Flat Morphism

Definition

A flat morphism is a type of morphism between schemes that intuitively represents a situation where the fibers of a map behave nicely in terms of dimension and structure. More specifically, it ensures that the pullback of a module is exact, meaning that flatness maintains properties when passing through a morphism, which is crucial in various areas such as deformation theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. Flat morphisms are characterized by the property that the fiber over each point behaves in a way that the dimension is constant or varies nicely.
2. In algebraic geometry, flatness helps in preserving properties like being integrally closed or Cohen-Macaulay when moving through morphisms.
3. A morphism is flat if and only if the corresponding sheaf of modules is flat as a module over the ring of regular functions on the base scheme.
4. Flatness is stable under base change, meaning that if you have a flat morphism and you base change it, the new morphism remains flat.
5. Flat morphisms play an important role in deformation theory by allowing us to analyze how objects vary smoothly within families.

Review Questions

• How does the property of flatness impact the behavior of fibers over a morphism in algebraic geometry?
  • Flatness ensures that fibers over points behave nicely, meaning that they do not have sudden jumps in dimension or structure. This property allows us to maintain certain algebraic characteristics across these fibers, making it easier to work with families of schemes. In particular, when you pull back modules along a flat morphism, the exactness is preserved, which is crucial for many applications like deformation theory.
• Discuss how flat morphisms relate to exact sequences in the context of sheaves and modules.
  • Flat morphisms preserve exactness when dealing with sheaves and modules. When you take an exact sequence of modules and apply a flat morphism, the resulting sequence remains exact. This property is significant because it allows us to transfer information about modules across morphisms without losing essential structural features, aiding in understanding how properties like Cohen-Macaulayness or integrally closed conditions are maintained.
• Evaluate the implications of flatness being stable under base change for algebraic varieties and their families.
  • The stability of flatness under base change has significant implications for algebraic varieties since it means that if we start with a family of varieties that are flat over a base scheme, pulling back to another base will keep this flatness intact. This characteristic allows mathematicians to study families of schemes through various perspectives without losing control over their properties. It opens up avenues for exploring deformation problems while ensuring that our geometric intuition remains grounded.

Related terms

Exact Sequence: A sequence of modules and homomorphisms where the image of one map equals the kernel of the next, reflecting a certain balance in structure.

Fiber Product: The categorical product of two schemes over a third scheme, giving a scheme that captures the idea of 'pulling back' the structure from both sources.

Locally Ringed Space: A space that has a stalk for each point, allowing local analysis using rings of functions defined near those points, essential for understanding schemes.