🧮Commutative Algebra Unit 5 – Localization and Local Rings

Key Concepts and Definitions

• Local ring defined as a ring with a unique maximal ideal
• Localization process creates a local ring from a given ring by inverting elements outside a prime ideal
• Quotient field obtained by localizing a domain at the zero ideal
• Residue field of a local ring defined as the quotient of the ring by its maximal ideal
• Prime spectrum of a ring consists of all prime ideals in the ring
• Localization at a prime ideal produces a local ring with that prime ideal as its maximal ideal
• Nakayama's lemma states that if $M$ is a finitely generated module over a local ring $R$ with maximal ideal $\mathfrak{m}$, and $\mathfrak{m}M = M$, then $M = 0$

Motivation and Historical Context

• Local rings arise naturally in algebraic geometry when studying the local behavior of algebraic varieties
• Localization allows for the study of rings and modules locally at a specific prime ideal
• Hensel's lemma, a key result in local algebra, has applications in number theory and algebraic geometry
• Local rings play a crucial role in the study of singularities and the classification of algebraic varieties
• The concept of localization was introduced by Wolfgang Krull in the 1930s
• Oscar Zariski's work on algebraic geometry in the 1940s and 1950s heavily relied on local rings and localization
• Grothendieck's development of scheme theory in the 1960s further emphasized the importance of local rings in algebraic geometry

Properties of Local Rings

• Local rings are Noetherian if and only if they satisfy the ascending chain condition on ideals
• Krull dimension of a local ring equals the supremum of lengths of chains of prime ideals
• Regular local rings are integral domains characterized by their Krull dimension equaling the minimal number of generators of the maximal ideal
• Completion of a local ring with respect to its maximal ideal produces a complete local ring
• Henselian local rings are local rings that satisfy Hensel's lemma
  • Hensel's lemma allows for the lifting of solutions of polynomial equations from the residue field to the ring
• Cohen structure theorem states that every complete Noetherian local ring is a quotient of a regular local ring

Localization Process

• Given a ring $R$ and a multiplicative subset $S$, the localization of $R$ at $S$ is denoted as $S^{-1}R$
• Elements of $S^{-1}R$ are equivalence classes of fractions $\frac{r}{s}$ with $r \in R$ and $s \in S$
• Two fractions $\frac{r_1}{s_1}$ and $\frac{r_2}{s_2}$ are equivalent if there exists $t \in S$ such that $t(s_2r_1 - s_1r_2) = 0$
• Localization is a functor from the category of rings to itself
  • Maps a ring homomorphism $f: R \to R'$ to the localized homomorphism $S^{-1}f: S^{-1}R \to S'^{-1}R'$, where $S' = f(S)$
• Localization at a prime ideal $\mathfrak{p}$ is denoted as $R_\mathfrak{p}$ and is obtained by inverting all elements outside $\mathfrak{p}$
• Localization preserves injectivity and surjectivity of ring homomorphisms

Examples and Applications

• Localizing the ring of integers $\mathbb{Z}$ at the prime ideal $(p)$ produces the local ring $\mathbb{Z}_{(p)}$, which consists of rational numbers with denominators not divisible by $p$
• Localizing the polynomial ring $k[x, y]$ at the maximal ideal $(x, y)$ results in the local ring $k[x, y]_{(x, y)}$, which is used to study the local behavior of plane curves at the origin
• Localization is used in the construction of the structure sheaf of an affine scheme
• Local rings appear in the study of étale morphisms and the étale topology in algebraic geometry
• Localization is employed in the definition of the sheaf of regular functions on an algebraic variety
• Local rings are used to define the tangent space and the cotangent space of an algebraic variety at a point
• Localization plays a role in the definition of the completion of a scheme along a closed subscheme

Theorems and Proofs

• Nakayama's lemma proof:
  • Let $M$ be a finitely generated module over a local ring $R$ with maximal ideal $\mathfrak{m}$, and suppose $\mathfrak{m}M = M$
  • Choose generators $x_1, \ldots, x_n$ of $M$, so $M = Rx_1 + \cdots + Rx_n$
  • By assumption, each $x_i \in \mathfrak{m}M$, so $x_i = \sum_{j=1}^n a_{ij}x_j$ with $a_{ij} \in \mathfrak{m}$
  • The matrix $(a_{ij} - \delta_{ij})$ has entries in $\mathfrak{m}$, so its determinant is in $\mathfrak{m}$
  • Since $R$ is local, the determinant is a unit, implying the matrix is invertible
  • This leads to a contradiction unless $M = 0$
• Proof of the characterization of Noetherian local rings:
  • If $R$ is Noetherian, the ascending chain condition holds for all ideals, including the maximal ideal
  • Conversely, if the maximal ideal satisfies the ascending chain condition, so does every ideal, as they are all contained in the maximal ideal
• Proof that the Krull dimension of a local ring equals the supremum of lengths of chains of prime ideals:
  • Every chain of prime ideals in a local ring $R$ is of the form $\mathfrak{p}_0 \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n \subset \mathfrak{m}$, where $\mathfrak{m}$ is the maximal ideal
  • The length of such a chain is at most the Krull dimension of $R$
  • Conversely, any maximal chain of prime ideals achieves the Krull dimension

Connections to Other Topics

• Local rings are fundamental in the study of commutative algebra and algebraic geometry
• Localization is a key tool in the construction of schemes and the study of their local properties
• Local rings are used in the definition of the étale topology and the study of étale morphisms
• Henselian local rings have applications in the study of valued fields and the theory of $p$-adic numbers
• Completions of local rings are related to the study of formal schemes and formal power series rings
• Local cohomology, a tool in commutative algebra, is defined using localization and the support of modules
• Localization is used in the construction of the derived category of a scheme and the study of derived functors
• The concept of localization can be generalized to non-commutative rings, leading to the theory of Ore localization

Common Pitfalls and FAQs

• Not every ring has a unique maximal ideal, so not every ring is local
• Localization at a prime ideal is not the same as taking the quotient by that ideal
• The localization of a ring at the complement of a prime ideal is not always a field
  • It is a field if and only if the prime ideal is maximal
• The localization of a Noetherian ring is not always Noetherian
  • However, the localization of a Noetherian ring at a prime ideal is always Noetherian
• The localization of an integral domain is always an integral domain, but the converse is not true
• The localization of a unique factorization domain (UFD) is always a UFD, but the converse is not true
• Localization does not always preserve the property of being a principal ideal domain (PID)
  • However, the localization of a PID at a prime ideal is always a PID