results from our experiment of merging two seemingly unrelated disciplines - higher algebraic K-theory of rings and the theory of lattice polytopes. The usual K-theory is the ``theory of a unit simplex''. A conjecture…

Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of…

Algebraic K-theory is the stable homotopy theory of homotopy theories, and it interacts with algebraic structures accordingly. In particular, we prove the Deligne Conjecture for algebraic K-theory.

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 consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field…

Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified…

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…

Higher twisted $K$-theory is an extension of twisted $K$-theory introduced by Ulrich Pennig which captures all of the homotopy-theoretic twists of topological $K$-theory in a geometric way. We give an overview of his formulation and key…

In this paper, we establish a multiplicative equivalence between two multiplicative algebraic $K$-theory constructions, Elmendorf and Mandell's version of Segal's $K$-theory and Blumberg and Mandell's version of Waldhausen's $S_\bullet$…

We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known…

In this paper, a new invariant was built towards the classification of separable C*-algebras of real rank zero, which we call latticed total K-theory. A classification theorem is given in terms of such an invariant for a large class of…

We prove a colimit formula for the K-theory spectra of reductive p-adic groups of rank one with regular coefficients in terms of the K-theory of certain compact open subgroups. Furthermore, in the complex case, we show, using the…

In this paper, we initiate the study of algebraic K-theory for non-commutative $\Gamma$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by…

We introduce filtered algebraic $K$-theory of a ring $R$ relative to a sublattice of ideals. This is done in such a way that filtered algebraic $K$-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge…

We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its $\pi_0$. We prove this as a consequence of a more general devissage result for stable infinity…

We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a…

Let $k$ be a field. In this paper, we define the notion of semi-fiber products of commutative $k$-algebras and show that the class of such rings contains several classes of commutative rings, including that of the fiber products of local…

We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N} = 2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level…

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group…