Regular functions and morphisms are the building blocks of algebraic geometry. They allow us to study varieties through their function rings and maps between them. Understanding these concepts is crucial for grasping how algebraic structures relate to geometric objects.

Morphisms between varieties are the natural maps that preserve algebraic structure. They're defined using regular functions and provide a way to compare and relate different varieties. Mastering morphisms is key to understanding the deeper connections between algebraic and geometric properties of varieties.

Regular functions on varieties

Defining regular functions on affine varieties

• A regular function on an affine variety is a function that can be expressed as a polynomial in the coordinate ring of the variety
• Regular functions on affine varieties form a ring called the coordinate ring of the variety
  • The coordinate ring consists of all polynomial functions on the affine variety
  • Example: For the affine variety $V(y - x^2)$, the coordinate ring is $k[x, y]/(y - x^2)$
• The set of regular functions on an affine variety is denoted by $\mathcal{O}(X)$, where $X$ is the affine variety

Defining regular functions on projective varieties

• A regular function on a projective variety is a function that can be expressed as a homogeneous polynomial of the same degree in each affine chart
  • Homogeneous polynomials have the property that all terms have the same total degree
  • Example: $x^2 + xy + y^2$ is a homogeneous polynomial of degree 2
• Regular functions on projective varieties are well-defined globally, meaning they are independent of the choice of affine charts
  • This ensures that the regular functions are consistent across different coordinate representations of the projective variety
• The set of regular functions on a projective variety forms a graded ring
  • The grading corresponds to the degree of the homogeneous polynomials
  • Example: For the projective variety $V(xz - y^2)$, the graded ring is $k[x, y, z]/(xz - y^2)$

Morphisms between varieties

Defining morphisms between algebraic varieties

• A morphism between two algebraic varieties is a map that preserves the structure of the varieties
  • Morphisms respect the algebraic properties and relations of the varieties
• Morphisms between affine varieties are defined by polynomial functions on the coordinate rings of the varieties
  • The polynomial functions induce a ring homomorphism between the coordinate rings
  • Example: A morphism from $V(y - x^2)$ to $V(w - z^3)$ can be defined by $x \mapsto z$ and $y \mapsto z^2$
• Morphisms between projective varieties are defined by homogeneous polynomial functions of the same degree on the homogeneous coordinate rings of the varieties
  • The homogeneous polynomial functions induce a graded ring homomorphism between the homogeneous coordinate rings
  • Example: A morphism from $V(xz - y^2)$ to $V(w^2 - uv)$ can be defined by $x \mapsto u$, $y \mapsto v$, and $z \mapsto w$
• Morphisms are required to be continuous with respect to the Zariski topology on the varieties
  • The preimage of a closed set under a morphism must be a closed set

Composition and identity morphisms

• The composition of two morphisms is also a morphism
  • If $f: X \to Y$ and $g: Y \to Z$ are morphisms, then $g \circ f: X \to Z$ is also a morphism
  • The composition of morphisms is associative: $(h \circ g) \circ f = h \circ (g \circ f)$
• The identity map is always a morphism
  • For any variety $X$, the identity map $\mathrm{id}_X: X \to X$ defined by $\mathrm{id}_X(x) = x$ for all $x \in X$ is a morphism
  • The identity morphism satisfies $f \circ \mathrm{id}_X = f$ and $\mathrm{id}_Y \circ f = f$ for any morphism $f: X \to Y$

Properties of morphisms

Injectivity of morphisms

• A morphism is injective (one-to-one) if distinct points in the domain map to distinct points in the codomain
  • Formally, a morphism $f: X \to Y$ is injective if for any $x_1, x_2 \in X$, $f(x_1) = f(x_2)$ implies $x_1 = x_2$
• Injectivity of a morphism can be determined by examining the kernel of the corresponding ring homomorphism between the coordinate rings of the varieties
  • The kernel of a ring homomorphism $\varphi: A \to B$ is the set $\ker(\varphi) = {a \in A \mid \varphi(a) = 0}$
  • A morphism is injective if and only if the kernel of the corresponding ring homomorphism is trivial (consists only of the zero element)

Surjectivity of morphisms

• A morphism is surjective (onto) if every point in the codomain is the image of some point in the domain
  • Formally, a morphism $f: X \to Y$ is surjective if for any $y \in Y$, there exists an $x \in X$ such that $f(x) = y$
• Surjectivity of a morphism can be determined by examining the image of the corresponding ring homomorphism between the coordinate rings of the varieties
  • The image of a ring homomorphism $\varphi: A \to B$ is the set $\mathrm{im}(\varphi) = {\varphi(a) \mid a \in A}$
  • A morphism is surjective if and only if the image of the corresponding ring homomorphism is equal to the codomain ring

Isomorphisms between varieties

• A morphism that is both injective and surjective is called an isomorphism, and the varieties are said to be isomorphic
  • Isomorphic varieties have the same algebraic structure and properties
  • Example: The affine variety $V(y - x^2)$ is isomorphic to the affine line $\mathbb{A}^1$ via the morphism defined by $x \mapsto x$
• The inverse of an isomorphism is also an isomorphism
  • If $f: X \to Y$ is an isomorphism, then there exists a unique morphism $f^{-1}: Y \to X$ such that $f \circ f^{-1} = \mathrm{id}_Y$ and $f^{-1} \circ f = \mathrm{id}_X$

Constructing morphisms with functions

Constructing morphisms between affine varieties

• To construct a morphism from an affine variety to another, define polynomial functions on the coordinate ring of the domain variety that map to the coordinate ring of the codomain variety
  • The polynomial functions induce a ring homomorphism between the coordinate rings
  • Example: To construct a morphism from $V(y - x^2)$ to $V(w - z^3)$, define $x \mapsto z$ and $y \mapsto z^2$
• The polynomial functions defining the morphism must satisfy the relations of the ideal defining the codomain variety
  • This ensures that the morphism is well-defined and maps points of the domain variety to points of the codomain variety
  • Example: In the previous example, the polynomial functions satisfy the relation $y - x^2 \mapsto z^2 - z^2 = 0$, which is in the ideal defining $V(w - z^3)$

Constructing morphisms between projective varieties

• To construct a morphism from a projective variety to another, define homogeneous polynomial functions of the same degree on the homogeneous coordinate ring of the domain variety that map to the homogeneous coordinate ring of the codomain variety
  • The homogeneous polynomial functions induce a graded ring homomorphism between the homogeneous coordinate rings
  • Example: To construct a morphism from $V(xz - y^2)$ to $V(w^2 - uv)$, define $x \mapsto u$, $y \mapsto v$, and $z \mapsto w$
• The homogeneous polynomial functions defining the morphism must satisfy the relations of the homogeneous ideal defining the codomain variety
  • This ensures that the morphism is well-defined and maps points of the domain variety to points of the codomain variety
  • Example: In the previous example, the homogeneous polynomial functions satisfy the relation $xz - y^2 \mapsto uw - v^2$, which is in the homogeneous ideal defining $V(w^2 - uv)$

Specifying morphisms by their action on generators

• Morphisms can be constructed by specifying their action on the generators of the coordinate ring or homogeneous coordinate ring of the domain variety
  • The action on the generators determines the action on all elements of the ring
  • Example: To construct a morphism from $V(y - x^2)$ to $V(w - z^3)$, it suffices to specify $x \mapsto z$ and $y \mapsto z^2$, as $x$ and $y$ generate the coordinate ring of $V(y - x^2)$
• The specified action on the generators must be compatible with the relations of the codomain variety
  • The relations of the codomain variety must be satisfied when the generators are mapped according to the specified action
  • Example: In the previous example, the relation $y - x^2 = 0$ is satisfied when $x \mapsto z$ and $y \mapsto z^2$, as $z^2 - z^2 = 0$ in the coordinate ring of $V(w - z^3)$