Skip to main content
SpringerLink
Log in

The six Grothendieck operations on o-minimal sheaves

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In topology, Borsuk asked in 1937 if the product of a normal space and the unit interval is normal and the question was only solved, negatively, without any set theoretic conditions beyond the axiom of choice, in 1971 by Rudin [43]. We thank the referee for pointing this to us. So Theorem 2.20 is yet another manifestation of tameness of o-minimal structures.

  2. Recall that a type \(\beta \) on X (i.e. an ultrafilter of definable subsets of X) is a definable type onX if and only if for every uniformly definable family \(\{Y_t\}_{t\in T}\) of definable subsets of X,  the set \(\{t\in T: Y_t\in \beta \}\) is a definable set.

  3. In topology it is trivial that \(g'^{-1}(\bullet )\) sends f-soft sheaves to \(f'\)-soft sheaves. Here this is not evident.

  4. As usual in category theory [32], to avoid set theoretic issues, we assumed throughout the paper that we are working in a (very big) fixed universe, so below by small sum we mean a sum of a family indexed by a set in this universe.

References

  1. Berarducci, A.: Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup. J. Symb. Logic 74(3), 891–900 (2009)

    Article  MathSciNet  Google Scholar 

  2. Berarducci, A., Fornasiero, A.: O-minimal cohomology: finiteness and invariance results. J. Math. Logic 9(2), 167–182 (2009)

    Article  MathSciNet  Google Scholar 

  3. Benoist, O., Wittenberg, O.: On the integral Hodge conjecture for real varieties, I. pp. 64. arXiv:1801.00872 (2018)

  4. Benoist, O., Wittenberg, O.: On the integral Hodge conjecture for real varieties, II. pp. 56. arXiv:1801.00873 (2018)

  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)

    MATH  Google Scholar 

  6. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Ergebnisse der Math. (3), vol. 36. Springer, Berlin (1998)

  7. Bredon, G.: Sheaf theory. In: Graduate Texts in Math., 2nd edn. vol. 170. Springer, New York (1997)

  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)

  9. D’Agnolo, A.: On the Laplace transform for tempered holomorphic functions. Int. Math. Res. Not. 16, 4587–4623 (2014)

    Article  MathSciNet  Google Scholar 

  10. D’Agnolo, A., Kashiwara, M.: Riemann-Hilbert correspondence for holonomic D-modules. Publ. Math. Inst. Hautes Étud. Sci. 123(1), 69–197 (2016)

    Article  MathSciNet  Google Scholar 

  11. Delfs, H.: Homology of locally semialgebraic spaces. In: Lecture Notes in Math, vol. 1484. Springer, Berlin (1991)

  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)

    Article  MathSciNet  Google Scholar 

  13. Denef, J., van den Dries, L.: \(p\)-adic and real subanalytic sets. Ann. Math. 128, 79–138 (1988)

    Article  MathSciNet  Google Scholar 

  14. Edmundo, M.: Structure theorems for o-minimal expansions of groups. Ann. Pure Appl. Log. 102(1–2), 159–181 (2000)

    Article  MathSciNet  Google Scholar 

  15. Edmundo, M., Jones, G., Peatfield, N.: Sheaf cohomology in o-minimal structures. J. Math. Log. 6(2), 163–179 (2006)

    Article  MathSciNet  Google Scholar 

  16. Edmundo, M., Mamino, M., Prelli, L.: On definably proper maps. Fund. Math. Fund. Math. 233(1), 1–36 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Edmundo, M., Prelli, L.: Poincaré–Verdier duality in o-minimal structures. Ann. Inst. Fourier Grenoble 60(4), 1259–1288 (2010)

    Article  MathSciNet  Google Scholar 

  19. Edmundo, M., Prelli, L.: Invariance of o-minimal cohomology with definably compact supports. Conflu. Math. 7(1), 35–53 (2015)

    Article  MathSciNet  Google Scholar 

  20. Edmundo, M., Terzo, G.: A note on generic subsets of definable groups. Fund. Math. 215(1), 53–65 (2011)

    Article  MathSciNet  Google Scholar 

  21. Godement, R.: Théorie des faisceaux. Hermann, Paris (1958)

    MATH  Google Scholar 

  22. Grothendieck, A., Dieudonné, J.: Elements de géometrie algébrique II. Inst. Hautes Études Sci. Publ. Math. 8, 5–222 (1961)

    Article  Google Scholar 

  23. Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

  24. Hrushovski, E.: Valued fields, metastable groups, draft (2004). http://www.ma.huji.ac.il/~ehud/mst.pdf

  25. Hrushovski, E., Loeser, F.: Non-archimedean tame topology and stably dominated types. In: Annals Mathematics Studies, vol. 192. Princeton University Press, Princeton (2016)

  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)

    Article  MathSciNet  Google Scholar 

  27. Hrushovski, E., Peterzil, Y., Pillay, A.: Groups, measures and the NIP. J. Am. Math. Soc. 21(2), 563–596 (2008)

    Article  MathSciNet  Google Scholar 

  28. Iversen, B.: Cohomology of Sheaves. Universitext. Springer, Berlin (1986)

    Book  Google Scholar 

  29. Kayal, T., Raby, G.: Ensemble sous-analytiques: quelques propriétés globales. C. R. Acad. Sci. Paris Ser. I Math. 308, 521–523 (1989)

  30. Kashiwara, M., Schapira, P.: Sheaves on manifolds. In: Grundlehren der Math., vol. 292. Springer, Berlin (1990)

  31. Kashiwara, M., Schapira, P.: Ind-sheaves. Astérisque 271, (2001)

  32. Kashiwara, M., Schapira, P.: Categories and sheaves. In: Grundlehren der Math., vol. 332. Springer, Berlin (2006)

  33. Lipshitz, L., Robinson, Z.: Overconvergent real closed quantifier elimination. Bull. Lond. Math. Soc. 38, 897–906 (2006)

    Article  MathSciNet  Google Scholar 

  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)

  35. Peterzil, Y., Steinhorn, C.: Definable compactness and definable subgroups of o-minimal groups. J. Lond. Math. Soc. 59(2), 769–786 (1999)

    Article  MathSciNet  Google Scholar 

  36. Pila, J.: O-minimality and the André-Oort conjecture for \({\mathbb{C}}^n\). Ann. Math. 173(3), 1779–1840 (2011)

    Article  MathSciNet  Google Scholar 

  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)

  38. Pila, J., Wilkie, A.J.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)

    Article  MathSciNet  Google Scholar 

  39. Pillay, A.: Sheaves of continuous definable functions. J. Symb. Log. 53(4), 1165–1169 (1988)

    Article  MathSciNet  Google Scholar 

  40. Pillay, A.: Type-definability, compact Lie groups and o-minimality. J. Math. Log. 4(2), 147–162 (2004)

    Article  MathSciNet  Google Scholar 

  41. Pillay, A., Steinhorn, C.: Definable sets in ordered structures I. Trans. Am. Math. Soc. 295(2), 565–592 (1986)

    Article  MathSciNet  Google Scholar 

  42. Prelli, L.: Sheaves on subanalytic sites. Rend. Sem. Mat. Univ. di Padova 120, 167–216 (2008)

    Article  MathSciNet  Google Scholar 

  43. Rudin, M.E.: A normal space \(X\) for which \(X\times I\) is not normal. Fund. Math. 73, 179–186 (1971)

    Article  MathSciNet  Google Scholar 

  44. Schwartz, N.: The basic theory of real closed spaces, vol. 77, no. 397. In: Memoirs of the AMS (1989)

  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)

  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)

  47. van den Dries, L., Macintyre, A., Marker, D.: The elementary theory of restricted analytic fields with exponentiation. Ann. Math. 140, 183–205 (1994)

    Article  MathSciNet  Google Scholar 

  48. van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)

    Article  MathSciNet  Google Scholar 

  49. Verdier, J.L.: Dualité dans la cohomologie des espaces localment compact. Seminaire Bourbaki 300, (1965)

  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)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6, Piso 1, 1749-016, Lisbon, Portugal

    Mário J. Edmundo

  2. Dipartimento di Matematica, Università degli Studi di Padova, via Trieste 63, 35121, Padua, Italy

    Luca Prelli

Authors
  1. Mário J. Edmundo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luca Prelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Prelli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by Fundação para a Ciência e a Tecnologia, Financiamento Base 2008-ISFL/1/209. The second author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work is part of the FCT project PTDC/MAT/101740/2008.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Edmundo, M.J., Prelli, L. The six Grothendieck operations on o-minimal sheaves. Math. Z. 294, 109–160 (2020). https://doi.org/10.1007/s00209-019-02274-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02274-0

Keywords

Mathematics Subject Classification

Search

Navigation