Ring of regular functions

Definition

The ring of regular functions on a variety is a collection of functions that are defined and behave well (like polynomials) on the variety, meaning they can be expressed as quotients of polynomials where the denominator does not vanish. This ring captures the algebraic structure of the variety and plays a critical role in understanding both its geometry and singularities, which is essential for working with concepts like blowing up and determining normality or Cohen-Macaulay properties.

5 Must Know Facts For Your Next Test

1. The ring of regular functions can be thought of as the coordinate ring of a variety, containing all functions that can be represented as polynomial expressions without denominators that vanish on the variety.
2. In blowing up, the ring of regular functions gets transformed to reflect the new geometry, allowing us to resolve singularities by creating new spaces with better properties.
3. For normal varieties, the ring of regular functions is integrally closed, meaning any element that satisfies a monic polynomial with coefficients from this ring must also belong to it.
4. A variety is Cohen-Macaulay if its ring of regular functions satisfies certain depth conditions, indicating that it has 'nice' geometric properties like being well-behaved at singular points.
5. Regular functions on varieties can be used to understand morphisms between varieties by relating their respective rings, highlighting how algebraic structures convey geometric relationships.

Review Questions

• How does the ring of regular functions relate to the process of blowing up in algebraic geometry?
  • The ring of regular functions plays a vital role during the process of blowing up because it transforms in such a way that reflects the new geometric structure being created. When we blow up a point on a variety, we are essentially creating an alternative space where the original singularity can be resolved. The ring gets restructured to capture this new local geometry, allowing us to analyze the resolution of singularities effectively through its algebraic properties.
• What are the implications of having an integrally closed ring of regular functions for the normality of varieties?
  • If the ring of regular functions on a variety is integrally closed, it indicates that the variety is normal. This means any function that satisfies a monic polynomial equation with coefficients from this ring must also belong to it, which reflects that there are no hidden singularities or irregular behaviors in its structure. Normal varieties have desirable properties like being smooth at most points and having well-defined local geometric features.
• Evaluate how the ring of regular functions contributes to establishing Cohen-Macaulay properties in varieties.
  • To establish Cohen-Macaulay properties, we look at the depth of the ring of regular functions associated with a variety. A Cohen-Macaulay variety will have its ring exhibit nice depth characteristics across its homogeneous ideals. This property ensures that the variety behaves well at its singular points and adheres to specific algebraic conditions that promote good geometric structure. Understanding these connections between rings and geometric properties helps in analyzing complex varieties and their applications in broader mathematical contexts.

Related terms

Affine Variety: An affine variety is a subset of affine space that is defined as the common zero set of a collection of polynomials.

Localization: Localization is a process that allows us to focus on a particular subset of a ring, enabling us to study local properties of functions defined on varieties.

Cohomology: Cohomology is a mathematical tool used to study topological spaces through algebraic means, often employed to analyze the properties of varieties.