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Operations and poly-operations in algebraic cobordism

Vishik, Alexander

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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, 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.

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