Residues on affine Grassmannians

Mathieu Florence; Philippe Gille

doi:10.1515/crelle-2021-0007

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Residues on affine Grassmannians

Mathieu Florence and Philippe Gille

Published/Copyright: March 13, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 776

Abstract

Given a linear group G over a field k, we define a notion of index and residue of an element g∈G⁢(k⁢((t))). The index r⁢(g) is a rational number and the residue a group homomorphism res(g):𝔾a⁢ or ⁢𝔾m→G. This provides an alternative proof of Gabber’s theorem stating that G has no subgroups isomorphic to 𝔾a or 𝔾m iff G⁢(k⁢[[t]])=G⁢(k⁢((t))). In the case of a reductive group, we offer an explicit connection with the theory of affine Grassmannians.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR Geolie

Award Identifier / Grant number: ANR-15-CE 40-0012

Funding statement: The authors are supported by the French National Research Agency, project ANR Geolie, ANR-15-CE 40-0012.

A Appendix: Descent

Lemma A.1.

Let S be a scheme. Let G be a S-group scheme flat of finite type. We are given a left action of G on an S-scheme X of finite type such that X admits a G-linearized line bundle L which is relatively ample over S. Let T be an S-scheme and let f:E→T be a GT-torsor. Then the contracted product E∧GXT over T is representable by a T-scheme. Furthermore, we have:

The line bundle ℳ=E∧Gℒ on E∧GXT is relatively ample over T.
If a S-group scheme J acts (on the left) on f:E→T such that T admits a J-linearized line bundle which is relatively ample over S, then E∧GXT admits a J-linearized line bundle which is relatively ample over T.

Proof.

The first part with property (i) is [2, Section 10.2, Lemme 6]. For establishing (ii), we are given a J-linearized line bundle 𝒩0 on T which is relatively ample over S. We consider the mapping h:E∧GXT→T, we know that there exists a positive integer n such that the line bundle 𝒩=h*⁢(𝒩0)⊗n⊗ℳ on E∧GXT→T is relatively ample over T (see [18, 4.6.13 (ii)]). This line bundle is J-linearized as desired. ∎

Lemma A.2.

Let S be a scheme and let G be a flat S-group scheme locally of finite type. We are given a G-morphism of S-schemes f:X→Y. We assume that G acts freely on X and on Y, that f is affine and that the fppf quotient Y/G is representable by a S-scheme. Then the fppf quotient X/G is representable by an S-scheme.

Proof.

Put Z:=Y/G. Assume first that the G-torsor Y→Z is trivial. Equivalently, the G-scheme Y is isomorphic, over S, to G×SZ. Choosing such an isomorphism allows to consider f as a G-morphism X→G×SZ. Put X0:=f-1⁢(e×SZ); it is a closed subscheme of X, which is transverse to the G-action: the natural G-morphism G×SX0→X is an isomorphism. Thus, X/G is represented by X0.

For the general case, we proceed by descent. Considering f as a morphism of Z-schemes, we may replace S by Z (and G by G×SZ), and assume for simplicity that Z=S. Hence, Y is a G-torsor over S. Put S′:=Y. We are going to base-change the situation via the morphism S′→S. Put

f′:=f×SS′:X′:=X×SS′→Y′:=Y×SS′.

Now, the G-torsor Y′→S′ is trivial, so that the quotient X′→X′/G exists (and is a trivial G-torsor) by the discussion above. Since X′→Y′ is affine, it follows that X′/G→Y′/G=S′ is affine as well. It is equipped with a canonical descent data for the fpqc morphism S′→S. Hence, this data is effective (by fpqc descent for affine schemes), yielding an arrow X~→S. Now, using descent for morphisms, the S′-arrow X′→X′/G descends to an S-arrow X→X~. This is the quotient X→X/G we sought for. This fact can, again, be checked by descent. ∎

Acknowledgements

It is a pleasure to thank Ofer Gabber and Laurent Moret-Bailly for useful conversations. We thank also Simon Riche and Xinwen Zhu for their expertise on affine Grassmannians and Alexis Bouthier for his reading. Finally, we would like to thank the reviewers for their helpful comments.

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Received: 2019-11-27

Revised: 2020-11-06

Published Online: 2021-03-13

Published in Print: 2021-07-01

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https://doi.org/10.1515/crelle-2021-0007