Types of Sheaves to Know for Sheaf Theory

Sheaf Theory studies how local data can be organized into global structures. Different types of sheaves, like constant, locally constant, and coherent sheaves, help us understand complex topological and algebraic concepts, making them essential in various mathematical fields.

1. Constant sheaves

   • Assigns the same set (or group) to every open set in a topological space.
   • The sections over any open set are isomorphic to a fixed set, typically denoted as ( A ).
   • Useful in algebraic topology and homological algebra for constructing cohomology theories.

2. Locally constant sheaves

   • Sections are locally constant functions, meaning they are constant on each connected component of the space.
   • Often arise in the study of covering spaces and fundamental groups.
   • Can be associated with local systems of coefficients in homology and cohomology theories.

3. Flasque sheaves

   • Any section over an open set can be extended to larger open sets, making them "flexible."
   • Useful for the application of the sheaf cohomology, particularly in the context of the sheaf's global sections.
   • They allow for the computation of cohomology groups via the derived functor approach.

4. Soft sheaves

   • Sections over smaller open sets can be extended to larger open sets, similar to flasque sheaves but with less strict conditions.
   • Important in the context of sheaf cohomology, particularly in the study of sheaves of modules.
   • They help in the analysis of the behavior of sheaves under various topological operations.

5. Fine sheaves

   • A refinement of soft sheaves, where sections can be extended from any open cover.
   • They are particularly useful in the context of differential forms and sheaf theory on manifolds.
   • Fine sheaves are often used in the context of de Rham cohomology.

6. Coherent sheaves

   • Sheaves of modules that satisfy certain finiteness conditions, such as being finitely generated.
   • Important in algebraic geometry, particularly in the study of varieties and schemes.
   • Coherent sheaves allow for the application of various cohomological techniques.

7. Quasi-coherent sheaves

   • Generalization of coherent sheaves, allowing for sheaves that may not be finitely generated.
   • They are essential in the study of schemes and algebraic geometry.
   • Quasi-coherent sheaves can be viewed as sheaves of modules over a ringed space.

8. Vector bundles (as sheaves of sections)

   • A vector bundle can be viewed as a sheaf of vector spaces over a topological space.
   • Sections of vector bundles correspond to smooth or continuous functions that take values in the vector space.
   • Fundamental in differential geometry and topology, particularly in the study of manifolds.

9. Étale sheaves

   • Sheaves defined in the context of étale topology, which is a generalization of the classical topology.
   • They are crucial in algebraic geometry for studying properties of schemes.
   • Étale sheaves allow for the application of Galois theory in a geometric setting.

10. Constructible sheaves

    • Sheaves that are locally constant on a stratification of the space, allowing for a piecewise structure.
    • Important in the study of perverse sheaves and intersection cohomology.
    • They play a significant role in the theory of motives and in the study of singularities.