2.3 Localization and local rings

Localization is a powerful tool in commutative algebra that zooms in on specific parts of a ring. By focusing on a subset of elements, we can study local properties and simplify complex structures.

Local rings, with their unique maximal ideal, are key players in algebraic geometry. They help us understand the behavior of algebraic varieties near specific points, bridging the gap between algebra and geometry.

Localization of rings

Definition and notation

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• The localization of a ring $R$ at a multiplicative subset $S$, denoted $S^{-1}R$, is the ring of fractions with denominators in $S$
  • Elements in $S^{-1}R$ are of the form $r/s$ where $r \in R$ and $s \in S$
• The localization of a ring $R$ at a prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is the localization of $R$ at the multiplicative set $R - \mathfrak{p}$
  • Elements in $R_{\mathfrak{p}}$ are of the form $r/s$ where $r \in R$ and $s \notin \mathfrak{p}$
• The localization of a ring $R$ at a maximal ideal $\mathfrak{m}$, denoted $R_{\mathfrak{m}}$, is called the local ring at $\mathfrak{m}$
• The localization of a ring $R$ at the multiplicative set $\{1, f, f^2, ...\}$ for some $f \in R$ is denoted $R_f$

Construction and properties

• The localization $S^{-1}R$ is constructed as the set of equivalence classes of pairs $(r, s)$ with $r \in R$ and $s \in S$, where $(r_1, s_1) \sim (r_2, s_2)$ if there exists $t \in S$ such that $t(s_1r_2 - s_2r_1) = 0$
  • The equivalence class of $(r, s)$ in $S^{-1}R$ is denoted by $r/s$
• Addition and multiplication in $S^{-1}R$ are defined by $(r_1/s_1) + (r_2/s_2) = (s_2r_1 + s_1r_2)/(s_1s_2)$ and $(r_1/s_1)(r_2/s_2) = (r_1r_2)/(s_1s_2)$
• The localization $S^{-1}R$ is a ring with identity element $1/1$
• The natural map $\phi: R \to S^{-1}R$ given by $r \mapsto r/1$ is a ring homomorphism
  • $\phi$ is injective if and only if $S$ contains no zero divisors
• If $R$ is an integral domain and $S = R - \{0\}$, then $S^{-1}R$ is the field of fractions of $R$ (e.g., $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$)

Properties of localization

Ideals and prime ideals

• If $I$ is an ideal of $R$, then $S^{-1}I = \{i/s : i \in I, s \in S\}$ is an ideal of $S^{-1}R$
• The map $I \mapsto S^{-1}I$ gives a bijection between the ideals of $R$ that do not intersect $S$ and the ideals of $S^{-1}R$
• If $\mathfrak{p}$ is a prime ideal of $R$, then $\mathfrak{p}R_{\mathfrak{p}}$ is the unique maximal ideal of $R_{\mathfrak{p}}$
• The map $\mathfrak{p} \mapsto \mathfrak{p}R_{\mathfrak{p}}$ gives a bijection between the prime ideals of $R$ that do not intersect $S$ and the prime ideals of $S^{-1}R$

Relationship between a ring and its localizations

• For any multiplicative subset $S$ of $R$, the ring $R$ can be viewed as a subring of $S^{-1}R$ via the natural map $\phi: R \to S^{-1}R$
• The localization $S^{-1}R$ can be viewed as a "local version" of $R$ where elements outside of $S$ are inverted
  • This allows for the study of local properties of $R$ (e.g., at a specific prime ideal)

Local rings and examples

Definition and properties

• A local ring is a ring with a unique maximal ideal
• The localization of a ring $R$ at a prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is a local ring with maximal ideal $\mathfrak{p}R_{\mathfrak{p}}$
• In a local ring $(R, \mathfrak{m})$, every element not in $\mathfrak{m}$ is a unit (invertible)
  • This is because $\mathfrak{m}$ is the only maximal ideal, so any proper ideal is contained in $\mathfrak{m}$

Examples of local rings

• The ring of germs of continuous functions at a point on a topological space is a local ring
• The ring of convergent power series over a field is a local ring
  • e.g., $\mathbb{R}[[x]]$, the ring of formal power series with real coefficients
• The ring of rational functions on an algebraic variety, localized at a point, is a local ring
  • e.g., $k[x, y]_{(x, y)}$, the localization of the polynomial ring $k[x, y]$ at the maximal ideal $(x, y)$

Examples of non-local rings

• The ring of integers $\mathbb{Z}$ is not a local ring, as it has infinitely many maximal ideals (one for each prime number)
• The ring of polynomials $k[x]$ over a field $k$ is not a local ring, as it has infinitely many maximal ideals (one for each irreducible polynomial)
  • However, localizing $k[x]$ at a specific maximal ideal (e.g., $(x-a)$ for some $a \in k$) yields a local ring

Ring vs localization relationship

Injective ring homomorphism

• The natural map $\phi: R \to S^{-1}R$ is an injective ring homomorphism if and only if $S$ contains no zero divisors
  • If $S$ contains a zero divisor $s$, then $\phi(s) = s/1$ is a zero divisor in $S^{-1}R$, contradicting injectivity
  • Conversely, if $S$ contains no zero divisors and $\phi(r) = 0$, then $r/1 = 0/1$, implying $tr = 0$ for some $t \in S$, which forces $r = 0$ since $t$ is not a zero divisor

Correspondence between ideals

• The map $I \mapsto S^{-1}I$ gives a bijection between the ideals of $R$ that do not intersect $S$ and the ideals of $S^{-1}R$
  • If $I \cap S \neq \emptyset$, then $S^{-1}I = S^{-1}R$, which corresponds to the improper ideal of $S^{-1}R$
• The map $\mathfrak{p} \mapsto \mathfrak{p}R_{\mathfrak{p}}$ gives a bijection between the prime ideals of $R$ that do not intersect $S$ and the prime ideals of $S^{-1}R$
  • This bijection preserves inclusions, i.e., if $\mathfrak{p} \subseteq \mathfrak{q}$, then $\mathfrak{p}R_{\mathfrak{p}} \subseteq \mathfrak{q}R_{\mathfrak{q}}$

Localization as a subring

• For any multiplicative subset $S$ of $R$, the ring $R$ can be viewed as a subring of $S^{-1}R$ via the natural map $\phi: R \to S^{-1}R$
  • This embedding allows for the transfer of properties from $R$ to $S^{-1}R$ and vice versa
  • For example, if $R$ is Noetherian, then so is $S^{-1}R$; if $S^{-1}R$ is an integral domain, then so is $R$