Morphism of varieties

Definition

A morphism of varieties is a map between two algebraic varieties that respects their structure, meaning it can be defined by regular functions on their coordinate rings. This concept establishes a relationship between the varieties, allowing for the transfer of geometric and algebraic properties. Morphisms can be thought of as the algebraic counterpart to continuous functions in topology, playing a crucial role in understanding how varieties interact and relate to each other.

5 Must Know Facts For Your Next Test

1. Morphism of varieties can be either morphisms of finite type or morphisms that are dominant, depending on the algebraic relationships they establish.
2. Every morphism induces a pullback on the coordinate rings, allowing for the study of how functions on one variety relate to functions on another.
3. If a morphism is surjective, it can also have interesting implications for the dimensions of the involved varieties.
4. Morphism of varieties can be used to classify varieties, as similar types of morphisms often reveal important geometric and algebraic properties.
5. In the case of projective varieties, morphisms can be defined with respect to their embeddings into projective space, leading to richer structures.

Review Questions

• How does a morphism of varieties relate to regular functions and coordinate rings?
  • A morphism of varieties is fundamentally linked to regular functions and coordinate rings because it is defined by these functions. Specifically, a morphism connects two varieties through maps that correspond to regular functions in their respective coordinate rings. This relationship allows us to analyze how properties of one variety can influence another, revealing deeper insights into their structure and interconnections.
• What is the significance of dominant morphisms in relation to the dimensions of varieties?
  • Dominant morphisms play a crucial role in understanding the dimensions of varieties because they indicate how one variety can cover another. When a morphism is dominant, it implies that the image of this morphism has dimension equal to or greater than that of the target variety. This connection helps in classifying varieties by their dimensional properties and reveals how higher-dimensional varieties might project onto lower-dimensional ones.
• Evaluate the impact that isomorphisms of varieties have on the classification and understanding of algebraic structures.
  • Isomorphisms of varieties significantly enhance our understanding and classification of algebraic structures by establishing a precise equivalence between them. When two varieties are isomorphic, they share identical algebraic and geometric properties, which allows mathematicians to transfer results and insights across seemingly different settings. This notion streamlines many concepts in algebraic geometry since it focuses attention on essential characteristics rather than superficial differences, facilitating deeper exploration into complex relationships among various geometric objects.

Related terms

Regular functions: Functions that can be expressed as quotients of polynomials, which are well-defined on the varieties they are associated with.

Coordinate ring: A ring that captures the algebraic structure of a variety, consisting of regular functions defined on that variety.

Isomorphism of varieties: A special type of morphism that establishes a one-to-one correspondence between two varieties, preserving their structure and properties.