Projectivity of the Witt vector affine Grassmannian. (English) Zbl 1397.14064
The authors prove that for a perfect field \(k\) of characteristic \(p > 0\), the Witt vector affine Grassmannian
\[
\mathrm{GL}_n(W(k)[1/p])/\mathrm{GL}_n(W(k))
\]
(analogous to the affine Grassmannian \(\mathrm{GL}_n(k(\!(t)\!))/\mathrm{GL}_n(k[\![t]\!])\) and classifying \(W(k)\)-lattices in \(W(k)[1/p]^n\), which is represented by an ind-projective ind-scheme) is representable by an inductive limit of the perfection of projective varieties \(\mathrm{Gr}^{W \mathrm{aff}, [a,b]}\) over \(\mathbb{F}_p\) for all integers \(a \leq b\) (the functor sending a perfect \(\mathbb{F}_p\)-algebra \(R\) to \(W(R)\)-lattices inside \(W(R)[1/p]^n\) between \(p^aW(R)^n\) and \(p^bW(R)^n\)) by constructing an ample line bundle, which improves independently previous results of X. Zhu [Ann. Math. (2) 185, No. 2, 403–492 (2017; Zbl 1390.14072)] who showed that it is represented by the perfection of a proper algebraic space over \(\mathbb{F}_p\). Note that the ring of Witt vectors is not well-behaved for non-perfect \(\mathbb{F}_p\)-algebras; for example, it can have \(p\)-torsion. The article also contains basic results on the perfection functor.
Reviewer: Timo Keller (Bayreuth)
Keywords:
Witt vector affine GrassmannianCitations:
Zbl 1390.14072References:
| [1] | Aberbach, I.M., Hochster, M.: Finite Tor dimension and failure of coherence in absolute integral closures. J. Pure Appl. Algebra 122(3), 171-184 (1997) · Zbl 0908.13006 · doi:10.1016/S0022-4049(97)00049-2 |
| [2] | Asgharzadeh, M.: Homological properties of the perfect and absolute integral closures of Noetherian domains. Math. Ann. 348(1), 237-263 (2010) · Zbl 1211.13008 · doi:10.1007/s00208-009-0474-x |
| [3] | Barwick, C.: On the algebraic K-theory of higher categories. J. Topol. (2012, to appear). arXiv:1204.3607 · Zbl 1364.19001 |
| [4] | Berthelot, P., Bloch, S., Esnault, H.: On Witt vector cohomology for singular varieties. Compos. Math. 143(2), 363-392 (2007) · Zbl 1213.14040 · doi:10.1112/S0010437X06002533 |
| [5] | Brylinski, J.-L., Deligne, P.: Central extensions of reductive groups by \[{ K}_2\] K2. Publ. Math. Inst. Hautes Études Sci. 94, 5-85 (2001) · Zbl 1093.20027 · doi:10.1007/s10240-001-8192-2 |
| [6] | Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261. Springer, Berlin (1984) (A systematic approach to rigid analytic geometry) · Zbl 0539.14017 |
| [7] | Bhatt, B.: Algebraization and Tannaka duality. Camb. J. Math. (2014, to appear). arXiv:1404.7483 · Zbl 1356.14006 |
| [8] | Bhatt, B., Halpern-Leistner, D.: Tannaka duality revisited (2015). arXiv:1507.01925 · Zbl 1401.14013 |
| [9] | Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions. Commun. Math. Phys. 164(2), 385-419 (1994) · Zbl 0815.14015 · doi:10.1007/BF02101707 |
| [10] | Bourbaki, N.: Éléments de mathématique. Masson, Paris (1985) (Algèbre commutative. Chapitres 5 à 7. [Commutative algebra. Chapters 5-7], Reprint) · Zbl 0547.13002 |
| [11] | Bhatt, B., Scholze, P.: The pro-étale topology for schemes. Astérisque (2013, to appear). arXiv:1309.1198 · Zbl 1351.19001 |
| [12] | Bhatt, B., Schwede, K., Takagi, S.: The weak ordinarity conjecture and F-singularities. In: Proceedings of the conference in honor of Professor Yujiro Kawamata on his 60th birthday (2013, to appear). arXiv:1307.3763 · Zbl 1390.14060 |
| [13] | Carlsson, G.: Derived completions in stable homotopy theory. J. Pure Appl. Algebra 212(3), 550-577 (2008) · Zbl 1146.55006 · doi:10.1016/j.jpaa.2007.06.015 |
| [14] | Chen, M., Kisin, M., Viehmann, E.: Connected components of affine Deligne-Lusztig varieties in mixed characteristic (2013). arXiv:1307.3845 · Zbl 1334.14017 |
| [15] | de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51-93 (1996) · Zbl 0916.14005 · doi:10.1007/BF02698644 |
| [16] | Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. (JEMS) 5(1), 41-68 (2003) · Zbl 1020.14002 · doi:10.1007/s10097-002-0045-x |
| [17] | Gabber, O.: Notes on some \[t\] t-structures. In: Geometric Aspects of Dwork Theory. vol. I, II, pp. 711-734. Walter de Gruyter GmbH & Co. KG, Berlin (2004) · Zbl 1074.14018 |
| [18] | Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. 11, 5-167 (1961) · doi:10.1007/BF02684273 |
| [19] | Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. 28, 5-255 (1966) · Zbl 0144.19904 · doi:10.1007/BF02684343 |
| [20] | Gepner, D., Groth, M., Nikolaus, T.: Universality of multiplicative infinite loop space machines (2013). arXiv:1305.4550 · Zbl 1336.55006 |
| [21] | Gabber, O., Ramero, L.: Almost ring theory. In: Lecture Notes in Mathematics, vol. 1800. Springer, Berlin (2003) · Zbl 1045.13002 |
| [22] | Greenberg, M.J.: Schemata over local rings. Ann. Math. 2(73), 624-648 (1961) · Zbl 0115.39004 · doi:10.2307/1970321 |
| [23] | Haboush, W.J.: Infinite dimensional algebraic geometry: algebraic structures on \[p\] p-adic groups and their homogeneous spaces. Tohoku Math. J. (2) 57(1), 65-117 (2005) · Zbl 1119.14004 · doi:10.2748/tmj/1113234835 |
| [24] | Hamacher, P.: The geometry of Newton strata in the reduction modulo p of Shimura varieties of PEL type (2013). arXiv:1312.0490 · Zbl 1335.14008 |
| [25] | Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness. (2014). arXiv:1402.3204 · Zbl 1521.14003 |
| [26] | Huber, R.: A generalization of formal schemes and rigid analytic varieties. Math. Z. 217(4), 513-551 (1994) · Zbl 0814.14024 · doi:10.1007/BF02571959 |
| [27] | Hartl, U., Viehmann, E.: The Newton stratification on deformations of local \[G\] G-shtukas. J. Reine Angew. Math. 656, 87-129 (2011) · Zbl 1225.14036 · doi:10.1515/crelle.2011.044 |
| [28] | Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic. Ann. Math. (2) 149(1), 253-286 (1999) · Zbl 0954.14004 · doi:10.2307/121025 |
| [29] | Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1), 19-55 (1976) · Zbl 0343.14008 · doi:10.7146/math.scand.a-11642 |
| [30] | Kreidl, M.: On \[p\] p-adic lattices and Grassmannians. Math. Z. 276(3-4), 859-888 (2014) · Zbl 1304.13010 · doi:10.1007/s00209-013-1225-y |
| [31] | Kruckman, A.: Notes on ultra filters (2012). https://math.berkeley.edu/ kruckman/ultrafilters.pdf. Accessed 27 Nov 2016 · Zbl 0815.14015 |
| [32] | Lazarsfeld, R.: Positivity in algebraic geometry. I. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48. Springer, Berlin (2004) (Classical setting: line bundles and linear series) · Zbl 1066.14021 |
| [33] | Lurie, J.: Higher topos theory. In: Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009) · Zbl 1175.18001 |
| [34] | Lurie, J.: Derived algebraic geometry viii: Quasi-coherent sheaves and tannaka duality theorems (2014). http://www.math.harvard.edu/ lurie/papers/DAG-VIII.pdf. Accessed 27 Nov 2016 · Zbl 0261.20025 |
| [35] | Lurie, J.: Higher algebra (2014). http://www.math.harvard.edu/ lurie/papers/higheralgebra.pdf. Accessed 27 Nov 2016 · Zbl 0391.55007 |
| [36] | Liu, Y., Zheng, W.: Enhanced six operations and base change theorem for Artin stacks. (2012). arXiv:1211.5948 · Zbl 0244.18008 |
| [37] | Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. 4(2), 1-62 (1969) · Zbl 0261.20025 · doi:10.24033/asens.1174 |
| [38] | Matthew, A.: The Galois group of a stable homotopy theory (2014). arXiv:1404.2156v1.pdf · Zbl 0535.13007 |
| [39] | May, J.\[P.: E_{\infty }\] E∞ spaces, group completions, and permutative categories. In: New Developments in Topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 61-93. London Math. Soc. Lecture Note Ser., No. 11. Cambridge University Press, London (1974) · Zbl 0815.14015 |
| [40] | MacLane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49(4), 28-46 (1963) · Zbl 0244.18008 |
| [41] | Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math. (2) 122(1), 27-40 (1985) · Zbl 0601.14043 · doi:10.2307/1971368 |
| [42] | May, J.P., Thomason, R.: The uniqueness of infinite loop space machines. Topology 17(3), 205-224 (1978) · Zbl 0391.55007 · doi:10.1016/0040-9383(78)90026-5 |
| [43] | Patel, D.: de Rham \[{\cal{E}}\] E-factors. Invent. Math. 190(2), 299-355 (2012) · Zbl 1275.14019 |
| [44] | Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118-198 (2008). (With an appendix by T. Haines and Rapoport) · Zbl 1159.22010 · doi:10.1016/j.aim.2008.04.006 |
| [45] | Quillen, D.: Higher algebraic \[KK\]-theory. I. In: Algebraic \[KK\]-Theory, I: Higher \[KK\]-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), vol. 341, pp. 85-147. Lecture Notes in Mathematics. Springer, Berlin (1973) · Zbl 0292.18004 |
| [46] | Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1-89 (1971) · Zbl 0227.14010 · doi:10.1007/BF01390094 |
| [47] | Rydh, D.: Submersions and effective descent of étale morphisms. Bull. Soc. Math. Fr. 138(2), 181-230 (2010) · Zbl 1215.14004 · doi:10.24033/bsmf.2588 |
| [48] | Segal, G.: Categories and cohomology theories. Topology 13, 293-312 (1974) · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6 |
| [49] | Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. In: Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., vol. 81, pp. 259-271. North-Holland, Amsterdam (1983) · Zbl 0528.58020 |
| [50] | Steinberg, R.: Générateurs, relations et revêtements de groupes algébriques. In: Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), pp. 113-127. Librairie Universitaire, Louvain, Gauthier-Villars, Paris (1962) · Zbl 0272.20036 |
| [51] | Segal, G., Wilson, G.: Loop groups and equations of KdV type. Inst. Hautes Études Sci. Publ. Math. 61, 5-65 (1985) · Zbl 0592.35112 |
| [52] | The Stacks Project Authors. Stacks project (2015). http://stacks.math.columbia.edu. Accessed 27 Nov 2016 · Zbl 0115.39004 |
| [53] | Thomason, R.W., Trobaugh, T.: Higher algebraic \[KK\]-theory of schemes and of derived categories. In: The Grothendieck Festschrift, Vol. III, vol. 88 of Progr. Math., pp. 247-435. Birkhäuser Boston, Boston, MA (1990) · Zbl 0731.14001 |
| [54] | Voevodsky, V.: Homology of schemes. Selecta Math. (N.S.) 2(1), 111-153 (1996) · Zbl 0871.14016 · doi:10.1007/BF01587941 |
| [55] | Voevodsky, V.: Homotopy theory of simplicial sheaves in completely decomposable topologies. (2000). http://www.math.uiuc.edu/K-theory/443/cdstructures.pdf. Accessed 27 Nov 2016 · Zbl 1194.55020 |
| [56] | Yanagihara, H.: Some results on weakly normal ring extensions. J. Math. Soc. Jpn. 35(4), 649-661 (1983) · Zbl 0535.13007 · doi:10.2969/jmsj/03540649 |
| [57] | Zhu, X.: Affine Grassmannians and the geometric satake in mixed characteristic. Ann. Math. (to appear, 2014). arXiv:1407.8519 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
