Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings $A\subset B$ if

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

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

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

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.