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Tate Duality in Positive Dimension over Function Fields
About this Title
Zev Rosengarten
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 290, Number 1444
ISBNs: 978-1-4704-6707-4 (print); 978-1-4704-7630-4 (online)
DOI: https://doi.org/10.1090/memo/1444
Published electronically: October 24, 2023
Table of Contents
Chapters
- 1. Introduction and Main Results
- 2. General fields
- 3. Local Fields
- 4. Local Integral Cohomology
- 5. Global Fields
- A. Products and Ultraproducts
- B. Valuation Rings
- C. Profinite Completions
- D. Duality Pairings and Weil Restriction
- E. Cohomology and Direct Limits
- F. Compatibility Between Čech and Derived Functor Constructions
- G. Characteristic $0$
Abstract
We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius (“Poitou-Tate without restrictions on the order,” 2015) extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.- Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 354653
- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI 10.1007/BF01390174
- A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164 (French). MR 181643, DOI 10.1007/BF02566948
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- N. Bourbaki, Topologie Générale, Springer-Verlag, Berlin, 2007.
- Lawrence S. Breen, On a nontrivial higher extension of representable abelian sheaves, Bull. Amer. Math. Soc. 75 (1969), 1249–1253. MR 255550, DOI 10.1090/S0002-9904-1969-12379-7
- Lawrence Breen, Un théorème d’annulation pour certains $E\textrm {xt}^{i}$ de faisceaux abéliens, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 339–352. MR 401768
- J.W.S. Cassels, A. Fröhlich, Algebraic Number Theory, Academic Press, 1993.
- Kęstutis Česnavičius, Topology on cohomology of local fields, Forum Math. Sigma 3 (2015), Paper No. e16, 55. MR 3482265, DOI 10.1017/fms.2015.18
- Kęstutis Česnavičius, Poitou-Tate without restrictions on the order, Math. Res. Lett. 22 (2015), no. 6, 1621–1666. MR 3507254, DOI 10.4310/MRL.2015.v22.n6.a5
- Ching-Li Chai, Brian Conrad, and Frans Oort, Complex multiplication and lifting problems, Mathematical Surveys and Monographs, vol. 195, American Mathematical Society, Providence, RI, 2014. MR 3137398, DOI 10.1090/surv/195
- Claude Chevalley, Deux théorèmes d’arithmétique, J. Math. Soc. Japan 3 (1951), 36–44 (French). MR 44570, DOI 10.2969/jmsj/00310036
- Brian Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18. MR 1906417
- Brian Conrad, Chow’s $K/k$-image and $K/k$-trace, and the Lang-Néron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108. MR 2255529
- B. Conrad Affine Quotients, available at http: //virtualmath1.stanford.edu/~conrad/252Page/handouts/affineqt.pdf.
- Brian Conrad, Finiteness theorems for algebraic groups over function fields, Compos. Math. 148 (2012), no. 2, 555–639. MR 2904198, DOI 10.1112/S0010437X11005665
- Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, 2nd ed., New Mathematical Monographs, vol. 26, Cambridge University Press, Cambridge, 2015. MR 3362817, DOI 10.1017/CBO9781316092439
- M. Demazure, P. Gabriel, Groupes Algébriques, Masson & Cie, Paris, 1970.
- M. Demazure, A. Grothendieck, Schémas en Groupes I, II, III, Lecture Notes in Math 151, 152, 153, Springer-Verlag, New York (1970).
- Ofer Gabber, Some theorems on Azumaya algebras, The Brauer group (Sem., Les Plans-sur-Bex, 1980) Lecture Notes in Math., vol. 844, Springer, Berlin, 1981, pp. 129–209. MR 611868
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
- A. Grothendieck, Eléments de Géométrie Algébrique, Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32, 1960–7.
- A. Grothendieck, D.S. Rim, Groupes de Monodromie en Géométrie Algébriques, Lecture Notes in Mathematics 288, Springer–Verlag, Heidelberg, 1972.
- Alexander Grothendieck, Le groupe de Brauer. II. Théorie cohomologique, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 67–87 (French). MR 244270
- Alexander Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 88–188 (French). MR 244271
- Kazuya Kato, Galois cohomology of complete discrete valuation fields, Algebraic $K$-theory, Part II (Oberwolfach, 1980) Lecture Notes in Math., vol. 967, Springer, Berlin-New York, 1982, pp. 215–238. MR 689394
- Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin-New York, 1973, pp. 69–190. MR 447119
- Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 302647
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1980. MR 575344
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. MR 347836
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
- J. S. Milne, Arithmetic duality theorems, 2nd ed., BookSurge, LLC, Charleston, SC, 2006. MR 2261462
- J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field. MR 3729270, DOI 10.1017/9781316711736
- S. Morel, Adic spaces, available at https://web.math.princeton.edu/~smorel/adic\_notes.pdf.
- Joseph Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique $p$, Invent. Math. 78 (1984), no. 1, 13–88 (French). MR 762353, DOI 10.1007/BF01388714
- Takashi Ono, Arithmetic of algebraic tori, Ann. of Math. (2) 74 (1961), 101–139. MR 124326, DOI 10.2307/1970307
- Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathematical Society, Providence, RI, 2017. MR 3729254, DOI 10.1090/gsm/186
- Paulo Ribenboim, The theory of classical valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999. MR 1677964, DOI 10.1007/978-1-4612-0551-7
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 152834
- Karl Schwede, Gluing schemes and a scheme without closed points, Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, 2005, pp. 157–172. MR 2182775, DOI 10.1090/conm/386/07222
- J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London-New York, 1977, pp. 193–268. MR 450201
- Jean-Pierre Serre, Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR 1466966, DOI 10.1007/978-3-642-59141-9
- The Stacks Project, https: //stacks.math.columbia.edu.
- Richard G. Swan, On seminormality, J. Algebra 67 (1980), no. 1, 210–229. MR 595029, DOI 10.1016/0021-8693(80)90318-X
- John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295. MR 175892
- Michael Temkin, Stable modification of relative curves, J. Algebraic Geom. 19 (2010), no. 4, 603–677. MR 2669727, DOI 10.1090/S1056-3911-2010-00560-7
- Robert Treger, On $p$-torsion in etale cohomology and in the Brauer group, Proc. Amer. Math. Soc. 78 (1980), no. 2, 189–192. MR 550491, DOI 10.1090/S0002-9939-1980-0550491-8
- André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 427267