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Cotorsion modules over tame finite-dimensional hereditary algebras

Part of the Lecture Notes in Mathematics book series (LNM,volume 903)

Abstract

All modules are over tame finite-dimensional algebras. A module M is said to be cotorsion if and only if Ext (F,M)=0 for all torsion-free modules F. It is shown that every cotorsion module can be put in the form M = M1 ∔ M2 ∔ M2 ∔ M3 where M1 is divisible, M2 has no torsion-free direct summand and M3 is reduced and torsion-free. The submodules M1 and M2 are uniquely determined by M. A module M is said to be pure injective if Pext (N,M)=0 for all modules N. The two classes of modules are connected by the following result: A torsion-free module with no preprojective direct summand is pure injective if and only if it is cotorsion. A pure injective module M can also be put in the form M = M1 ∔ M2 ∔ M2 ∔ M3 where M1 is divisible, M2 is reduced and nonsingular and M3 is preprojective. As in abelian group theory M1 and M2 are completely described by sets of cardinal numbers. Under a restriction satisfied by Kronecker modules M3 is also described by cardinal numbers.

Keywords

  • Direct Summand
  • Cardinal Number
  • Artin Algebra
  • Indecomposable Direct Summand
  • Pure Submodule

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1981 Springer-Verlag

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Okoh, F. (1981). Cotorsion modules over tame finite-dimensional hereditary algebras. In: Auslander, M., Lluis, E. (eds) Representations of Algebras. Lecture Notes in Mathematics, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092998

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  • DOI: https://doi.org/10.1007/BFb0092998

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11179-5

  • Online ISBN: 978-3-540-38963-7

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