Assassins and Torsion Functors


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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    This shows that the converse of 3.3 need not hold.

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    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.


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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).

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

Mathematics Subject Classification (2010)

  • Primary 13C12
  • Secondary 13D30
  • 13D45