Completions and Nakayama’s lemma

For an ideal I in R, the I-adic completion of M is the inverse limit of M/I^nM. An element is a compatible list of residue classes, with each reducing to the previous one. The natural map from M sends an element to all its residues, and its kernel is the intersection of I^nM. Thus injectivity requires separation and should not be inferred for an arbitrary ring and module. For k[x] or its localization at (x), x-adic completion gives k[[x]], whose coefficients are determined one finite truncation at a time.

A formal series can be inverted when its constant coefficient is a unit, by recursively solving for coefficients. This is an algebraic compatibility construction rather than an analytic radius-of-convergence claim. The topology depends on the chosen ideal, so completing at different ideals can retain different local information. For a finite module over a Noetherian local ring, maximal-ideal separation follows from Krull’s intersection theorem. That is an additional theorem with finiteness hypotheses, not a property of every inverse limit by definition.

Let M be finite over a commutative unital ring R and let I lie in the Jacobson radical, the intersection of all maximal ideals. If IM=M, Nakayama’s lemma gives M=0. To see its mechanism, write each chosen generator as an I-linear combination of the generators. The determinant of identity minus the coefficient matrix annihilates M and lies in 1+I. Every element of 1+I is a unit under the radical hypothesis, forcing M to vanish. Finite generation is needed to form this finite determinant argument.

Apply the lemma to a quotient to obtain the generator version: if residues of elements m₁,…,m_r generate M/IM, then those elements generate M. For a local ring with maximal ideal m, the minimum number of generators of a finite module equals the residue-field dimension of M/mM. This does not say that a lifted minimal generating set is a basis; relations may remain. If I is not in the Jacobson radical, IM=M can hold for a nonzero finite module. If M is not finite, it can also fail even over a local ring.

For a Noetherian ring R and any ideal I, completion is exact on finite R-modules, and the completion of a finite module is M⊗_R R̂. The Artin–Rees lemma controls the induced filtration on a submodule, which is the key reason injectivity survives after taking the inverse limit. The ring map R→R̂ is flat. These statements do not identify arbitrary infinite-module completion with tensoring by R̂: flatness of the ring map and exactness of completion on the finite category are distinct assertions with distinct scopes.

If I also lies in the Jacobson radical, the Noetherian completion map is faithfully flat. In particular, maximal-ideal completion of a Noetherian local ring has this property. Without that condition, completion can discard modules away from the chosen ideal. For example, Z→Z₂ is flat, but a module Z/3Z becomes zero because three is a unit in Z₂; the map is not faithfully flat. By contrast Z_(2)→Z₂ is the local completion and is faithfully flat. The base ring matters when deciding what information the completion detects.