Theorem of transition

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

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

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

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

Notes

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.