Math 2.0

Commutative algebra Definition

Localization of a module

Builds on Module over a ringMultiplicative setLocalizationEquivalence relation

Definition

Let RR be a commutative ring, let S⊆RS \subseteq R be a multiplicative set, and let MM be an RR-module. On M×SM \times S, let (m,s)∼(n,t)(m, s) \sim (n, t) if there is u∈Su \in S with u(tm−sn)=0u(tm - sn) = 0. This is an equivalence relation; write m/sm/s for the class of (m,s)(m, s). The localization of MM at SS is the set S−1MS^{-1}M of these classes with the operations ms+nt=tm+snst,at⋅ms=amts\frac{m}{s} + \frac{n}{t} = \frac{tm + sn}{st}, \qquad \frac{a}{t} \cdot \frac{m}{s} = \frac{am}{ts} for a∈Ra \in R. These operations are well defined and make S−1MS^{-1}M a module over the localization S−1RS^{-1}R. For a prime ideal 𝔭\mathfrak p of RR, one writes M𝔭M_{\mathfrak p} for S−1MS^{-1}M with S=R∖𝔭S = R \setminus \mathfrak p.

Remarks

There is a natural isomorphism of S−1RS^{-1}R-modules S−1R⊗RM≅S−1MS^{-1}R \otimes_R M \cong S^{-1}M, given by (a/s)⊗m↦am/s(a/s) \otimes m \mapsto am/s; see tensor product of modules. The functor M↦S−1MM \mapsto S^{-1}M is exact, so S−1RS^{-1}R is a flat RR-module.

Example: let R=ℤR = \Z, M=ℤ/3ℤM = \Z/3\Z and 𝔭=2ℤ\mathfrak p = 2\Z. Then 3∈S=ℤ∖2ℤ3 \in S = \Z \setminus 2\Z, and m/s=3m/(3s)=0m/s = 3m/(3s) = 0 for every mm, so M(2)=0M_{(2)} = 0. For 𝔭=3ℤ\mathfrak p = 3\Z, the elements of SS act invertibly on ℤ/3ℤ\Z/3\Z, and M(3)≅ℤ/3ℤM_{(3)} \cong \Z/3\Z.

Non-example: the relation "tm=sntm = sn" without the factor uu is in general not transitive, as for the localization of a ring, which is the case M=RM = R.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Equivalence relation, concept, no evidence levelEquivalence relationconceptLocalization, concept, no evidence levelLocalizationconceptModule over a ring, concept, no evidence levelModule over a ringconceptMultiplicative set, concept, no evidence levelMultiplicative setconceptLocalization of a module, definition, no evidence levelLocalization of a mod…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Equivalence relation concept
  2. Localization concept
  3. Module over a ring concept
  4. Multiplicative set concept
  5. Localization of a module definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
About this page
TagS357810Z25
URIm2://S357810Z25
Kinddefinition
WritingW0 stub
MSC13B30, 13C99
DefinesLocalization of a module
UsesModule over a ring, Multiplicative set, Localization, Equivalence relation
Changed2 November 2026

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