Skip to main content
SpringerLink
Log in

Model Theory for Sheaves of Modules

  • Conference paper
  • First Online:
Logic and Its Applications (ICLA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11600))

Included in the following conference series:

  • 387 Accesses

Abstract

We describe how the model theory of modules is adapted to deal with sheaves of modules.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    One could let the ring vary by using a two-sorted language: one sort for the ring, one for the module, so that the structures are (ring, module) pairs \((R,M_R)\). The model theory of such pairs is, however, much less well-behaved than that for modules over a fixed ring, and not at all as amenable to useful analysis.

  2. 2.

    In fact, [18], see [9, 2.1.6], it is enough to check for \(n=1\).

  3. 3.

    We will not present background on abelian category theory here but there are many suitable references, for example [4, 20].

  4. 4.

    In category-theoretic terms it is the restriction of the contravariant functor F to the full subcategory on the objects with a morphism to U.

  5. 5.

    It is a right module via the left action of R on the module \(R_R\).

References

  1. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  2. Borceux, F., Van den Bossche, G.: Algebra in a Localic Topos with Applications to Ring Theory. LNM, vol. 1038. Springer, Heidelberg (1983). https://doi.org/10.1007/BFb0073030

    Book  MATH  Google Scholar 

  3. Eklof, P., Sabbagh, G.: Model-completions and modules. Ann. Math. Logic 2(3), 251–295 (1971)

    Article  MathSciNet  Google Scholar 

  4. Freyd, P.: Abelian Categories. Harper and Row, New York (1964)

    MATH  Google Scholar 

  5. Garavaglia, S.: Dimension and rank in the model theory of modules. Preprint, University of Michigan (1979). Revised 1980

    Google Scholar 

  6. Gruson, L., Jensen, C.U.: Modules algébriquement compact et foncteurs \(\mathop {\varprojlim }\nolimits ^{(i)}\). C. R. Acad. Sci. Paris 276, 1651–1653 (1973)

    MathSciNet  MATH  Google Scholar 

  7. Iversen, B.: Cohomology of Sheaves. Springer, Heidelberg (1986). https://doi.org/10.1007/978-3-642-82783-9

    Book  MATH  Google Scholar 

  8. Prest, M.: Model Theory and Modules. London Mathematical Society Lecture Note Series, vol. 130. Cambridge University Press, Cambridge (1988)

    Book  Google Scholar 

  9. Prest, M.: Purity, Spectra and Localisation. Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  10. Prest, M.: Definable Additive Categories: Purity and Model Theory. Memoirs of the American Mathematical Society, vol. 210/no. 987. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  11. Prest, M.: Abelian categories and definable additive categories. arXiv:1202.0426

  12. Prest, M.: Modules as exact functors. In: Proceedings of 2016 Auslander Distinguished Lectures and Conference, Contemporary Mathematics, vol. 716. American Mathematical Society. arXiv:1801.08015 (to appear)

  13. Prest, M.: Multisorted modules and their model theory. In: Contemporary Mathematics. arXiv:1807.11889 (to appear)

  14. Prest, M., Puninskaya, V., Ralph, A.: Some model theory of sheaves of modules. J. Symbolic Logic 69(4), 1187–1199 (2004)

    Article  MathSciNet  Google Scholar 

  15. Prest, M., Ralph, A.: On sheafification of modules. Preprint, University of Manchester (2001). Revised 2004. https://personalpages.manchester.ac.uk/staff/mike.prest/publications.html

  16. Prest, M., Ralph, A.: Locally finitely presented categories of sheaves of modules. Preprint, University of Manchester (2001). Revised 2004 and 2018. https://personalpages.manchester.ac.uk/staff/mike.prest/publications.html

  17. Prest, M., Slávik, A.: Purity in categories of sheaves. Preprint. arXiv:1809.08981 (2018)

  18. Rothmaler, Ph.: A trivial remark on purity. In: Proceedings of the 9th Easter Conference on Model Theory, Gosen 1991, Seminarber. 112, Humboldt-Univ. zu Berlin, p. 127 (1991)

    Google Scholar 

  19. Sabbagh, G.: Sous-modules purs, existentiellement clos et élementaires. C. R. Acad. Sci. Paris 272, 1289–1292 (1971)

    MathSciNet  MATH  Google Scholar 

  20. Stenström, B.: Rings of Quotients. Springer, Heidelberg (1975). https://doi.org/10.1007/978-3-642-66066-5

    Book  MATH  Google Scholar 

  21. Tennison, B.R.: Sheaf Theory. London Mathematical Society Lecture Note Series, vol. 20. Cambridge University Press, Cambridge (1975)

    Book  Google Scholar 

  22. Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26(2), 149–213 (1984)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Mike Prest

Authors
  1. Mike Prest
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mike Prest .

Editor information

Editors and Affiliations

  1. Indian Institute of Technology Indore, Madhya Pradesh, India

    Md. Aquil Khan

  2. Indian Institute of Technology Goa, Goa, India

    Amaldev Manuel

Rights and permissions

Reprints and permissions

Copyright information

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

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Prest, M. (2019). Model Theory for Sheaves of Modules. In: Khan, M., Manuel, A. (eds) Logic and Its Applications. ICLA 2019. Lecture Notes in Computer Science(), vol 11600. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-58771-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-58771-3_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-58770-6

  • Online ISBN: 978-3-662-58771-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation