## Riemann-Roch for equivariant Chow groups

Citations: | 16 - 2 self |

### BibTeX

@MISC{Edidin_riemann-rochfor,

author = {Dan Edidin and William Graham},

title = {Riemann-Roch for equivariant Chow groups},

year = {}

}

### OpenURL

### Abstract

The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group G. For a G-space X, this theorem gives an isomorphism

### Citations

1002 |
Intersection Theory
- Fulton
- 1984
(Show Context)
Citation Context ...ight square also commutes. This proves Proposition 3.2. 4. The Riemann-Roch isomorphism All spaces considered here are again assumed to be separated, so that we can apply the Riemann-Roch theorems of =-=[Fu]-=-. Let G ⊂ GLn and let I ⊂ R(GLn) be the augmentation ideal. In this section we will use the notation ̂ G G (X) to denote the I-adic completion of G G (X) ⊗ Q, and continue to denote CH i G (X) ⊗ Q by ... |

323 | Éléments de géométrie algébrique - Grothendieck, Dieudonné - 1971 |

144 | Revêtements Étales et Groupe Fondamental - Grothendieck, Raynaud - 1971 |

124 | Algebraic Spaces - Knutson - 1971 |

109 | Equivariant intersection theory
- Edidin, Graham
- 1998
(Show Context)
Citation Context ...G (X) → ̂ G G (X) Q ≃ → ∞∏ CH i G(X)Q. Here ̂ GG (X) is the completion alon the augmentation ideal of the representation ring R(G), and he groups CH i G (X) are the equivariant Chow groups defined in =-=[EG2]-=-. The map τG has the same functorial properties as the non-equivariant Riemann-Roch map of [BFM], [Fu, Theorem 18.3] and if G acts freely then τG can be identified with the non-equivariant Todd class ... |

88 |
Equivariant K-Theory and Completion
- Atiyah, Segal
- 1969
(Show Context)
Citation Context ...(V, U) = (X × H V, X × H U) to define τ G X/H as ρU. Hence the maps coincide. In K-theory, part (a) of the next proposition is due to Thomason [Tho3], following ideas that go back to Atiyah and Segal =-=[AS]-=-. Part (c) has apparently been used implicitly by Thomason ([Tho4]), but we do not know of an explicit statement or proof. Proposition 3.2. Let H be a subgroup of G, and let X be an H-space. (a) There... |

44 | Quotient spaces modulo reductive algebraic groups - Seshadri - 1972 |

36 | Equivariant Chow Groups for Torus Actions - Brion - 1997 |

34 |
Intersection theory on algebraic stacks and Q-varieties
- Gillet
- 1984
(Show Context)
Citation Context ... X is separated, so is X × G U. If Z is a separated algebraic space, then there is a Riemann-Roch map τZ : G(Z) → CH ∗ (Z)Q with the same properties as the RiemannRoch map for schemes. This fact (cf. =-=[Gi]-=-) can be deduced from the Riemann-Roch theorem for quasi-projective schemes and the existence of Chow envelopes for separated algebraic spaces. Equivariant Chow groups We will use the notation AG k (X... |

31 |
Algebraic K-theory of group scheme actions. In Algebraic topology and algebraic K-theory
- Thomason
- 1983
(Show Context)
Citation Context ... 3.1 we can use vector bundles and open sets of the form (V, U) = (X × H V, X × H U) to define τ G X/H as ρU. Hence the maps coincide. In K-theory, part (a) of the next proposition is due to Thomason =-=[Tho3]-=-, following ideas that go back to Atiyah and Segal [AS]. Part (c) has apparently been used implicitly by Thomason ([Tho4]), but we do not know of an explicit statement or proof. Proposition 3.2. Let H... |

26 |
The representation ring of a compact Lie group
- Segal
- 1968
(Show Context)
Citation Context ...n map G G (X) → G G (X)I.26 DAN EDIDIN AND WILLIAM GRAHAM 5.1. The case G is diagonalizable. Let P = P0, P1, . . .Pk be the prime ideals in the support of GG i (X) as an R(G)-module. Following Segal =-=[Segal]-=- each prime Pi corresponds to a finite subgroup (called the support of Pi) Hi ⊂ G. It is defined as the minimal element of the set of subgroups H ⊂ G such that Pi ∈ Im(Spec R(H) → Spec R(G)). Note tha... |

21 | Groupes de Grothendieck des schemas en groupes reductifs deployes - Serre - 1968 |

21 | Geometric Invariant Theory (3rd enlarged edtion), Ergebnisse der Mathematik und ihrere Grenzgebiete 34 - Mumford, Fogarty, et al. - 1994 |

20 |
The index theorem for homogeneous differential operators
- Bott
- 1965
(Show Context)
Citation Context ...RIEMANN-ROCH 3 behaves naturally with respect to restriction to a subgroup (Section 3.2). In Section 3.1 we illustrate the use of this theorem by deriving the Weyl character formula for SL2 following =-=[B]-=-. In Section 4 we prove that the Riemann-Roch map induces an isomorphism on the completions. Section 5 contains results, mentioned above, on actions with finite stabilizers, and proves Vistoli’s conje... |

18 | Une formule de Lefschetz en K-théorie équivariante algébrique - Thomason - 1992 |

17 | Characteristic classes in the Chow ring
- Edidin, Graham
- 1997
(Show Context)
Citation Context ...U be an open subset of V 0 . A quotient (X ×U)/G exists; such a quotient is usually written as X × G U. Choose V and U such that the codimension of V − U is greater than k (this is always possible by =-=[EG1]-=-); then by definition, CH k G (X) = CHk (X × G U). With this as motivation, let V be a collection of pairs (V, U) of Gmodules and invariant open sets with the following properties: (i) G acts freely o... |

15 | The Grothendieck-Riemann-Roch Theorem for group scheme actions - Kock - 1998 |

15 | Linear Algebraic Groups, 2nd Enlarged Edition, Graduate Texts - Borel - 1991 |

9 |
Equivariant Grothendieck groups and equivariant Chow groups, in Classification of irregular varieties
- Vistoli
- 1990
(Show Context)
Citation Context ...d and G acts with finite stabilizers. There is a map τ : G G (X) → G G (X)I satisfying properties (a)-(e) of Theorem 3.1. ≃ → CH ∗ G (X)Q When G acts on X with finite reduced stabilizers then Vistoli =-=[Vi]-=- stated a theorem which asserted the existence of a map τX : G G (X) ⊗ Q → CH ∗ ([X/G] ⊗ Q) where here [X/G] is the Deligne-Mumford quotient stack. By [EG2, Proposition 14], CH ∗ ([X/G]) ⊗ Q = CH ∗ G ... |

7 | Lefschetz-Riemann-Roch theorem and coherent trace formula - Thomason - 1986 |

4 | Riemann-Roch type theorems for arithmetic schemes with a finite group action, preprint - Chinburg, Erez, et al. |

3 | Intersection Theory, Ergebnisse, 3 - Fulton - 1984 |

3 | Algebraic Spaces - Knuston - 1971 |

3 |
Riemann-Roch Theorems for Deligne-Mumford Stacks, preprint math.AG/9803076
- Toen
- 1990
(Show Context)
Citation Context ... give a simple proof of a conjecture of Vistoli (Corollary 5.2). If G is diagonalizable, then we can express GG (X) in terms of the equivariant Chow groups (an unpublished result of Vistoli, cf. also =-=[To]-=-). Actions with finite stabilizers are particularly important because quotients by these actions arise naturally in geometric invariant theory. In a subsequent paper, we will use these results to expr... |

3 |
The Chow Ring of the Symmetric Group
- Totaro
- 1994
(Show Context)
Citation Context ...bset U of a representation V , where G acts freely on U, and V − U is a finite union of linear subspaces. Approximations to EG by open sets in representations were introduced by Totaro in Chow theory =-=[T]-=-, and used in [EG2] to define equivariant Chow groups. However, in these papers, V − U is only required to have large codimension: because Chow groups are naturally graded we can identify ⊕N 0 CH i G ... |

3 |
Riemann-Roch for singular varieties Publ
- Baum, Fulton, et al.
- 1975
(Show Context)
Citation Context ...l of the representation ring R(G), and the groups CH i G (X) are the equivariant Chow groups defined in [EG2]. The map τG has the same functorial properties as the non-equivariant Riemann-Roch map of =-=[BFM]-=-, [Fu, Theorem 18.3] and if G acts freely then τG can be identified with the non-equivariant Todd class map τX/G : G(X/G) → CH ∗ (X/G)Q. The key to proving this isomorphism is a geometric description ... |

2 | Linear Algebra (Springer–Verlag - Lang - 1987 |

2 |
Equivariant algebraic vs
- Thomason
- 1988
(Show Context)
Citation Context ...aps coincide. In K-theory, part (a) of the next proposition is due to Thomason [Tho3], following ideas that go back to Atiyah and Segal [AS]. Part (c) has apparently been used implicitly by Thomason (=-=[Tho4]-=-), but we do not know of an explicit statement or proof. Proposition 3.2. Let H be a subgroup of G, and let X be an H-space. (a) There is a natural isomorphism of R(G)-modules G G (G× H X) ≃ G H (X), ... |

2 | Notes on the construction of the moduli space of curves, to appear - Edidin |

1 |
Riemann-Roch for singular varieties I.H.E.S
- Baum, Fulton, et al.
- 1975
(Show Context)
Citation Context ...al of the representation ring R(G), and he groups CH i G (X) are the equivariant Chow groups defined in [EG2]. The map τG has the same functorial properties as the non-equivariant Riemann-Roch map of =-=[BFM]-=-, [Fu, Theorem 18.3] and if G acts freely then τG can be identified with the non-equivariant Todd class map τX/G : G(X/G) → CH ∗ (X/G)Q. The key to proving this isomorphism is a geometric description ... |

1 | Elements de Geometrie Algebraique - Grothendieck, Dieudonné - 1965 |

1 | Equivariant and topological K-theory - Thomason - 1986 |