8.1 Zariski topology

The Zariski topology is a key concept in algebraic geometry, providing a framework to study geometric properties of algebraic sets using tools from topology and algebra. It defines closed sets as common zeros of polynomial equations, allowing us to analyze algebraic varieties through a topological lens.

This topology bridges the gap between algebra and geometry, enabling the study of affine and projective varieties as topological spaces. It possesses unique properties like being Noetherian and quasi-compact, which are crucial for understanding the structure and behavior of algebraic sets in modern algebraic geometry.

Zariski topology fundamentals

• The Zariski topology is a fundamental concept in algebraic geometry that provides a topological structure on algebraic varieties
• It allows us to study geometric properties of algebraic sets using tools from topology and commutative algebra

Definition of Zariski topology

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• The Zariski topology on an affine space $\mathbb{A}^n$ is defined by specifying the closed sets
• A subset $V \subseteq \mathbb{A}^n$ is Zariski closed if it is the set of common zeros of a collection of polynomial equations $f_1(x_1, \ldots, x_n) = \cdots = f_k(x_1, \ldots, x_n) = 0$
• The Zariski topology is the topology generated by taking the Zariski closed sets as the closed sets

Zariski closed sets

• Zariski closed sets are the algebraic subsets of an affine or projective space
• The Zariski closure of a set $S$ is the smallest Zariski closed set containing $S$, denoted by $\overline{S}$
• The Zariski closure of a set can be obtained by taking the vanishing set of the ideal generated by the polynomials that vanish on $S$

Zariski open sets

• A subset $U \subseteq \mathbb{A}^n$ is Zariski open if its complement $\mathbb{A}^n \setminus U$ is Zariski closed
• Zariski open sets are the complements of algebraic subsets
• The Zariski topology is not Hausdorff, as two distinct points can have the same Zariski closure ($\mathbb{A}^1$ with Zariski topology)

Irreducible vs reducible sets

• A Zariski closed set $V$ is irreducible if it cannot be expressed as the union of two proper Zariski closed subsets
• Equivalently, $V$ is irreducible if and only if its coordinate ring $\mathbb{C}[V]$ is an integral domain
• A Zariski closed set that is not irreducible is called reducible
• Every algebraic set can be uniquely decomposed into a finite union of irreducible components

Zariski topology on affine varieties

• Affine varieties are algebraic sets in affine space equipped with the Zariski topology
• The Zariski topology on affine varieties allows us to study their geometric and topological properties

Affine varieties as Zariski topological spaces

• An affine variety $V \subseteq \mathbb{A}^n$ is an irreducible Zariski closed subset of $\mathbb{A}^n$
• The Zariski topology on $V$ is the subspace topology induced by the Zariski topology on $\mathbb{A}^n$
• Affine varieties are the building blocks of algebraic geometry and are studied using the Zariski topology

Zariski closed subsets of affine space

• Zariski closed subsets of affine space $\mathbb{A}^n$ are algebraic sets defined by polynomial equations
• The Zariski closure of a subset $S \subseteq \mathbb{A}^n$ is the smallest algebraic set containing $S$
• The Zariski closure of a set can be computed using Groebner bases and the Nullstellensatz

Zariski closure of a set

• The Zariski closure of a set $S \subseteq \mathbb{A}^n$, denoted by $\overline{S}$, is the smallest Zariski closed set containing $S$
• $\overline{S}$ can be obtained by taking the vanishing set of the ideal $I(S) = \{f \in \mathbb{C}[x_1, \ldots, x_n] : f(p) = 0 \text{ for all } p \in S\}$
• The Zariski closure captures the "algebraic" part of a set and is a fundamental concept in the Zariski topology

Irreducible components of affine varieties

• An affine variety $V$ can be uniquely decomposed into a finite union of irreducible components $V = V_1 \cup \cdots \cup V_k$
• Each $V_i$ is an irreducible affine variety, and $V_i \not\subseteq V_j$ for $i \neq j$
• The irreducible components of an affine variety correspond to the minimal prime ideals of its coordinate ring

Zariski topology on projective varieties

• Projective varieties are algebraic sets in projective space equipped with the Zariski topology
• The Zariski topology on projective varieties is similar to the affine case but takes into account the homogeneous nature of projective space

Projective varieties as Zariski topological spaces

• A projective variety $V \subseteq \mathbb{P}^n$ is an irreducible Zariski closed subset of $\mathbb{P}^n$
• The Zariski topology on $V$ is the subspace topology induced by the Zariski topology on $\mathbb{P}^n$
• Projective varieties are important objects in algebraic geometry and are studied using the Zariski topology

Zariski closed subsets of projective space

• Zariski closed subsets of projective space $\mathbb{P}^n$ are algebraic sets defined by homogeneous polynomial equations
• A subset $V \subseteq \mathbb{P}^n$ is Zariski closed if it is the set of common zeros of a collection of homogeneous polynomials
• The Zariski topology on $\mathbb{P}^n$ is defined by taking the Zariski closed sets as the closed sets

Homogeneous ideals and projective varieties

• A homogeneous ideal $I \subseteq \mathbb{C}[x_0, \ldots, x_n]$ is an ideal generated by homogeneous polynomials
• The vanishing set of a homogeneous ideal $I$ in $\mathbb{P}^n$ is a projective variety $V(I)$
• Conversely, every projective variety is the vanishing set of a homogeneous ideal (projective Nullstellensatz)

Irreducible components of projective varieties

• A projective variety $V$ can be uniquely decomposed into a finite union of irreducible components $V = V_1 \cup \cdots \cup V_k$
• Each $V_i$ is an irreducible projective variety, and $V_i \not\subseteq V_j$ for $i \neq j$
• The irreducible components of a projective variety correspond to the minimal prime ideals of its homogeneous coordinate ring

Properties of Zariski topology

• The Zariski topology has several important properties that distinguish it from other topologies and make it suitable for studying algebraic varieties

Noetherian property of Zariski topology

• The Zariski topology is a Noetherian topology, meaning that every descending chain of Zariski closed sets stabilizes
• Equivalently, every Zariski open set is a finite union of basic open sets (sets of the form $D(f) = \{p : f(p) \neq 0\}$ for some polynomial $f$)
• The Noetherian property is a consequence of the Hilbert Basis Theorem and is crucial for many results in algebraic geometry

Quasi-compactness of Zariski topology

• The Zariski topology is quasi-compact, meaning that every open cover of a Zariski closed set has a finite subcover
• This property follows from the Noetherian property of the Zariski topology
• Quasi-compactness is a weaker notion than compactness, as the Zariski topology is not Hausdorff

Zariski topology and dimension

• The dimension of an irreducible Zariski closed set $V$ is the transcendence degree of its function field $\mathbb{C}(V)$ over $\mathbb{C}$
• The dimension of a reducible Zariski closed set is the maximum of the dimensions of its irreducible components
• The Zariski topology is well-behaved with respect to dimension, as Zariski closed subsets have smaller or equal dimension than the ambient space

Zariski topology and regular functions

• A regular function on a Zariski open set $U \subseteq \mathbb{A}^n$ is a function that can be expressed as a polynomial on each Zariski open subset of $U$
• The set of regular functions on $U$ forms a ring, denoted by $\mathcal{O}(U)$
• Regular functions are the algebraic analog of continuous functions and play a central role in the study of algebraic varieties

Morphisms in Zariski topology

• Morphisms between algebraic varieties are the maps that preserve the algebraic structure and are continuous with respect to the Zariski topology

Continuous maps in Zariski topology

• A map $f: X \to Y$ between two Zariski topological spaces is continuous if the preimage of every Zariski open set in $Y$ is Zariski open in $X$
• Equivalently, $f$ is continuous if the preimage of every Zariski closed set in $Y$ is Zariski closed in $X$
• Continuous maps in the Zariski topology are not necessarily continuous in the classical sense, as the Zariski topology is coarser than the Euclidean topology

Zariski closed maps

• A map $f: X \to Y$ between two Zariski topological spaces is Zariski closed if the image of every Zariski closed set in $X$ is Zariski closed in $Y$
• Zariski closed maps are the algebraic analog of closed maps in topology
• Polynomial maps between affine varieties are examples of Zariski closed maps

Zariski open maps

• A map $f: X \to Y$ between two Zariski topological spaces is Zariski open if the image of every Zariski open set in $X$ is Zariski open in $Y$
• Zariski open maps are the algebraic analog of open maps in topology
• Isomorphisms of algebraic varieties are examples of Zariski open maps

Zariski homeomorphisms

• A map $f: X \to Y$ between two Zariski topological spaces is a Zariski homeomorphism if it is a bijective continuous map with continuous inverse
• Zariski homeomorphisms are the isomorphisms in the category of Zariski topological spaces
• Isomorphisms of algebraic varieties are examples of Zariski homeomorphisms

Applications of Zariski topology

• The Zariski topology is a fundamental tool in algebraic geometry and has numerous applications in related fields

Zariski topology in algebraic geometry

• The Zariski topology is the foundation of modern algebraic geometry
• It allows the study of geometric properties of algebraic varieties using topological and algebraic methods
• Many important results in algebraic geometry, such as the Nullstellensatz and the correspondence between ideals and varieties, rely on the Zariski topology

Zariski topology and sheaf theory

• Sheaves are a central concept in modern algebraic geometry and are defined using the Zariski topology
• The structure sheaf of an algebraic variety is a sheaf of rings that encodes the regular functions on the variety
• Sheaf cohomology, a powerful tool in algebraic geometry, is defined using the Zariski topology and sheaves

Zariski topology and schemes

• Schemes are a generalization of algebraic varieties that allow the study of more general spaces, such as non-reduced or non-irreducible spaces
• The Zariski topology is used to define the underlying topological space of a scheme
• The structure sheaf of a scheme is defined using the Zariski topology and local rings

Zariski topology in commutative algebra

• The Zariski topology has important applications in commutative algebra, particularly in the study of prime ideals and spectra of rings
• The spectrum of a commutative ring $R$, denoted by $\text{Spec}(R)$, is the set of prime ideals of $R$ equipped with the Zariski topology
• Many results in commutative algebra, such as the Hilbert Nullstellensatz and the correspondence between ideals and algebraic sets, have topological interpretations using the Zariski topology