An arithmetic Lefschetz–Riemann–Roch theorem. Issue 3 (25th June 2020)

• Record Type: Journal Article
• Authors: Tang, Shun
• Is Part Of: Proceedings of the London Mathematical Society. Volume 122:Issue 3(2021)
• Page Start: 377
• Page End: 433
• Publication Date: 2020-06-25
• Subjects: 14C40 -- 14G40 -- 14L30 -- 19E08 -- 58J52 (primary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12349
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalizable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fiber, we define a direct image map between corresponding higher equivariant arithmetic K‐groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K‐groups and the higher arithmetic concentration theorem developed in Tang ( Math. Z . 290 (2018) 307–346) to prove an arithmetic Lefschetz‐Riemann‐Roch theorem. This theorem can be viewed as a generalization, to the higher equivariant arithmetic K‐theory, of the fixed‐point formula of Lefschetz type proved by Köhler and Roessler ( Invent. Math . 145 (2001) 333–396).