The Riemann–Roch theorem for the Adams operations

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2023-08-01 DOI:10.1016/j.exmath.2023.07.002

A. Navarro , J. Navarro

{"title":"The Riemann–Roch theorem for the Adams operations","authors":"A. Navarro , J. Navarro","doi":"10.1016/j.exmath.2023.07.002","DOIUrl":null,"url":null,"abstract":"<div>We prove the classical Riemann–Roch theorems for the Adams operations <math><mrow><mspace></mspace><msup><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi></mrow></msup><mspace></mspace></mrow></math> on <math><mi>K</mi></math>-theory: a statement with coefficients on <math><mrow><mi>Z</mi><mrow><mo>[</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow></mrow></math>, that holds for arbitrary projective morphisms, as well as another statement 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 paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.</div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Expositiones Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0723086923000622","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove the classical Riemann–Roch theorems for the Adams operations $ψ^{j}$ on $K$ -theory: a statement with coefficients on $Z [j^{- 1}]$ , that holds for arbitrary projective morphisms, as well as another statement 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 paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

查看原文本刊更多论文

Adams运算的Riemann-Roch定理

我们证明了k理论上Adams运算的经典Riemann-Roch定理:一个在Z[j−1]上有系数的命题，它适用于任意射影态射，以及另一个具有积分系数的命题，它适用于闭浸入。在有理系数存在的情况下，我们还分析了一个Adams运算对应的Riemann-Roch公式与chen特征的类似公式之间的关系。为了做到这一点，我们完成了对Panin-Smirnov的工作的基本阐述，这是由第一作者在前一篇论文中发起的。他们关于代数变异的有向上同论的概念允许使用经典论证来证明一般和整齐的陈述，这意味着所有上述结果都是特殊情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

期刊介绍： Our aim is to publish papers of interest to a wide mathematical audience. Our main interest is in expository articles that make high-level research results more widely accessible. In general, material submitted should be at least at the graduate level.Main articles must be written in such a way that a graduate-level research student interested in the topic of the paper can read them profitably. When the topic is quite specialized, or the main focus is a narrow research result, the paper is probably not appropriate for this journal. Most original research articles are not suitable for this journal, unless they have particularly broad appeal.Mathematical notes can be more focused than main articles. These should not simply be short research articles, but should address a mathematical question with reasonably broad appeal. Elementary solutions of elementary problems are typically not appropriate. Neither are overly technical papers, which should best be submitted to a specialized research journal.Clarity of exposition, accuracy of details and the relevance and interest of the subject matter will be the decisive factors in our acceptance of an article for publication. Submitted papers are subject to a quick overview before entering into a more detailed review process. All published papers have been refereed.