Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings ${\displaystyle A\subset B}$ if^[1][2]

1. ${\displaystyle B}$ dominates ${\displaystyle A}$; i.e., for each proper ideal I of A, ${\displaystyle IB}$ is proper and for each maximal ideal ${\displaystyle {\mathfrak {n}}}$ of B, ${\displaystyle {\mathfrak {n}}\cap A}$ is maximal
2. for each maximal ideal ${\displaystyle {\mathfrak {m}}}$ and ${\displaystyle {\mathfrak {m}}}$-primary ideal ${\displaystyle Q}$ of ${\displaystyle A}$, ${\displaystyle \operatorname {length} _{B}(B/QB)}$ is finite and moreover

   ${\displaystyle \operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).}$

Given commutative rings ${\displaystyle A\subset B}$ such that ${\displaystyle B}$ dominates ${\displaystyle A}$ and for each maximal ideal ${\displaystyle {\mathfrak {m}}}$ of ${\displaystyle A}$ such that ${\displaystyle \operatorname {length} _{B}(B/{\mathfrak {m}}B)}$ is finite, the natural inclusion ${\displaystyle A\to B}$ is a faithfully flat ring homomorphism if and only if the theorem of transition holds between ${\displaystyle A\subset B}$.^[2]

Notes

1. Nagata 1975, Ch. II, § 19.
2. Matsumura 1986, Ch. 8, Exercise 22.1.

References

• Nagata, M. (1975). Local Rings. Interscience tracts in pure and applied mathematics. Krieger. ISBN 978-0-88275-228-0.
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.