Integral extensions and going up

For an inclusion A⊆B of commutative unital rings, b is integral over A if it satisfies b^n+a_{n−1}b^{n−1}+⋯+a₀=0 with coefficients in A. Higher powers can then be reduced to the first n powers, so A[b] is finite as an A-module. Conversely, a finite A-algebra is integral: multiplication by an element acts on a finite generating list, and the determinant trick yields a monic annihilating polynomial. Applying that polynomial to the algebra’s identity turns the module relation into the desired element relation.

Finitely many integral elements generate a finite A-module algebra by adjoining them one at a time. Thus an integral algebra of finite type is finite as a module. A finite-type algebra need not be integral: A[x] has one algebra generator but arbitrarily many independent powers as an A-module when A is a nonzero ring. Integrality is also transitive and behaves well under localization and quotient. These properties allow prime arguments to reduce to local or residue situations without discarding the monic equations that supplied the original finiteness control.

For an integral inclusion A⊆B, lying over says every prime p of A is the contraction of some prime q of B. Going up says that if p⊆p′ and q over p is already chosen, there exists q′ over p′ containing that chosen q. The direction matters: we keep the lower prime upstairs and extend the chain upward. These are theorem statements; a standard proof localizes and passes to suitable quotients, using the fact that an algebra integral over a field has maximal prime ideals. No Noetherian hypothesis is needed for these integral-extension results.

Incomparability supplies a further constraint: two comparable distinct primes upstairs cannot contract to the same prime downstairs in an integral extension. It prevents an integral map from hiding a nontrivial upstairs chain entirely over one base point. These facts do not imply each prime has only one lift, or that the lift is unramified. For example, arithmetic extensions can split a base prime into several primes or give a repeated factor. Lying over asserts existence, not uniqueness, and going up preserves a chosen chain rather than selecting a canonical lift.

Going down starts with p⊆p′ and a chosen q′ over the upper prime p′, then seeks q⊆q′ over p. One sufficient theorem assumes A and B are domains, B integral over A, and A integrally closed in its fraction field. Under those hypotheses going down holds. The normality requirement here means integrally closed domain; it is not a claim that every base ring in every going-down result must be normal. Flat maps provide another route to going down, introduced later.

An integral extension without this extra condition still has going up, but one cannot simply invoke the normal-domain going-down theorem. Failure of that sufficient hypothesis does not by itself prove going down fails in the particular example. This distinction is especially relevant for cusp subrings, whose normalization adjoins a missing integral parameter. Tracking which ring is the base, which ring is integrally closed and which prime endpoint is prescribed avoids a large class of incorrect chain-lifting arguments in dimension theory.