Abstract
Let A be an associative and unital K-algebra sheaf, where K is a commutative ring sheaf, and ε an (A, A)-bimodule, that is, a sheaf of (A, A)-bimodules. We construct an (A, A)-bimodulc which is K-isomorphic with the K-module D K (A, ε) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module D K (A, ε) with the A-module Hom A (Ω K (A), ε). where A, in addition of being associative and unital, is assumed to be commutative, and Ω K (A) denotes the A-module of germs of Kähler differentials. Finally, we expound on functoriality of Kähler differentials.
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References
T. S. Blyth, Module Theory. An Approach to Linear Algebra, Second edition, Oxford Science Publications, Clarendon Press, Oxford, 1990.
N. Bourbaki, Elements of Mathematics, Algebra I, Chapters 1–3, Springer-Verlag, 1989.
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, 2004.
S. Lang, Algebra, Revised Third Edition, Springer, 2002.
A. Mallios, Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry. Volume I: Vector Sheaves. General Theory, Kluwer Academic Publishers, Dordrecht, 1998.
A. Mallios, Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry. Volume II: Geometry. Examples and Applications, Kluwer Academic Publishers, Dordrecht, 1998.
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I would like to express my gratitude to Professor A. Mallios for suggesting the topic.
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Abel, M., Ntumba, P.P. Universal problem for Kähler differentials in A-modules: Non-commutative and commutative cases. Indian J Pure Appl Math 45, 497–511 (2014). https://doi.org/10.1007/s13226-014-0077-4
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DOI: https://doi.org/10.1007/s13226-014-0077-4