Theorem of transition

Short description: Theorem about commutative rings and subrings

In algebra, the theorem of transition is said to hold between commutative rings [math]\displaystyle{ A \subset B }[/math] if

• (i) [math]\displaystyle{ B }[/math] dominates [math]\displaystyle{ A }[/math]; i.e., for each proper ideal I of A, [math]\displaystyle{ IB }[/math] is proper and for each maximal ideal [math]\displaystyle{ \mathfrak n }[/math] of B, [math]\displaystyle{ \mathfrak n \cap A }[/math] is maximal
• (ii) for each maximal ideal [math]\displaystyle{ \mathfrak m }[/math] and [math]\displaystyle{ \mathfrak m }[/math]-primary ideal [math]\displaystyle{ Q }[/math] of [math]\displaystyle{ A }[/math], [math]\displaystyle{ \operatorname{length}_B (B/ Q B) }[/math] is finite and moreover

  [math]\displaystyle{ \operatorname{length}_B (B/ Q B) = \operatorname{length}_B (B/ \mathfrak{m} B) \operatorname{length}_A(A/Q). }[/math]

Given commutative rings [math]\displaystyle{ A \subset B }[/math] such that [math]\displaystyle{ B }[/math] dominates [math]\displaystyle{ A }[/math] and for each maximal ideal [math]\displaystyle{ \mathfrak m }[/math] of [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ \operatorname{length}_B (B/ \mathfrak{m} B) }[/math] is finite, the natural inclusion [math]\displaystyle{ A \to B }[/math] is a faithfully flat ring homomorphism if and only if the theorem of transition holds between [math]\displaystyle{ A \subset B }[/math].

References

• Nagata, Local Rings
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. ISBN 0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC&pg=PA123.