Localisation, Local Rings and Local–Global Methods
Commutative algebra studies ideals, prime structure, factorisation, localisation, algebraic dependence and dimension. It is most reliable when global questions are reduced to ideal-theoretic or local statements and then reassembled. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Learning pathCommutative Algebra
LevelAdvanced
FormatHandbook guide
Read time13 min
Executive summary
This chapter develops localisation, local rings and local–global methods as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the ring, its units and its relevant ideals.
Check finiteness or chain conditions before using finite-generation arguments.
Use prime or maximal ideals to convert multiplicative questions into quotient-domain or quotient-field questions.
Localise when the property can be tested near a prime or maximal ideal.
Track integrality, dimension or ideal factorisation with the exact hypotheses stated.
Return from local or quotient data to the original ring only through a justified correspondence.
Core definitions
Definition
Let R be a domain with Q = Frac(R). If M is an R-module, define rank(M) = dimQ(Q ⊗R M). For example, the rank of an abelian group G is defined as dimQ(Q ⊗Z G). Recall that if R is a domain, then an R-module M is torsion-free if it has no nonzero elements of finite order; that is, if r ∈R and m ∈M are nonzero, then rm is nonzero. Local and Global
Definition
A discrete valuation ring, abbreviated DVR, is a local PID that is not a field. For example, Z(p) is a DVR. 1Recall that a local ring is a commutative ring having a unique maximal ideal. Most authors insist that local rings are ascending-chain-finite [Z(p) is even a PID]. Other authors allow local rings to be noncommutative, defining a ring R to be local if it has a unique maximal left ideal m. In this case, m = J(R), the maximal-ideal intersection radical, so that it is a two-sided ideal. Local and Global
Definition
Let G be an abelian group. If x ∈G and p is a prime, we say that x is divisible by pn in G if there exists yn ∈G with pnyn = x. Define the p-height of x, denoted by h p(x), by h p(x) = ∞ if x is divisible by pn in G for all n ≥0 k if x is divisible by pk in G but not by pk+1. The height sequence (or characteristic) of x in G, where x is nonzero, is the sequence χ(x) = χG(x) = (h2(x), h3(x), h5(x), . . . , h p(x), . . .). Thus, χ(x) is a sequence (h p), where h p = ∞or h p ∈N. Let G ⊆Q and let x ∈G be nonzero. If χ(x) = (h p) and a = p f1 1 · · · p fn n , then 1 a x ∈G if and only if f pi ≤h pi for i = 1, . . . , n.
Definition
Two height sequences (h2, h3, . . . , h p, . . .) and (k2, k3, . . . , kp, . . .) are equivalent, denoted by (h2, h3, . . . , h p, . . .) ∼(k2, k3, . . . , kp, . . .), if there are only finitely many p for which h p ̸= kp and, for such primes p, neither h p nor kp is ∞.
Definition
Let R be a commutative ring and let S be any subset of R. A localization of R is an R-algebra S−1R and an R-algebra map h : R →S−1R, called the localization map, such that h(s) is invertible in S−1R, for every s ∈S, and S−1R is a solution to the following universal mapping problem. R h ϕ → → → → → → → → S−1R Δϕ R′ If R′ is a commutative R-algebra and ϕ : R →R′ is an R-algebra map for which ϕ(s) is invertible in R′ for all s ∈S, then there exists a unique R-algebra map Δϕ : S−1R →R′ with Δϕh = ϕ. The localization S−1R, as any solution to a universal mapping problem, is unique up to isomorphism if it exists.
Definition
A subset S of a commutative ring R is multiplicatively closed if 1 ∈S and s, s′ ∈S implies ss′ ∈S. Every commutative ring is a multiplicative monoid. If S is any (possibly empty) subset of R, then S = the submonoid of R generated by S. We call S the multiplicatively closed subset generated by S.
Definition
If p is a prime ideal in a commutative ring R, then the complement S = R−p is multiplicatively closed, and S−1R is denoted by Rp.
Definition
Let R be a commutative ring and let S be any subset of R. A localization of an R-module M is an S−1R-module S−1M (i.e.., µs : S−1m →S−1M is invertible for all s ∈S) and an R-map hM : M →S−1M, called the localization map, which is a solution to the following universal mapping problem: M h ϕ + + + + + + + + S−1M Δϕ { M′ If ϕ : M →M′ is an R-map, where M′ is an S−1R-module, then there is a unique S−1Rmap Δϕ : S−1M →M′ making the diagram commute. The obvious candidate for S−1M—namely, S−1R ⊗R M—is, in fact, its localization.
Principal results and structural facts
Key result
Let R be a domain with Q = Frac(R) and let M be a torsion-free Rmodule. Then M has rank 1 if and only if it is isomorphic to a nonzero R-submodule of Q.
Key result
(i) For each prime p, the ring Z(p) is a local1 PID. (ii) If G is a torsion-free abelian group of rank 1, then Z(p) ⊗Z G is a torsion-free Z(p)-module of rank 1. (iii) If M is a torsion-free Z(p)-module of rank 1, then M ∼= Z(p) or M ∼= Q.
Key result
If G and H are torsion-free abelian groups of rank 1, then G ∼= H if and only if τ(G) = τ(H).
Key result
(i) There are uncountably many nonisomorphic subgroups of Q. (ii) If R is a subring of Q, then the height sequence of 1 consists of 0’s and ∞’s. (iii) There are uncountably many nonisomorphic subrings of Q. In fact, distinct subrings of Q are not isomorphic as rings.
Key result
Let S be a subset of a commutative ring R. If r/σ,r′/σ ′ ∈S−1R, where σ, σ ′ ∈S, then r/σ = r′/σ ′ if and only if there exists σ ′′ ∈S with σ ′′(rσ ′ −r′σ) = 0 in R. Remark. If S contains no zero divisors, then σ ′′(rσ ′−r′σ) = 0 if and only if rσ ′−r′σ = 0, because σ ′′ is a unit, and so rσ ′ = r′σ. ◀
Key result
Let S be a subset of a commutative ring R. (i) Every ideal J in S−1R is of the form S−1I for some ideal I in R. In fact, if R is a domain and I = J ∩R, then J = S−1I; in the general case, if I = h−1(h(R) ∩J), then J = S−1I. (ii) If I is an ideal in R, then S−1I = S−1R if and only if I ∩S ̸= ∅. (iii) If q is a prime ideal in R with q ∩S = ∅, then S−1q is a prime ideal in S−1R. (iv) The function q ↦S−1q is a bijection from the family of all prime ideals in R that are disjoint from S to Spec(S−1R). (v) If R is ascending-chain-finite, then S−1R is also ascending-chain-finite.
Key result
Let R be a local ring with maximal ideal m. An element r ∈R is a unit if and only if r /∈m.
Key result
gives K = mK ⊕(K ∩mB′). But K ∩mB′ ⊆K ∩B′ = {0}, so that K = mK. The submodule K is finitely generated, being a summand (and hence a homomorphic image) of the finitely generated module F, so that local-generation lemma (Corollary 8.32) gives K = {0}. Therefore, ϕ is an isomorphism and B is free. • Having localized a commutative ring, we now localize its modules. If M is an R-module and s ∈R, let µs denote the multiplication map M →M defined by m ↦sm. Note that if S is a subset of R, then µs : M →M is invertible for every s ∈S if and only if M is an S−1R-module.
Key result
Let S be a subset of a commutative ring R and let M be an R-module. (i) Every element u ∈S−1M has the form u = σ −1m for some σ ∈S and some m ∈M. (ii) s−1 1 m1 = s−1 2 m2 in S−1M if and only if σ(s−1 1 m1 −s−1 2 m2) in M for some σ ∈S. Local and Global
Key result
If S is a subset of a commutative ring R, then localization M ↦ S−1M = S−1R ⊗R M defines an exact functor RMod →S−1RMod.
Key result
Let S be a subset of a commutative ring R, and let M and A be R-modules with A finitely presented. Then there is a natural isomorphism τA : S−1 HomR(A, M) →HomS−1R(S−1 A, S−1M).
Key result
If S is a subset of a commutative ACC rings R, and if E is an injective R-module, then S−1E is an injective (S−1R)-module. Remark. This result can fail if R is not ascending-chain-finite. ◀
Key result
If R is a left ACC rings and A is a finitely generated left R-module, then there is a projective resolution P• of A in which each Pn is finitely generated.
Key result
Let A be a finitely generated R-module over a commutative ACC rings R. Then Am is a projective Rm-module for every maximal ideal m if and only if A is a projective R-module.
Source-grounded examples
Worked source example
The following abelian groups are torsion-free of rank 1: (i) The group Z of integers; (ii) The additive group Q; (iii) the set of all rationals having a finite decimal expansion; (iv) the set of all rationals having squarefree denominator. ◀
Worked source example
(i) If S is a subset of a commutative ring R, and if I is an ideal in R containing an element σ ∈S—that is, I ∩S ̸= ∅, then S−1I contains σ/σ = 1, and so S−1I = S−1R. (ii) Let S consist of all the odd integers [that is, S is the complement of the prime ideal (2)], let I = (3), and let I ′ = (5). Then S−1I = S−1Z = S−1I ′. Therefore, the function from the ideals in Z to the ideals in S−1Z = Z(2), given by I ↦S−1I, is not injective. ◀
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
Failure mode
Control
Confusing prime and maximal ideals.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Assuming localisation preserves every property automatically.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Using finite-generation conclusions without a chain condition.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Treating radicals, integral closure or dimension as elementwise notions only.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Assuming geometric statements over arbitrary fields without checking the field hypotheses.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Verification checklist
The ambient set, ring, field, group, module or category has been stated.
Every operation and map used is well-defined in that setting.
The hypotheses of each structural result have been checked before use.
Representatives, coordinates or generators have not been confused with the underlying object.
Existence and uniqueness have been separated where both matter.
The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about localisation, local rings and local–global methods?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.
Connections within the handbook
Source basis: supplied advanced algebra reference. Source-identifying authorship, publisher information, acknowledgements and biographical material are intentionally omitted. Mathematical terminology and results are retained in handbook form.
