Abstract
We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.
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Notes
This is an important assumption that we keep throughout the paper.
In [49], affine flag/Grassmannian varieties for groups that are not necessarily constant are referred to as “twisted”. Here we omit this adjective.
See the forthcoming thesis of B. Levin.
The torus Z ∗ is induced because the Galois group permutes \(\varDelta _{G^{*}}\setminus \varDelta _{M^{*}}\).
Note that G is not always connected and so the set-up differs slightly from the previous paragraph where it was assumed that G is connected.
We emphasize here that, in general, G is not connected and that \(G={\mathbf {G}}_{\mathbb {Q}_{p}}^{\circ}\).
For this, we need to allow in our formalism Shimura varieties for some non-connected reductive groups.
Note that under the sign convention of the Kottwitz homomorphism in [37], t λ acts on \(\mathcal {A}(G',S', F')\) by v↦v−λ.
As usual, this category is not the “real” derived category of \(\operatorname {Sh}_{c}(X\times_{s}\eta,\overline{\mathbb {Q}}_{\ell})\), but is defined via a limit process. See [25, Footnote 2].
References
Anantharaman, S.: Schémas en Groupes, Espaces Homogènes et Espaces Algébriques sur Une Base de Dimension 1, sur les groupes algébriques, Soc. Math. France, vol. 33, pp. 5–79. Bull. Soc. Math. France, Paris (1973)
Arkhipov, S., Bezrukavnikov, R.: Perverse sheaves on affine flags and Langlands dual group. Isr. J. Math. 170, 135–183 (2009). With an appendix by Bezrukavrikov and Ivan Mirković
Artin, M.: Versal deformations and algebraic stacks. Invent. Math. 27, 165–189 (1974)
Beauville, A., Laszlo, Y.: Un lemme de descente. C. R. Math. Acad. Sci. Paris Sér. I Math. 320(3), 335–340 (1995)
Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. Preprint. http://www.math.utexas.edu/users/benzvi/BD/hitchin.pdf
Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578. Springer, Berlin (1994)
Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Publ. Math. IHÉS 60, 197–376 (1984)
Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. Fr. 112(2), 259–301 (1984)
Bruhat, F., Tits, J.: Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 34(3), 671–698 (1987)
Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires. Bull. Soc. Math. Fr. 115(2), 141–195 (1987)
Chernousov, V., Gille, P., Pianzola, A.: Torsors over the punctured affine line. Am. J. Math. 134(6), 1541–1583 (2012)
Colliot-Thélène, J.-L.: Résolutions flasques des groupes linéaires connexes. J. Reine Angew. Math. 618, 77–133 (2008)
Conrad, B.: Reductive group schemes. Notes for the SGA 3 Summer School, Luminy 2011. http://math.stanford.edu/~conrad/papers/luminysga3.pdf
Deligne, P.: Travaux de Shimura. In: Séminaire Bourbaki, 23ème année (1970/71), Exp. No 389, Lecture Notes in Math., vol. 244, pp. 123–165. Springer, Berlin (1971)
Edixhoven, B.: Néron models and tame ramification. Compos. Math. 81(3), 291–306 (1992)
Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5(1), 41–68 (2003)
Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144(2), 253–280 (2001)
Gille, P.: Torseurs sur la droite affine. Transform. Groups 7(3), 231–245 (2002)
Görtz, U.: On the flatness of models of certain Shimura varieties of PEL-type. Math. Ann. 321(3), 689–727 (2001)
Görtz, U.: On the flatness of local models for the symplectic group. Adv. Math. 176(1), 89–115 (2003)
Görtz, U., Haines, T.: The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties. J. Reine Angew. Math. 609, 161–213 (2007)
Haines, T.: Test functions for Shimura varieties: the Drinfeld case. Duke Math. J. 106(1), 19–40 (2001)
Haines, T.: Introduction to Shimura varieties with bad reduction of parahoric type. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Math. Proc., vol. 4, pp. 583–642. Am. Math. Soc., Providence (2005)
Haines, T.: The base change fundamental lemma for central elements in parahoric Hecke algebras. Duke Math. J. 149(3), 569–643 (2009)
Haines, T., Ngô, B.C.: Nearby cycles for local models of some Shimura varieties. Compos. Math. 133(2), 117–150 (2002)
Haines, T., Rapoport, M.: On parahoric subgroups (2008). Appendix to [49]
Haines, T., Rostami, S.: The Satake isomorphism for special maximal parahoric Hecke algebras. Represent. Theory 14, 264–284 (2010)
He, X.: Normality and Cohen-Macaulayness of local models of Shimura varieties. Preprint (2012). arXiv:1202.4119
Illusie, L.: Autour du théorème de monodromie locale. In: Périodes p-adiques, Bures-sur-Yvette, 1988, Astérisque, vol. 223, pp. 9–57 (1994)
Jacobson, N.: A note on hermitian forms. Bull. Am. Math. Soc. 46, 264–268 (1940)
Kisin, M., Pappas, G.: Integral models for Shimura varieties with parahoric level structure, in preparation
Kneser, M.: Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II. Math. Z. 89, 250–272 (1965)
Knutson, D.: Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Springer, Berlin (1971)
Kottwitz, R.: Shimura varieties and twisted orbital integrals. Math. Ann. 269(3), 287–300 (1984)
Kottwitz, R.: Stable trace formula: cuspidal tempered terms. Duke Math. J. 51(3), 611–650 (1984)
Kottwitz, R.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5(2), 373–444 (1992)
Kottwitz, R.: Isocrystals with additional structure. II. Compos. Math. 109(3), 255–339 (1997)
Kottwitz, R., Rapoport, M.: Minuscule alcoves for \({\rm GL}_{n}\) and \(G{\rm Sp}_{2n}\). Manuscr. Math. 102(4), 403–428 (2000)
Landvogt, E.: A Compactification of the Bruhat-Tits Building. Lecture Notes in Mathematics, vol. 1619. Springer, Berlin (1996)
Landvogt, E.: Some functorial properties of the Bruhat-Tits building. J. Reine Angew. Math. 518, 213–241 (2000)
Larsen, M.: Maximality of Galois actions for compatible systems. Duke Math. J. 80(3), 601–630 (1995)
Laszlo, Y., Sorger, C.: The line bundles on the moduli of parabolic G-bundles over curves and their sections. Ann. Sci. Éc. Norm. Super. (4) 30(4), 499–525 (1997)
Laumon, G.: Vanishing cycles over a base of dimension ≥1. In: Algebraic Geometry, Tokyo/Kyoto, 1982, Lecture Notes in Math., vol. 1016, pp. 143–150. Springer, Berlin (1983)
Lusztig, G.: Singularities, character formulas, and a q-analog of weight multiplicities. In: Analysis and Topology on Singular Spaces, II, III, Luminy, 1981. Astérisque, vol. 101, pp. 208–229. Soc. Math. France, Paris (1983)
Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007)
Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Algebr. Geom. 9(3), 577–605 (2000)
Pappas, G., Rapoport, M.: Local models in the ramified case. I. The EL-case. J. Algebr. Geom. 12(1), 107–145 (2003)
Pappas, G., Rapoport, M.: Local models in the ramified case. II. Splitting models. Duke Math. J. 127(2), 193–250 (2005)
Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118–198 (2008). With an appendix by Haines, T. and Rapoport, M.
Haines, T., Rapoport, M.: Local models in the ramified case. III. Unitary groups. J. Inst. Math. Jussieu 8(3), 507–564 (2009)
Gille, P., Rapoport, M., Smithling, B.: Local models of Shimura varieties, I. Geometry and combinatorics. Handbook of Moduli (to appear). arXiv:1011.5551
Philippe, G., Pianzola, A.: Torsors, reductive group schemes and extended affine Lie algebras. Preprint. arXiv:1109.3405 (to appear, Mem. AMS)
Prasad, G., Yu, J.-K.: On finite group actions on reductive groups and buildings. Invent. Math. 147(3), 545–560 (2002)
Rapoport, M.: A guide to the reduction modulo p of Shimura varieties. Astérisque 298, 271–318 (2005). Automorphic forms. I
Rapoport, M., Zink, Th.: Period Spaces for p-Divisible Groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)
Raynaud, M.: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics, vol. 119. Springer, Berlin (1970)
Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)
Richarz, T., Zhu, X.: Appendix to [76]
Rostami, S.: Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties. Preprint. arXiv:1112.0074
Scholze, P.: The Langlands-Kottwitz method and deformation spaces of p-divisible groups. J. Am. Math. Soc. 26(1), 227–259 (2013)
Scholze, P., Shin, S.W.: On the cohomology of compact unitary group Shimura varieties at ramified split places. J. Am. Math. Soc. 26(1), 261–294 (2013)
Serre, J.-P.: Groupes de Grothendieck des schémas en groupes réductifs déployés. Publ. Math. Inst. Hautes Études Sci. 34, 37–52 (1968)
Serre, J.-P.: Galois Cohomology, 5th edn. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1994)
Seshadri, C.S.: Triviality of vector bundles over the affine space K 2. Proc. Natl. Acad. Sci. USA 44, 456–458 (1958)
SGA3: Schémas en groupes. III: In: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, vol. 153. Springer, Berlin (1962/1964)
SGA7I: Groupes de monodromie en géométrie algébrique. I. In: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D.S. Rim. Lecture Notes in Mathematics, vol. 288. Springer, Berlin (1972) (in French)
SGA7II: Groupes de monodromie en géométrie algébrique. II. In: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics, vol. 340. Springer, Berlin (1973) (in French)
Smithling, B.: Topological flatness of orthogonal local models in the split, even case. I. Math. Ann. 350(2), 381–416 (2011)
Springer, T.A.: Linear Algebraic Groups, 2nd edn. Modern Birkhäuser Classics. Birkhäuser, Boston (2009)
Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of the Am. Math. Soc., vol. 80. Am. Math. Soc., Providence (1968)
Thomason, R.W.: Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes. Adv. Math. 65(1), 16–34 (1987)
Tits, J.: Reductive groups over local fields. In: Automorphic Forms, Representations and L-Functions, Oregon State Univ, Corvallis, Ore, 1977. Proc. Sympos. Pure Math., vol. XXXIII, pp. 29–69. Am. Math. Soc., Providence (1979)
Tsukamoto, T.: On the local theory of quaternionic anti-hermitian forms. J. Math. Soc. Jpn. 13, 387–400 (1961)
Waterhouse, W.C., Weisfeiler, B.: One-dimensional affine group schemes. J. Algebra 66(2), 550–568 (1980)
Yu, J.-K.: Smooth models associated to concave functions in Bruhat-Tits theory. Preprint (2002). http://www.math.purdue.edu/~jyu/prep/model.pdf
Zhu, X.: The geometric Satake correspondence for ramified groups. Preprint. arXiv:1107.5762
Zhu, X.: On the coherence conjecture of Pappas and Rapoport. Preprint. arXiv:1012.5979
Acknowledgements
The authors would like to warmly thank M. Rapoport, B. Conrad, T. Haines and B. Levin for useful discussions and comments. G.P. is partially supported by NSF grant DMS11-02208. X.Z. is partially supported by NSF grant DMS10-01280.
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Appendix: Homogeneous spaces
Appendix: Homogeneous spaces
In this section, we study the representability of certain quotients of group schemes over a two-dimensional base. The results are used in Chap. 6 to show the ind-representability of the global affine Grassmannian \({\rm Gr}_{\mathcal {G}, X}\).
11.1
We assume that A is an excellent Noetherian regular ring of Krull dimension 2. If G, H are smooth group schemes over A and H↪G is a closed group scheme immersion then by Artin’s theorem [3], the fppf quotient G/H is represented by an algebraic space which is separated of finite presentation and in fact smooth over A.
Lemma 11.1
Suppose that G 1↪G 2↪G 3 are closed group scheme immersions and G 1, G 2, G 3 are smooth over A. The natural morphism
is an fppf fibration with fibers isomorphic to G 2/G 1. Suppose that G 2/G 1 is quasi-affine (affine). If G 3/G 2 is a scheme, then so is G 3/G 1. If in addition G 3/G 2 is quasi-affine (resp. affine), then so is G 3/G 1.
Proof
The first statement follows from the fact that fppf descent is effective for quasi-affine schemes. In fact, by our assumption, G 3/G 1→G 3/G 2 is a quasi-affine morphism. If G 3/G 2 is quasi-affine, then G 3/G 1 is also quasi-affine (affine) since its structure morphism to A is a composition of quasi-affine (resp. affine) morphisms and as such is also quasi-affine (resp. affine). □
Proposition 11.2
Let \(\mathcal {G}, \mathcal {H}\to S=\mathrm {Spec}(A)\) be two smooth affine group schemes with connected fibers. Assume that \(\mathcal {H}\) is a closed subgroup scheme of \(\mathcal {G}\).
Set \(\mathcal {G}=\mathrm {Spec}(B)\), so that B is an A-Hopf algebra. Then there is a free finitely generated A-module M=A n with \(\mathcal {G}\)-action (i.e. a B-comodule) and a projective A-submodule W⊂M which is a locally a direct summand, such that:
-
(i)
There is a \(\mathcal {G}\)-equivariant surjection \({\rm Sym}^{\bullet}_{A}(M) \to B \) and the \(\mathcal {G}\)-action on M gives a group scheme homomorphism \(\rho: \mathcal {G}\hookrightarrow {\rm GL}(M)\) which is a closed immersion.
-
(ii)
The representation ρ identifies \(\mathcal {H}\) with the subgroup scheme of \(\mathcal {G}\) that stabilizes W.
Proof
Write \(\mathcal {G}=\mathrm {Spec}(B)\), \(\mathcal {H}=\mathrm {Spec}(B')\) and let p:B→B′ be the ring homomorphism that corresponds to \(\mathcal {H}\subset \mathcal {G}\). Observe that B, B′ are Hopf algebras over A. We will often refer to the B-comodules for the Hopf algebra B as “modules with \(\mathcal {G}\)-action”. Since \(\mathcal {G}\), \(\mathcal {H}\) are smooth with connected geometrical fibers both B and B′ are projective A-modules by Raynaud-Gruson [57, Proposition 3.3.1]. We start with two lemmas. □
Lemma 11.3
Let P be a projective A-module and N⊂P be a finitely generated A-submodule. Then the A-torsion submodule of P/N is finitely generated.
Proof
As P is a direct summand of a free A-module, we can assume that P=A I itself is free, with a basis {e i ;i∈I}. Let n 1,…,n t be a set of generators of N, and write n i =∑a ij e j . Then J={j∈I∣∃i such that a ij ≠0} is a finite set and A I/N=A J/N⊕A I−J. The conclusion follows. □
Lemma 11.4
Let P be a projective A-module and suppose that N⊂P is a finitely generated A-submodule. If P/N is torsion free, then N is a projective A-module.
Proof
Observe that N is A-torsion free. Consider the double dual N ∨∨. We have N⊂N ∨∨⊂P, and N ∨∨/N⊂P/N is torsion. Therefore, N=N ∨∨, which is projective by our assumption that A is regular and two-dimensional. □
Now let us prove the proposition. Observe first [71, Corollary 3.2] and its proof imply (i) of the proposition. (In other words, [71] implies that such a \(\mathcal {G}\) is linear, i.e. a closed subgroup scheme of \({\rm GL}_{n}\).) To obtain the proof of the whole proposition we have to refine this construction from [71] to account for the subgroup scheme \(\mathcal {H}\).
Let V⊂B be a finitely generated A-submodule with \(\mathcal {G}\) action that contains both a set of generators of I=ker(B→B′) and a set of generators of B as an A-algebra. Let p(V) be the image of V under p:B→B′. The following diagram is a commutative diagram of modules with \(\mathcal {G}\)-action
where \(\mathcal {G}\) acts on V⊗ A B, p(V)⊗ A B, B⊗ A B, B′⊗ A B via the actions on the second factors.
The image of \(N:=(p\otimes1)\cdot{\rm coact}(V)\) in p(V)⊗ A B is a finite A-submodule with \(\mathcal {G}\)-action. Let ϵ:B→A be the unit map which splits the natural A⊂B. Let M be the image of N under \(B'\otimes B\stackrel{1\otimes\epsilon}{\to} B'\). Observe that M is a finite A-module, but is not necessarily \(\mathcal {G}\)-stable. By [62, Proposition 2], we can choose a finite \(\mathcal {G}\)-stable A-module \(\tilde{M}\) in B′ containing M. By Lemma 11.3, we can enlarge \(\tilde{M}\) if necessary to assume that \(B'/\tilde{M}\) is torsion free (so \(\tilde{M}\) is projective over A by Lemma 11.4). We regard \(\tilde{M}\) as a \(\mathcal {G}\)-stable submodule of B′⊗ A B via \(\tilde{M}\subset B'=B'\otimes_{A}A\subset B'\otimes_{A}B\) (this is indeed a \(\mathcal {G}\)-stable submodule since the inclusion A⊂B is \(\mathcal {G}\)-equivariant). Let \(\tilde{N}=\tilde{M}+N\). Then \(\tilde{N}\) is a finite \(\mathcal {G}\)-stable submodule of B′⊗ A B, and under the map 1⊗ϵ:B′⊗ A B→B′, \((1\otimes\epsilon)(\tilde{N})=\tilde{M}\). Observe that the torsion submodule \(t((B'\otimes_{A}B)/\tilde{N})\subset(B'\otimes_{A}B)/\tilde{N}\) is a module with \(\mathcal {G}\)-action and maps to zero under \((B'\otimes_{A}B)/\tilde{N}\stackrel{1\otimes \epsilon}{\to} B'/\tilde{M}\). Let \(\tilde{N}'\) be the preimage of \(t((B'\otimes_{A}B)/\tilde{N})\) under \(B'\otimes_{A}B\to(B'\otimes_{A}B)/\tilde{N}\). From Lemma 11.3, \(\tilde{N}'\) is finite \(\mathcal {G}\)-stable A-module, and \((1\otimes \epsilon)(\tilde{N}')=\tilde{M}\). In addition, \(\tilde{N}'\) is locally free since \((B'\otimes_{A}B)/\tilde{N}'\) is torsion free.
Let \(\tilde{V}\) be the \(\mathcal {G}\)-stable A-submodule of B given by the fiber product
Observe that \((p\otimes1)\cdot{\rm comult}: B' \to B'\otimes_{A}B\) is injective. Therefore, \(\tilde{V}\) is an A-submodule of \(\tilde{N}'\) and therefore it is finitely generated over A. In addition, \(\tilde{N}'\supset N\), \(V\subset\tilde{V}\). Since \(B/\tilde{V}\hookrightarrow(B'\otimes_{A}B)/\tilde{N}'\) is torsion free, \(\tilde{V}\) is projective. Observe that
is just the projection p. Therefore, \(p(\tilde{V})=\tilde{M}\).
Therefore, we obtain the following commutative diagram
with the first row finitely generated projective A-modules. Notice that \(\tilde{V}\supset V\) contains a set of generators of the B-ideal I and a set of A-algebra generators of B. Hence, we obtain a closed immersion \(\mathcal {G}\xrightarrow{}{\rm GL}(\tilde{V})\) of group schemes and we can see that \(\mathcal {H}\) can be identified with the closed subgroup scheme of \(\mathcal {G}\) that preserves the direct summand \(W\subset M:=\tilde{V}\). By replacing M by M⊕M′ and W by W⊕M′ where M′ is a finitely generated projective A-module with trivial \(\mathcal {G}\)-action, we can assume that M is A-free as desired.
Corollary 11.5
Suppose that \(\mathcal {H}\subset \mathcal {G}\) are as in Proposition 11.2. Then the fppf quotient \(\mathcal {G}/\mathcal {H}\) is representable by a quasi-projective scheme over A.
Proof
By Artin’s theorem the fppf quotient \(\mathcal {G}/\mathcal {H}\) is represented by an algebraic space over A. The algebraic space \(\mathcal {G}/\mathcal {H}\) is separated of finite type and even smooth over A, the quotient \(\mathcal {G}\to \mathcal {G}/\mathcal {H}\) is also smooth. Take M and W as in Proposition 11.2 and set \(P:=\bigwedge^{\operatorname {rk}W}M\) and \(L=\bigwedge^{\operatorname {rk}W}W\subset\bigwedge^{\operatorname {rk}W}M=A^{r}\), where \(r={{\rm rank}(M)\choose{\rm rank}(W)}\). Then, \(\mathcal {H}\) is the stabilizer of [L] in \({\rm Proj}( \bigwedge^{\operatorname {rk}W}M)={\mathbb{P}}^{r-1}_{A}\). We obtain a morphism \(f: \mathcal {G}\to {\mathbb{P}}^{r-1}_{A}\). This gives a monomorphism \(\bar{f}: \mathcal {G}/\mathcal {H}\to {\mathbb{P}}^{r-1}_{A}\) which is a separated quasi-finite morphism of algebraic spaces. By [33, 6.15], \(\mathcal {G}/\mathcal {H}\) is a scheme and we can now apply Zariski’s main theorem to \(\bar{f}\). We obtain that \(\bar{f}\) is a composition of an open immersion with a finite morphism and we can conclude that \(\mathcal {G}/\mathcal {H}\) is quasi-projective. (See [13, proof of Theorem 2.3.1] for a similar argument.) □
Remark 11.6
General homogeneous spaces over Dedekind rings are schemes [1], but this is not always the case when the base is a Noetherian regular ring of dimension 2; see [56, X]. In loc. cit. Raynaud asks if \(\mathcal {G}/\mathcal {H}\) is a scheme when both \(\mathcal {G}\) and \(\mathcal {H}\) are smooth and affine over a normal base and \(\mathcal {H}\) has connected fibers. The above Corollary gives a partial answer to this question.
Corollary 11.7
Suppose that \(\mathcal {G}\) is a smooth affine group scheme with connected fibers over A. Then there n≥1 and a closed subgroup scheme embedding \(\mathcal {G}\hookrightarrow{\rm GL}_{n}\), such that the fppf quotient \({\rm GL}_{n}/\mathcal {G}\) is represented by a smooth quasi-affine scheme over A.
Proof
By Proposition 11.2 applied to \(\mathcal {G}\) and \(\mathcal {H}=\{e\}\), we see that there is a closed subgroup scheme embedding \(\rho: \mathcal {G}\hookrightarrow{\rm GL}_{m}\) (this follows also directly from [71]). Now apply Corollary 11.5 and its proof to the pair of the group \({\rm GL}_{m}\) with its closed subgroup scheme \(\mathcal {G}\). We obtain a \({\rm GL}_{m}\)-representation ρ′:GL m →GL r =GL(M) that induces a locally closed embedding \(\mathrm {GL}_{m}/\mathcal {G}\hookrightarrow{\mathbb{P}}^{r-1}\). Denote by \(\chi: \mathcal {G}\to \mathbb {G}_{\mathrm {m}}={\rm Aut}_{A}(L)\) the character giving the action of \(\mathcal {G}\) on the A-line L (as in the proof of Corollary 11.5) and consider \(\mathcal {G}\to \mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}\) given by g↦(ρ′(g),χ −1(g)). Consider the quotient \((\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}\); to prove it is quasi-affine it is enough to reduce to the case that A is local. Then L is free, L=A⋅v, and \(\mathcal {G}\) is the subgroup scheme of \(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}\) (acting by (g,a)⋅m=aρ′(g)(m)) that fixes v. This gives a quasi-finite separated monomorphism \((\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}\rightarrow{\mathbb{A}}^{r}\) and so by arguing as in the proof of Corollary 11.5 we see that \((\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}\) is quasi-affine. Consider now the standard diagonal block embedding \(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}\hookrightarrow \mathrm {GL}_{m+1}\). The quotient \(\mathrm {GL}_{m+1}/(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})\) is affine and we can conclude using Lemma 11.1. □
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Pappas, G., Zhu, X. Local models of Shimura varieties and a conjecture of Kottwitz. Invent. math. 194, 147–254 (2013). https://doi.org/10.1007/s00222-012-0442-z
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DOI: https://doi.org/10.1007/s00222-012-0442-z