Assassins and Torsion Functors

Abstract

Let R be a ring, let \(\mathfrak {a}\subseteq R\) be an ideal, and let M be an R-module. Let \({\Gamma }_{\mathfrak {a}}\) denote the \(\mathfrak {a}\)-torsion functor. Conditions are given for the (weakly) associated primes of \({\Gamma }_{\mathfrak {a}}(M)\) to be the (weakly) associated primes of M containing \(\mathfrak {a}\), and for the (weakly) associated primes of \(M/{\Gamma }_{\mathfrak {a}}(M)\) to be the (weakly) associated primes of M not containing \(\mathfrak {a}\).

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Notes

  1. 1.

    Throughout what follows, rings are understood to be commutative. In general, notation and terminology follow Bourbaki’s Éléments de mathématique.

  2. 2.

    This shows that the converse of 3.3 need not hold.

  3. 3.

    It is seen from the proof that in fact we do not need\(\mathfrak {b}\)to be weakly proregular, but rather to have a finite generating familyb such that thecanonical morphism \(\gamma ^{1}_{\mathbf {b}}\colon H_{\mathfrak {b}}^{1}\rightarrow \check {H}^{1}_{\mathbf {b}}\)is an isomorphism.

References

  1. 1.

    Alonso Tarrío, L., Jeremías López, A., Lipman, J.: Local homology and cohomology on schemes. Ann. Sci. Éc. Norm. Supér. (4) 30, 1–39 (1997). Corrections available at http://www.math.purdue.edu/∼lipman/papers/homologyfix.pdf

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bourbaki, N.: Éléments de Mathématique. Algèbre Commutative. Chapitres 1 à 4. Masson, Paris (1985)

  3. 3.

    Brodmann, M., Fumasoli, S., Rohrer, F.: First lectures on local cohomology. (In preparation)

  4. 4.

    Brodmann, M.P., Sharp, R.Y.: Local Cohomology, 2nd Edn. Cambridge Stud. Adv Math, vol. 136. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  5. 5.

    Cahen, P.-J.: Torsion theory and associated primes. Proc. Am. Math. Soc. 38, 471–476 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Čoupek, P.: Tilting Theory for Quasicoherent Sheaves. Diploma Thesis. Charles University Prague, Prague (2016)

    Google Scholar 

  7. 7.

    Porta, M., Shaul, L., Yekutieli, A.: On the homology of completion and torsion. Algebr. Represent. Theory 17, 31–67 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Quy, P.H., Rohrer, F.: Injective modules and torsion functors. Comm. Algebra 45, 285–298 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Rohrer, F.: Torsion functors with monomial support. Acta Math. Vietnam. 38, 293–301 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Schenzel, P.: Proregular sequences, local cohomology, and completion. Math. Scand. 92, 161–180 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    The Stacks Project Authors: Stacks project. http://stacks.math.columbia.edu (2017)

  12. 12.

    Yassemi, S.: A survey of associated and coassociated primes. Vietnam J. Math. 28, 195–208 (2000)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

I thank the referee for his careful reading and his suggestions, and – as so often – Pham Hung Quy for suggesting nice counterexamples.

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Correspondence to Fred Rohrer.

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Rohrer, F. Assassins and Torsion Functors. Acta Math Vietnam 43, 125–136 (2018). https://doi.org/10.1007/s40306-017-0233-0

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Keywords

  • Torsion functor
  • Assassin
  • Weak assassin
  • Non-noetherian ring
  • Well-centered torsion theory
  • Weakly proregular ideal

Mathematics Subject Classification (2010)

  • Primary 13C12
  • Secondary 13D30
  • 13D45