Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1104.2761

Tensor triangular geometry of non-commutative motives

Ivo Dell'Ambrogio·Goncalo Tabuada
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we initiate the study of the tensor triangular geometry of the categories Mot(k)_a and Mot(k)_l of non-commutative motives (over a base ring k). Since the full computation of the spectrum of Mot(k)_a and Mot(k)_l seems completely out of reach, we provide some information about the spectrum of certain subcategories. More precisely, we show that when k is a finite field (or its algebraic closure) the spectrum of the monogenic cores Core(k)_a and Core(k)_l (i.e. the thick triangulated subcategories generated by the tensor unit) is closely related to the Zariski spectrum of the integers. Moreover, we prove that if we slightly enlarge Core(k)_a to contain the non-commutative motive associated to the ring of polynomials k[t], and assume that k is a field of characteristic zero, then the corresponding spectrum is richer than the Zariski spectrum of the integers.

Explore related subjects

math.KTmath.AGmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ivo Dell'Ambrogio, Goncalo Tabuada. 2011-12-06. Tensor triangular geometry of non-commutative motives. https://arxiv.org/abs/1104.2761

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

General linear and Steinberg groups over the Leavitt algebra $L_{\mathbb F_2}(1,2)$

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The unit group $R^\times$ is finitely presented, and we describe an explicit finite presentation. The homology calculation uses leaf coordinates, simultaneous extensions of ordered frames, and finite-field actions on stabilizers. The Steinberg argument lifts relations from a simply connected frame complex and refines coordinates. We also state separate criteria for the two arguments over rings of characteristic two.

math.KT↗
arXiv2026

Around Segal conjecture in p-adic geometry

This article records multiple results coming from interplay between de-completed topological periodic cyclic homology, Segal conjecture, and F-smoothness. We establish completeness of motivic filtration on de-completed topological periodic cyclic homology of commutative rings with weakly finitely generated absolute cotangent complex. When the ring in question is in addition F-smooth, we show that Segal conjecture holds for its topological Hochschild homology. We also identify our de-completed topological periodic cyclic homology with Manam's Frobenius untwisted topological periodic cyclic homology for quasiregular semiperfectoid rings. We find a crystalline degeneration of Segal conjecture which corresponds to such a statement for F-smoothness. On the other hand, inspired by constructions for topological Hochschild homology, the theory of cyclotomic synthetic spectra allows us to produce a relative conjugate filtration on Hodge--Tate cohomology and its variants, and in the same time, a relative conjugate filtration on topological Hochschild homology and its variants. As a consequence, we deduce transitivity of weak and strong F-smoothness.

math.KT↗
arXiv2026

Solvability of isotropic $ \mathrm{K}_1 $-functor over semilocal rings

We show that the $ \mathrm{K}_1 $-functor modeled on simple reductive groups over semilocal rings is solvable if the isotropic rank is at least $ 2 $ and that the Tits index is neither $ {}^{2} \mathsf{E}_{6, 2}^{16'} $ nor $ \mathsf{E}_{8, 2}^{78} $. For these two Tits indices the result is already known, but assuming that the base ring contains a field. Our result implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

math.KT↗