Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2211.10178

The Riemann-Roch theorem for the Adams operations

A. Navarro·J. Navarro
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the classical Riemann-Roch theorems for the Adams operations $\,ψ^j\,$ on $K$-theory: a statement with coefficients on $\mathbb{Z}[j^{-1}]$, that holds for arbitrary projective morphisms, as well as another one with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann-Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin-Smirnov that was initiated by the first author in a previous work. Their notion of oriented cohomology theory of algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

Explore related subjects

math.KTmath.AGmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. Navarro, J. Navarro. 2022-11-18. The Riemann-Roch theorem for the Adams operations. https://arxiv.org/abs/2211.10178

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

Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra

Let $A$ be a finite dimensional algebra and let $\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\calN$ (resp. $G$, $\calG$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\rmHH^*(A)/\calN$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\rmHH^*(A)/G$ is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\calN$; hence $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ is not a finitely generated algebra. Furthermore, we show that $\rmHH^*(A)/\calG\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\rmHH^*(A)/\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.

math.KT↗