Operations and poly-operations in algebraic cobordism

Vishik, Alexander

Abstract

© 2020 Elsevier Inc. In the case of a field of characteristic zero, we describe all operations (including non-additive ones) from a theory A⁎ obtained from Algebraic Cobordism Ω⁎ of M. Levine-F. Morel by change of coefficients to any oriented cohomology theory B⁎ (in the sense of Definition 2.1). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done in [24], as well as the topological one obtained by T. Kashiwabara - see [6]. The key new ingredients which permit us to treat the non-additive operations are: the use of poly-operations and the “Discrete Taylor expansion”. As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p - see [23], which permits to sharpen results on the structure of Algebraic Cobordism - see [22]. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the case of multiplicative operations proved by I. Panin in [13].

Citation

Vishik, A. (2020). Operations and poly-operations in algebraic cobordism. Advances in Mathematics, 366, Article 107066. https://doi.org/10.1016/j.aim.2020.107066

• Journal Article Type: Article
• Acceptance Date: Feb 11, 2020
• Online Publication Date: Feb 26, 2020
• Publication Date: Jun 3, 2020
• Deposit Date: Nov 19, 2018
• Publicly Available Date: Feb 27, 2021
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Electronic ISSN: 1090-2082
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 366
• Article Number: 107066
• DOI: https://doi.org/10.1016/j.aim.2020.107066
• Keywords: Cohomological operations ; Riemann-Roch theorem ; Algebraic cobordism ; Symmetric operations ; Poly-operations ; Discrete Taylor expansion
• Public URL: https://nottingham-repository.worktribe.com/output/1173988
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S000187082030092X
• Additional Information: This article is maintained by: Elsevier; Article Title: Operations and poly-operations in algebraic cobordism; Journal Title: Advances in Mathematics; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.aim.2020.107066; Content Type: article; Copyright: © 2020 Elsevier Inc. All rights reserved.