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Pure Morphisms are Effective for Modules

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Abstract

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk’s descent criterion.

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References

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

  2. Barr, M.: On Categories with Effective Unions. Lecture Notes Math., vol. 1348, pp. 19–35. Springer, Berlin (1988)

    Google Scholar 

  3. Borceux, F.: Handbook of Categorical Algebra, vol. 2. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  4. Chase, E., Sweedler, M.: Hopf Algebras and Galois Theory. Lecture Notes Math. vol. 97. Springer, Berlin (1969)

    MATH  Google Scholar 

  5. Janelidze, G., Tholen, W.: Facets of Descent, I. Appl. Categ. Struct. 2, 245–281 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Janelidze, G., Tholen, W.: Facets of descent, II. Appl. Categ. Struct. 5, 229–248 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Janelidze, G., Tholen, W.: Facets of descent, III: monadic descent for rings and algebras. Appl. Categ. Struct. 12, 461–477 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  9. Mac Lane, S., Paré, R.: Coherence in bicategories and indexed categories. J. Pure Appl. Algebra 37, 59–80 (1985)

    Article  MathSciNet  Google Scholar 

  10. Mesablishvili, B.: Pure morphisms of commutative rings are effective descent morphisms for modules − a new proof. Theory Appl. Categ. 7, 38–42 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Mesablishvili, B.: Monads of effective descent type and comonadicity. Theory Appl. Categ. 16, 1–45 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Moerdijk, I.: Descent theory for toposes. Bull. Soc. Math. Belg. 41, 373–391 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Paré, R., Schumacher, D.: Abstract Families and the Adjoint Functor Theorem. Lecture Notes Math, vol. 661, pp. 1–125. Springer, Berlin (1978)

    Google Scholar 

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Authors and Affiliations

  1. A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 2, University St., Tbilisi, 0186, Republic of Georgia

    Bachuki Mesablishvili

  2. Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3/35, Tbilisi, 0168, Republic of Georgia

    Bachuki Mesablishvili

Authors
  1. Bachuki Mesablishvili
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Correspondence to Bachuki Mesablishvili.

Additional information

The work was partially supported by Volkswagen Foundation (Ref.: I/85989) and Shota Rustaveli National Science Foundation Grant DI/12/ 5-103/11.

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Mesablishvili, B. Pure Morphisms are Effective for Modules. Appl Categor Struct 21, 801–809 (2013). https://doi.org/10.1007/s10485-012-9283-6

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  • DOI: https://doi.org/10.1007/s10485-012-9283-6

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