Regular Morphism

Definition

A regular morphism is a type of morphism between algebraic varieties that locally looks like a polynomial function, meaning it can be expressed as the vanishing of a collection of polynomials. This concept plays a significant role in understanding the structure of algebraic varieties, particularly in affine and projective schemes, where it helps characterize how one variety maps to another and the nature of singularities and smooth points.

5 Must Know Facts For Your Next Test

1. Regular morphisms are characterized by being locally given by polynomial functions, ensuring smooth transitions between varieties.
2. In the context of affine schemes, a regular morphism corresponds to a ring homomorphism that preserves the structure of the underlying coordinate rings.
3. A morphism is considered regular if it can be described by finitely many equations in terms of polynomial functions.
4. The existence of regular morphisms allows for the study of smooth points on varieties, providing insights into their geometric properties.
5. Regular morphisms can help establish when varieties can be understood through simpler pieces, leading to important results in intersection theory and resolution of singularities.

Review Questions

• How does the definition of regular morphism relate to the concept of smoothness in algebraic varieties?
  • Regular morphisms are closely tied to the notion of smoothness because they ensure that locally, the mapping behaves like a polynomial function. This means that at smooth points, the morphism does not introduce singularities and allows for nice geometric properties. If a morphism is regular at all points of an algebraic variety, this implies that the variety itself is smooth at those points.
• Discuss how regular morphisms contribute to the understanding of affine and projective schemes in algebraic geometry.
  • Regular morphisms play a critical role in linking affine and projective schemes by establishing connections through polynomial functions. In affine schemes, they correspond to ring homomorphisms that maintain the integrity of coordinate rings. For projective schemes, regular morphisms help describe how varieties can be embedded and interact within projective space, enabling mathematicians to analyze their properties under various transformations.
• Evaluate how regular morphisms influence the resolution of singularities and the study of geometric properties in algebraic varieties.
  • Regular morphisms are instrumental in resolving singularities because they provide a framework for understanding how singular points relate to smooth structures. By analyzing these morphisms, mathematicians can often find ways to 'smooth out' singular points or understand them better through their local properties. This understanding is vital for broader investigations into the geometric characteristics of varieties, allowing for deeper insights into their structure and behavior under various operations.

Related terms

Affine Variety: An affine variety is a subset of affine space that can be defined as the common zeros of a set of polynomials, serving as the basic building block in algebraic geometry.

Projective Variety: A projective variety is a type of algebraic variety that is defined in projective space, allowing for the study of properties that remain invariant under projective transformations.

Locally Ringed Space: A locally ringed space is a topological space equipped with a sheaf of rings such that each stalk is a local ring, providing a framework for studying regular morphisms in algebraic geometry.