4.3 Local rings and localization

Local rings are crucial in algebraic geometry, focusing on properties around specific points on varieties. They have a unique maximal ideal containing all non-units, allowing us to study local behavior. Localization creates new rings by inverting elements, often used to analyze coordinate rings at prime ideals corresponding to points on varieties.

Localizing coordinate rings at prime ideals produces local rings that capture a variety's behavior near specific points. This process helps us understand geometric properties like dimension, regularity, and singularities. The correspondence between points and maximal ideals allows us to study variety geometry through coordinate ring algebra.

Local rings and their properties

Definition and basic properties of local rings

• A local ring is a commutative ring with a unique maximal ideal
• The unique maximal ideal consists of all non-units in the local ring
  • Units are elements that have multiplicative inverses within the ring
  • Non-units are elements that do not have multiplicative inverses
• The quotient of a local ring by its maximal ideal is a field, called the residue field
  • This residue field captures the behavior of the ring "locally" around the maximal ideal
  • Example: For the local ring $\mathbb{Q}[x]_{(x)}$ (localization of $\mathbb{Q}[x]$ at the prime ideal $(x)$), the residue field is $\mathbb{Q}$

Additional properties and examples of local rings

• Local rings are Noetherian if they satisfy the ascending chain condition on ideals
  • Ascending chain condition: Every ascending chain of ideals $I_1 \subseteq I_2 \subseteq \cdots$ eventually stabilizes, i.e., there exists an $N$ such that $I_n = I_N$ for all $n \geq N$
  • Noetherian rings have favorable properties, such as the Hilbert basis theorem and the existence of primary decompositions
• Localization of a ring at a prime ideal produces a local ring
  • The localization of a ring $R$ at a prime ideal $\mathfrak{p}$ is denoted as $R_{\mathfrak{p}}$
  • Example: The localization of $\mathbb{Z}$ at the prime ideal $(p)$ is the local ring $\mathbb{Z}_{(p)}$, consisting of fractions with denominators not divisible by $p$

Localization of coordinate rings

The process of localization

• Localization is a process that creates a new ring from an existing ring by inverting a subset of elements
  • Given a ring $R$ and a multiplicatively closed subset $S$ (i.e., $1 \in S$ and $a, b \in S \implies ab \in S$), the localization of $R$ at $S$ is the ring $S^{-1}R = \{\frac{r}{s} \mid r \in R, s \in S\}$
  • The elements of $S$ become units in the localized ring $S^{-1}R$
• For a prime ideal $\mathfrak{p}$ in a ring $R$, the localization of $R$ at $\mathfrak{p}$ is the ring $R_{\mathfrak{p}}$ obtained by inverting all elements of $R$ not in $\mathfrak{p}$
  • The multiplicatively closed subset in this case is $S = R \setminus \mathfrak{p}$
  • Example: For the ring $\mathbb{Z}$ and the prime ideal $(2)$, the localization $\mathbb{Z}_{(2)}$ consists of fractions with odd denominators

Localization of coordinate rings at prime ideals

• The coordinate ring of an affine variety can be localized at a prime ideal corresponding to a point on the variety
  • For an affine variety $V \subseteq \mathbb{A}^n$, its coordinate ring is $A(V) = k[x_1, \ldots, x_n] / I(V)$, where $I(V)$ is the ideal of polynomials vanishing on $V$
  • Each point $P \in V$ corresponds to a maximal ideal $\mathfrak{m}_P$ in $A(V)$
• Localizing the coordinate ring at a prime ideal produces the local ring at that point
  • The local ring at a point $P \in V$ is $\mathcal{O}_{V,P} = A(V)_{\mathfrak{m}_P}$
  • This local ring captures the local behavior of the variety near the point $P$
• The maximal ideal of the localized ring consists of functions vanishing at the corresponding point
  • For a point $P = (a_1, \ldots, a_n) \in V$, the maximal ideal $\mathfrak{m}_P$ of $\mathcal{O}_{V,P}$ is generated by $\{x_1 - a_1, \ldots, x_n - a_n\}$
  • Functions in $\mathfrak{m}_P$ have a zero at $P$, while functions outside $\mathfrak{m}_P$ are non-vanishing at $P$

Local rings and geometric points

Correspondence between points and maximal ideals

• There is a one-to-one correspondence between points on an affine variety and maximal ideals in its coordinate ring
  • Each point $P \in V$ corresponds to a maximal ideal $\mathfrak{m}_P$ in $A(V)$
  • Conversely, each maximal ideal $\mathfrak{m}$ in $A(V)$ corresponds to a point $P_{\mathfrak{m}} \in V$
  • This correspondence allows us to study the geometry of the variety through the algebraic properties of its coordinate ring

Properties of local rings and geometric implications

• The dimension of the local ring at a point is equal to the dimension of the variety at that point
  • The dimension of a local ring $\mathcal{O}_{V,P}$ is the Krull dimension, i.e., the supremum of the lengths of chains of prime ideals in $\mathcal{O}_{V,P}$
  • The dimension of a variety $V$ at a point $P$ is the dimension of the tangent space $T_P V$, which is isomorphic to the Zariski cotangent space $\mathfrak{m}_P / \mathfrak{m}_P^2$
• The residue field of the local ring at a point $P$ is isomorphic to the field of rational functions on $V$ defined at $P$
  • The residue field $k(P) = \mathcal{O}_{V,P} / \mathfrak{m}_P$ is the field of fractions of the quotient ring $A(V) / \mathfrak{m}_P$
  • Elements of $k(P)$ can be viewed as rational functions on $V$ that are well-defined at $P$
• Properties of the local ring, such as regularity and normality, reflect geometric properties of the variety at the corresponding point
  • A point $P \in V$ is regular (or smooth) if the local ring $\mathcal{O}_{V,P}$ is a regular local ring, i.e., its maximal ideal is generated by a regular sequence
  • A variety $V$ is normal if all of its local rings are integrally closed domains

Applications of local rings in algebraic geometry

Studying local properties of varieties

• Determine whether a given ring is local by checking for a unique maximal ideal
  • A ring $R$ is local if and only if it has a unique maximal ideal
  • To check if a ring is local, find all maximal ideals and verify that there is only one
• Compute the localization of a given ring at a specified prime ideal
  • To localize a ring $R$ at a prime ideal $\mathfrak{p}$, form the multiplicatively closed subset $S = R \setminus \mathfrak{p}$ and construct the localized ring $R_{\mathfrak{p}} = S^{-1}R$
  • Example: To localize $\mathbb{Z}[x]$ at the prime ideal $(x)$, invert all polynomials with a non-zero constant term to obtain $\mathbb{Z}[x]_{(x)}$
• Determine the maximal ideal and residue field of a local ring
  • For a local ring $R$ with maximal ideal $\mathfrak{m}$, the residue field is the quotient $R / \mathfrak{m}$
  • Example: For the local ring $\mathbb{Q}[x, y]_{(x, y)}$, the maximal ideal is $(x, y)$, and the residue field is $\mathbb{Q}$

Applying local rings to solve geometric problems

• Use local rings to study the local properties of a variety at a specific point
  • Compute the dimension of the variety at a point by finding the Krull dimension of the local ring
  • Determine the regularity or singularity of a point by examining the local ring
• Apply localization to solve problems related to the local structure of varieties, such as determining the dimension or singularity type at a point
  • Example: To find the dimension of a variety $V$ at a point $P$, localize the coordinate ring $A(V)$ at the maximal ideal $\mathfrak{m}_P$ and compute the Krull dimension of $\mathcal{O}_{V,P}$
• Use local rings to compute the tangent space and tangent cone of a variety at a point
  • The tangent space $T_P V$ is isomorphic to the Zariski cotangent space $\mathfrak{m}_P / \mathfrak{m}_P^2$
  • The tangent cone $C_P V$ is the spectrum of the associated graded ring $\operatorname{gr}_{\mathfrak{m}_P}(\mathcal{O}_{V,P}) = \bigoplus_{i=0}^{\infty} \mathfrak{m}_P^i / \mathfrak{m}_P^{i+1}$
• Solve problems involving the relationship between local rings and the geometry of varieties, such as determining the multiplicity of a point or the intersection multiplicity of curves
  • The multiplicity of a point $P$ on a variety $V$ is the length of the local ring $\mathcal{O}_{V,P}$ as a module over itself
  • The intersection multiplicity of two curves at a point can be computed using the Hilbert-Samuel multiplicity of the local ring at the intersection point