Krull Dimension

Definition

Krull dimension is a fundamental concept in algebraic geometry that refers to the maximum length of chains of prime ideals in a given ring. It provides insight into the 'size' and structure of algebraic varieties, linking the geometric properties of varieties to the algebraic properties of the rings associated with them. This notion is particularly useful in understanding affine and projective schemes, as well as the dimensions and degrees of varieties.

5 Must Know Facts For Your Next Test

1. The Krull dimension of a ring can be determined by finding the longest ascending chain of prime ideals within that ring.
2. In algebraic geometry, the Krull dimension is an important tool for understanding the dimensionality of both affine and projective varieties.
3. For a field, the Krull dimension is 0 because it has no non-trivial prime ideals other than itself.
4. The Krull dimension relates closely to the concept of irreducibility in varieties, as irreducible varieties have Krull dimension 1.
5. In projective geometry, the Krull dimension helps characterize projective spaces, indicating that projective space of dimension n has Krull dimension n+1.

Review Questions

• How does Krull dimension relate to chains of prime ideals, and what does this reveal about the structure of a ring?
  • Krull dimension is defined as the maximum length of chains of prime ideals in a ring. This length gives insight into how complex or layered the algebraic structure is; a longer chain indicates a more intricate structure. Understanding these chains allows mathematicians to classify rings based on their dimensional properties and can reveal whether certain geometric features exist within associated varieties.
• Discuss how Krull dimension impacts the study of affine varieties in algebraic geometry.
  • The Krull dimension directly impacts affine varieties by determining their dimensionality based on the associated coordinate ring. For an affine variety defined by polynomial equations, its Krull dimension gives a measure of its geometric complexity. Essentially, if an affine variety has higher Krull dimension, it suggests that it can accommodate more complicated structures or relationships among its points defined by these equations.
• Evaluate how understanding Krull dimension can enhance our comprehension of projective schemes and their properties.
  • Understanding Krull dimension enhances comprehension of projective schemes by linking geometric properties with algebraic characteristics. Since projective spaces are constructed through homogeneous coordinates, knowing their Krull dimension allows us to grasp how many parameters define them. For instance, in projective space of dimension n, recognizing that it has a Krull dimension of n+1 reveals not just how many dimensions we visually perceive but also how these dimensions relate to underlying algebraic structures and ideals in their defining rings.

Related terms

Prime Ideal: A prime ideal in a ring is an ideal such that if the product of two elements is in the ideal, at least one of those elements must also be in the ideal.

Affine Varieties: Affine varieties are subsets of affine space defined as the common zeros of a set of polynomials, and they correspond to coordinate rings in algebraic geometry.

Dimension Theory: Dimension theory studies the different notions of dimension for spaces and algebraic objects, providing tools to understand their topological and algebraic structure.