A regular function is a type of function defined on a variety that behaves nicely, meaning it is represented by a polynomial and does not have any singularities. These functions are crucial in algebraic geometry because they allow for a well-defined structure on the varieties, facilitating the study of their properties. Regular functions can be thought of as the 'nice' functions that can be computed at every point of a variety, offering important insights into the geometric properties of the space.
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Regular functions can be globally defined on affine varieties and locally defined on more general varieties.
These functions can be added, subtracted, multiplied, and divided (except by zero) while still remaining regular.
The set of all regular functions on an affine variety corresponds to the coordinate ring of that variety.
Regular functions can be used to determine the properties of varieties, such as dimension and singularities.
In the context of projective varieties, regular functions can be extended to include some types of limits or behaviors at infinity.
Review Questions
How do regular functions contribute to our understanding of affine varieties?
Regular functions are essential for studying affine varieties because they define the coordinate ring, which encodes all the algebraic information about the variety. By examining these functions, we gain insights into the geometry and structure of the variety, such as its dimensionality and singular points. The behavior and interactions of regular functions inform us about how the variety can be manipulated and understood algebraically.
Discuss the differences between regular functions and rational maps, including their implications for geometric properties.
Regular functions are well-defined everywhere on a variety, while rational maps may not be defined at certain points due to potential singularities. This distinction has significant implications: regular functions provide a consistent way to analyze a variety's structure, whereas rational maps can highlight specific behaviors or transitions between varieties. Understanding this difference allows mathematicians to distinguish between global properties described by regular functions and more nuanced behaviors captured by rational maps.
Evaluate how the concept of local rings enhances our understanding of regular functions in various contexts.
Local rings play a vital role in enhancing our understanding of regular functions by focusing on behavior around specific points on a variety. They consist of regular functions defined in small neighborhoods, allowing for precise analysis of singularities and other local phenomena. By studying these local properties through local rings, we can apply powerful techniques like localization, which help elucidate global properties while retaining important local information about regularity and behavior near particular points.
A rational map is a function between varieties that may not be defined everywhere, often arising from the quotient of two regular functions.
Affine variety: An affine variety is a subset of affine space that is defined as the solution set to a system of polynomial equations, where regular functions can be studied.
Local ring: A local ring is a type of ring associated with a point on a variety, consisting of regular functions that are defined in a neighborhood around that point.