Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2505.12364 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918242731163648 |
|---|---|
| author | Sala, Francesco |
| author_facet | Sala, Francesco |
| contents | B. Toen defined a Riemann-Roch map from the rational algebraic K-theory of a tame Deligne-Mumford quotient stack to the étale K-theory of its inertia. He proved that this map is an isomorphism and that it is covariant with respect to proper maps. Moreover G. Vezzosi and A. Vistoli proved a decomposition theorem for the equivariant K-theory of a noetherian scheme. In this paper we give a geometric definition of the Vezzosi-Vistoli decomposition, interpreting the pieces as corresponding to the components of the cyclotomic inertia. When the map from the cyclotomic inertia to the stack is finite, we can define a Riemann-Roch map in Toen's style. We prove that this map is an isomorphism and it is covariant with respect to proper relatively tame maps; moreover in some favourable circumstances we explicitly compute its inverse map, and show that we can recover Toen's one when the stack is tame Deligne-Mumford. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Riemann-Roch formula for stacks with finite cyclotomic inertia Sala, Francesco Algebraic Geometry B. Toen defined a Riemann-Roch map from the rational algebraic K-theory of a tame Deligne-Mumford quotient stack to the étale K-theory of its inertia. He proved that this map is an isomorphism and that it is covariant with respect to proper maps. Moreover G. Vezzosi and A. Vistoli proved a decomposition theorem for the equivariant K-theory of a noetherian scheme. In this paper we give a geometric definition of the Vezzosi-Vistoli decomposition, interpreting the pieces as corresponding to the components of the cyclotomic inertia. When the map from the cyclotomic inertia to the stack is finite, we can define a Riemann-Roch map in Toen's style. We prove that this map is an isomorphism and it is covariant with respect to proper relatively tame maps; moreover in some favourable circumstances we explicitly compute its inverse map, and show that we can recover Toen's one when the stack is tame Deligne-Mumford. |
| title | On a Riemann-Roch formula for stacks with finite cyclotomic inertia |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2505.12364 |