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On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme

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Abstract

In this paper, we prove that for a noetherian formal scheme \(\mathfrak X\), its derived category of sheaves of modules with quasi-coherent torsion homologies \(\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X)\) is generated by a single compact object. In an Appendix we prove that the category of compact objects in \(\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X)\) is skeletally small.

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References

  1. Alonso Tarrío, L., Jeremías López, A., Lipman, J.: Duality and flat base change on formal schemes. In: Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes. Contemp. Math., vol. 244, pp. 3–90. American Mathematical Society, Providence (1999)

    Google Scholar 

  2. Alonso Tarrío, L., Jeremías López, A., Lipman, J.: Greenlees-May duality of formal schemes. In: Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes. Contemp. Math., vol. 244, pp. 93–112. American Mathematical Society, Providence (1999)

    Google Scholar 

  3. Alonso Tarrío, L., Jeremías López, A., Souto Salorio, M.J.: Localization in categories of complexes and unbounded resolutions. Can. J. Math. 52(2), 225–247 (2000)

    Article  MATH  Google Scholar 

  4. Alonso Tarrío, L., Jeremías López, A., Souto Salorio, M.J.: Construction of t −structures and equivalences of derived categories. Trans. Am. Math. Soc. 355, 2523–2543 (2003)

    Article  MATH  Google Scholar 

  5. Alonso Tarrío, L., Jeremías López, A., Souto Salorio, M.J.: Bousfield localization on formal schemes. J. Algebra 278(2), 585–610 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alonso Tarrío, L., Jeremías López, A., Pérez Rodríguez, M., Vale Gonsalves M.J.: The derived category of quasi-coherent sheaves and axiomatic stable homotopy. Adv. Math. 218, 1224–1252 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86, 209–234 (1993)

    MATH  Google Scholar 

  8. Bondal, A., van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003).

    MathSciNet  MATH  Google Scholar 

  9. Dwyer, W.G., Greenlees, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. 9(2), 119–221 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grothendieck, A.: Géométrie formelle et géométrie algébrique. Sémin. Bourbaki 5(182), 28 (1958–1960)

    Google Scholar 

  12. Grothendieck, A., Dieudonné, J.A.: Eléments de Géométrie Algébrique I. Grundlehren der Math. Wiss., vol. 166. Springer, Heidelberg (1971)

    MATH  Google Scholar 

  13. Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic stable homotopy theory. Mem. Am. Math. Soc. 128(610), 1224–1252 (1997)

    MathSciNet  Google Scholar 

  14. Illusie, L.: Généralités sur las conditions de finitude dans les catégories dérivées. In: Théorie des Intersections et Théorème de Riemann-Roch (SGA 6). Lecture Notes in Math., vol. 225, pp. 78–159. Springer, New York (1971)

    Chapter  Google Scholar 

  15. Lipman, J.: Notes on derived functors and Grothendieck duality. In: Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics, no. 1960, pp. 1–259. Springer-Verlag, New York (2009)

    Chapter  Google Scholar 

  16. Lipman, J., Neeman, A.: Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor. Ill. J. Math. 51(1), 209–236 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Lipman, J., Nayak, S., Sastry P.: Variance and duality for Cousin complexes on formal schemes. In: Pseudofunctorial Behavior of Cousin Complexes on Formal Schemes. Contemp. Math., vol. 375, pp. 3–133. American Mathematical Society, Providence (2005)

    Google Scholar 

  18. Neeman, A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. Éc. Norm. Super. (4) 25(5), 547–566 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Neeman, A.: Triangulated categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001)

    MATH  Google Scholar 

  21. Thomason, R.W., Trobaugh, T.: Higher Algebraic K-theory of Schemes and of Derived Categories. The Grothendieck Festschrift, vol. III. Progr. Math., vol. 88, pp. 247–435. Birkhäuser Boston, Boston (1990)

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Álxebra, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Leovigildo Alonso Tarrío, Ana Jeremías López & María J. Vale Gonsalves

  2. Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, Universidade de Vigo, 32004, Ourense, Spain

    Marta Pérez Rodríguez

Authors
  1. Leovigildo Alonso Tarrío
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  2. Ana Jeremías López
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  3. Marta Pérez Rodríguez
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  4. María J. Vale Gonsalves
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Corresponding author

Correspondence to Leovigildo Alonso Tarrío.

Additional information

This work has been partially supported by Spain’s MEC and E.U.’s FEDER research projects MTM2005-05754 and MTM2008-03465.

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Alonso Tarrío, L., Jeremías López, A., Pérez Rodríguez, M. et al. On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme. Appl Categor Struct 19, 865–877 (2011). https://doi.org/10.1007/s10485-009-9204-5

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