##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?

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Discussion Overview

The discussion revolves around the isomorphism between the localization of a quotient ring and its associated structures, specifically focusing on Proposition 2.1.1 and its proof. Participants explore the application of the first isomorphism theorem, the construction of surjective homomorphic maps, and the implications of various mappings between ideals and localization in the context of commutative algebra.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions whether the classic method of the first isomorphism theorem can be applied to Proposition 2.1.1 by constructing a surjective homomorphic map and finding its kernel.
  • Another participant suggests that the isomorphism in Proposition 3.5 is proved using the first isomorphism theorem and implies that applying it to the case of Proposition 2.1.1 is a valid approach.
  • There is a query about the nature of the map from prime ideals of A to prime ideals of A_p, specifically whether it is from A\setminus p to A_p.
  • Participants seek clarification on the contraction map along the localization map A to A_p and its role in proving statements about the maximality of the ideal in Proposition 2.1.1.
  • Questions arise regarding the precise mathematical notation of the phrase "runs once through the prime ideals" as stated in Proposition 2.1.1.
  • Further inquiries are made about the correct formulation of the map f used in Proposition 3.5 and its surjectivity.
  • One participant references a statement from a text regarding the kernel of a composite mapping and its implications for the isomorphism of modules, seeking to understand its relevance to the current discussion.

Areas of Agreement / Disagreement

Participants express uncertainty and seek clarification on various aspects of the propositions and mappings discussed. There is no consensus on the interpretations of certain statements or the application of theorems, indicating that multiple competing views remain.

Contextual Notes

Participants highlight the need for careful consideration of definitions and the implications of various mappings in the context of localization and ideal extensions. Unresolved questions about the nature of certain mappings and their proofs suggest limitations in the current understanding.

  • #31
@fresh_42 your post #6 points 2.)-4.) is where you try to shown that statement?
 
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  • #32
Yes.
 

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