Reduced Scheme

Definition

A reduced scheme is a type of scheme in algebraic geometry where the structure sheaf has no nilpotent elements. This means that the local rings of a reduced scheme do not have any elements that, when raised to some power, become zero. Reduced schemes are significant because they correspond to spaces that do not have 'infinitesimally small' points, making them easier to work with when considering geometric properties.

5 Must Know Facts For Your Next Test

1. In a reduced scheme, for every local ring associated with a point, there are no nilpotent elements, which simplifies many algebraic computations.
2. Every reduced scheme can be thought of as corresponding to an underlying topological space that is 'nice' and behaves well with respect to geometric intuition.
3. Reduced schemes can be considered as the 'classical' parts of more complicated schemes that might have nilpotent elements, making them essential for studying singularities.
4. The notion of being reduced is crucial when discussing morphisms between schemes, as it helps identify when morphisms are 'well-behaved' in a geometric sense.
5. Examples of reduced schemes include affine schemes given by reduced rings, such as $ ext{Spec}(k[x,y]/(xy))$, where $k$ is a field.

Review Questions

• What is the significance of having no nilpotent elements in the structure sheaf of a reduced scheme?
  • The absence of nilpotent elements in the structure sheaf of a reduced scheme means that the local rings at each point are integral domains. This property simplifies the analysis of functions defined on the scheme, allowing for better geometric interpretation and leading to cleaner results in algebraic computations. It also means that reduced schemes behave more like classical varieties, making them crucial for understanding more complex structures in algebraic geometry.
• How does the concept of a reduced scheme relate to irreducibility in algebraic geometry?
  • While reduced schemes focus on the absence of nilpotent elements, irreducibility pertains to the inability to decompose a scheme into simpler components. A reduced scheme can still be reducible if it can be expressed as the union of closed subsets. However, being reduced guarantees that each component behaves 'nicely' at the level of its local rings. Therefore, while both concepts deal with different aspects of the structure of schemes, they collectively help classify and understand the underlying geometry.
• Evaluate how reduced schemes facilitate understanding singularities in algebraic geometry.
  • Reduced schemes provide a clearer framework for examining singularities since they exclude nilpotent elements that often complicate such analyses. By focusing on these 'classical' aspects, mathematicians can isolate singular points from nonsingular ones more effectively. This simplification allows researchers to utilize tools from commutative algebra and topology without the interference of infinitesimal behaviors inherent in non-reduced schemes. Thus, studying reduced schemes is fundamental for identifying and understanding various types of singularities within broader algebraic structures.

Related terms

Nilpotent Element: An element of a ring that becomes zero when raised to some power.

Scheme: A mathematical object that generalizes algebraic varieties and allows for the use of tools from both algebra and geometry.

Irreducible Scheme: A scheme that cannot be expressed as the union of two proper closed subschemes, indicating it is 'whole' in a certain sense.