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The six Grothendieck operations on o-minimal sheaves

  • Mário J. Edmundo
  • Luca PrelliEmail author
Article
  • 6 Downloads

Abstract

In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Künneth formula; (v) local and global Verdier duality.

Keywords

o-Minimal structures Proper direct image Sheaves Cohomology Semi-algebraic Globally sub-analytic 

Mathematics Subject Classification

03C64 55N30 

Notes

References

  1. 1.
    Berarducci, A.: Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup. J. Symb. Logic 74(3), 891–900 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berarducci, A., Fornasiero, A.: O-minimal cohomology: finiteness and invariance results. J. Math. Logic 9(2), 167–182 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benoist, O., Wittenberg, O.: On the integral Hodge conjecture for real varieties, I. pp. 64. arXiv:1801.00872 (2018)
  4. 4.
    Benoist, O., Wittenberg, O.: On the integral Hodge conjecture for real varieties, II. pp. 56. arXiv:1801.00873 (2018)
  5. 5.
    Bierstone, E., Milman, P.: Sub-analytic geometry. In: Haskell, D., Pillay, A., Steinhorn, C. (eds.) Model Theory, Algebra and Geometry, vol. 39, pp. 151–172. MSRI Publications, Cambridge (2000)Google Scholar
  6. 6.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Ergebnisse der Math. (3), vol. 36. Springer, Berlin (1998)Google Scholar
  7. 7.
    Bredon, G.: Sheaf theory. In: Graduate Texts in Math., 2nd edn. vol. 170. Springer, New York (1997)Google Scholar
  8. 8.
    Coste, M.: An introduction to o-minimal geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000)Google Scholar
  9. 9.
    D’Agnolo, A.: On the Laplace transform for tempered holomorphic functions. Int. Math. Res. Not. 16, 4587–4623 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D’Agnolo, A., Kashiwara, M.: Riemann-Hilbert correspondence for holonomic D-modules. Publ. Math. Inst. Hautes Étud. Sci. 123(1), 69–197 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Delfs, H.: Homology of locally semialgebraic spaces. In: Lecture Notes in Math, vol. 1484. Springer, Berlin (1991)Google Scholar
  12. 12.
    Delfs, H., Knebush, M.: Semi-algebraic topology over a real closed field II: basic theory of semi-algebraic spaces. Math. Z. 178, 175–213 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Denef, J., van den Dries, L.: \(p\)-adic and real subanalytic sets. Ann. Math. 128, 79–138 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Edmundo, M.: Structure theorems for o-minimal expansions of groups. Ann. Pure Appl. Log. 102(1–2), 159–181 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Edmundo, M., Jones, G., Peatfield, N.: Sheaf cohomology in o-minimal structures. J. Math. Log. 6(2), 163–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Edmundo, M., Mamino, M., Prelli, L.: On definably proper maps. Fund. Math. Fund. Math. 233(1), 1–36 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Edmundo, M., Mamino, M., Prelli, L., Ramakrishnan, J., Terzo, G.: On Pillay’s conjecture in the general case. Adv. Math. 310, 940–992 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Edmundo, M., Prelli, L.: Poincaré–Verdier duality in o-minimal structures. Ann. Inst. Fourier Grenoble 60(4), 1259–1288 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Edmundo, M., Prelli, L.: Invariance of o-minimal cohomology with definably compact supports. Conflu. Math. 7(1), 35–53 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Edmundo, M., Terzo, G.: A note on generic subsets of definable groups. Fund. Math. 215(1), 53–65 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Godement, R.: Théorie des faisceaux. Hermann, Paris (1958)zbMATHGoogle Scholar
  22. 22.
    Grothendieck, A., Dieudonné, J.: Elements de géometrie algébrique II. Inst. Hautes Études Sci. Publ. Math. 8, 5–222 (1961)CrossRefGoogle Scholar
  23. 23.
    Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)Google Scholar
  24. 24.
    Hrushovski, E.: Valued fields, metastable groups, draft (2004). http://www.ma.huji.ac.il/~ehud/mst.pdf
  25. 25.
    Hrushovski, E., Loeser, F.: Non-archimedean tame topology and stably dominated types. In: Annals Mathematics Studies, vol. 192. Princeton University Press, Princeton (2016)Google Scholar
  26. 26.
    Hrushovski, E., Peterzil, Y.: A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard. J. Symb. Log. 72, 119–122 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hrushovski, E., Peterzil, Y., Pillay, A.: Groups, measures and the NIP. J. Am. Math. Soc. 21(2), 563–596 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Iversen, B.: Cohomology of Sheaves. Universitext. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kayal, T., Raby, G.: Ensemble sous-analytiques: quelques propriétés globales. C. R. Acad. Sci. Paris Ser. I Math. 308, 521–523 (1989)Google Scholar
  30. 30.
    Kashiwara, M., Schapira, P.: Sheaves on manifolds. In: Grundlehren der Math., vol. 292. Springer, Berlin (1990)Google Scholar
  31. 31.
    Kashiwara, M., Schapira, P.: Ind-sheaves. Astérisque 271, (2001)Google Scholar
  32. 32.
    Kashiwara, M., Schapira, P.: Categories and sheaves. In: Grundlehren der Math., vol. 332. Springer, Berlin (2006)Google Scholar
  33. 33.
    Lipshitz, L., Robinson, Z.: Overconvergent real closed quantifier elimination. Bull. Lond. Math. Soc. 38, 897–906 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Peterzil, Y., Starchenko, S.: Uniform definability of the Weierstrass \(\wp \)-functions and generalized tori of dimension one. Sel. Math. (N.S.) 10(4), 525–550 (2004)Google Scholar
  35. 35.
    Peterzil, Y., Steinhorn, C.: Definable compactness and definable subgroups of o-minimal groups. J. Lond. Math. Soc. 59(2), 769–786 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pila, J.: O-minimality and the André-Oort conjecture for \({\mathbb{C}}^n\). Ann. Math. 173(3), 1779–1840 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pila, J., Zannier, U.: Rational points in periodic analytic sets and the Manin-Mumford conjecture. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 19(2), 149–162 (2008)Google Scholar
  38. 38.
    Pila, J., Wilkie, A.J.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pillay, A.: Sheaves of continuous definable functions. J. Symb. Log. 53(4), 1165–1169 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pillay, A.: Type-definability, compact Lie groups and o-minimality. J. Math. Log. 4(2), 147–162 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pillay, A., Steinhorn, C.: Definable sets in ordered structures I. Trans. Am. Math. Soc. 295(2), 565–592 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Prelli, L.: Sheaves on subanalytic sites. Rend. Sem. Mat. Univ. di Padova 120, 167–216 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rudin, M.E.: A normal space \(X\) for which \(X\times I\) is not normal. Fund. Math. 73, 179–186 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schwartz, N.: The basic theory of real closed spaces, vol. 77, no. 397. In: Memoirs of the AMS (1989)Google Scholar
  45. 45.
    van den Dries, L.: A generalization of Tarski-Seidenberg theorem and some nondefinability results. Bull. Am. Math. Soc. (N.S) 15, 189–193 (1986)Google Scholar
  46. 46.
    van den Dries, L.: Tame topology and o-minimal structures. In: London Math. Soc. Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)Google Scholar
  47. 47.
    van den Dries, L., Macintyre, A., Marker, D.: The elementary theory of restricted analytic fields with exponentiation. Ann. Math. 140, 183–205 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Verdier, J.L.: Dualité dans la cohomologie des espaces localment compact. Seminaire Bourbaki 300, (1965)Google Scholar
  50. 50.
    Wilkie, A.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9, 1051–1094 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisbonPortugal
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PadovaPaduaItaly

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