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On the Injective and Projective Limit of Complexes

  • Krishna Tewari (a1)

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Let R be a commutative ring with unit. Let (Aα, φ βα) (α ≤ β) (resp. (Bα, ψαβ) (α ≤ β)) be an infective (resp. projective) system of R-algebras indexed by a directed set I; let ((Xα, dα), fβα) (α ≤ β) (resp. ((Yα, δα), gαβ) (α≤β)) be an injective (resp. projective) system of complexes, indexed by the same set I, such that for each αϵI, (Xα, dα) (resp. (Yα, δα)) is a complex over Aα (resp. over Bα). The purpose of this paper is to show that the covariant functor from the category of all such injective systems of complexes and complex homomorphisms over the R-algebra Aα is such that it associates with an injective system ((Uα, dα), hβα) of universal complexes a universal complex over Aα whereas the same is not true of the covariant functor the category of all such projective systems of complexes and their maps.

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References

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1. Berger, R., Űber Verschiedene Differentenbegriffe. Sitzungsberichte der Heidelberger Akademie der Wissenschaften Mathematischnaturwissenschaftliche Klasse, 196.
2. Bourbaki, N., Algèbre Commutative, Chapter II, (Hermann and Cie, Paris).
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6. Kähler, E., Algebra und Differentialrechnung. Bericht űber die Mathematiker Tagung in Berlin, Vol. 14, Bis. 18, January 1953.
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On the Injective and Projective Limit of Complexes

  • Krishna Tewari (a1)

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