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Skyscraper SheafEdit

Skyscraper sheaf is a basic construction in sheaf theory and algebraic geometry that packages data so that it lives entirely at a single point of a space. Given a point x in a topological space X and a algebraic object A (typically an abelian group or a module over a ring), the skyscraper sheaf δx(A) concentrates all of A at x and is zero away from x. Concretely, δ_x(A) is the pushforward Pushforward (sheaf theory) i{x*} A along the inclusion i_x: {x} → X, so its sections on an open set U are A if x ∈ U and 0 if x ∉ U. The stalk of δ_x(A) at a point y is A if y = x and 0 otherwise, making its support the single point {x}.

Definition

Let X be a topological space, x ∈ X a fixed point, and A a module (for example an abelian group) over a ring R. Let i_x: {x} → X denote the inclusion of the one-point subspace. The skyscraper sheaf at x with value A is

δx(A) = i{x*} A,

a sheaf of A-valued objects on X whose sections satisfy Γ(U, δ_x(A)) ≅ A if x ∈ U and Γ(U, δ_x(A)) ≅ 0 if x ∉ U. The stalk at any y ∈ X is

(δ_x(A))_y ≅

  • A if y = x,
  • 0 if y ≠ x.

This construction is standard in the study of sheaves on spaces Sheaf and is frequently used in both topology and algebraic geometry, including in the setting of schemes Scheme and closed points Closed immersion.

Basic properties

  • Global sections: Because x ∈ X, the skyscraper sheaf has Γ(X, δ_x(A)) ≅ A.
  • Support: The support of δ_x(A) is precisely the singleton {x}, making it a natural test object for local phenomena at a point.
  • Hom and Ext tests: For any sheaf F on X, there is a natural adjunction

Hom_X(δ_x(A), F) ≅ Hom(A, F_x),

where F_x is the stalk of F at x. Concretely, maps from the skyscraper to F correspond to choosing a map from A into the fiber data at x. In higher Ext groups, one has Ext^i_X(δ_x(A), F) ≅ Ext^i_R(A, F_x) for i > 0 in appropriate settings, illustrating how δ_x(A) encodes pointwise information.

  • Functoriality: δx(A) behaves functorially in A via the functoriality of i{x*} and in x under the action of the space’s topology.

Examples

  • Topological line: On X = R with its usual topology and a fixed abelian group A, δ_0(A) assigns A to any open set containing 0 and 0 to any open set missing 0. The stalk at 0 is A, while the stalks at other points are 0.
  • Discrete spaces: If X is a discrete space, δ_x(A) looks like the sheaf that assigns A to every open set that contains x (which, in the discrete topology, is exactly those that include x) and 0 otherwise; this behaves like a simple component in the direct sum decomposition of constructible sheaves on X.
  • Schemes and residue fields: On a scheme X, the skyscraper sheaf at a closed point x with value the residue field k(x) is δx(k(x)) ≅ i{x*} k(x). Such objects generalize the idea of “functions supported at a point” to the setting of sheaves on schemes Scheme and are used to model local information at x.

In algebraic geometry and schemes

Skyscraper sheaves arise naturally in the category of sheaves on a scheme X. For a closed point x ∈ X, the inclusion i_x: Spec k(x) → X (where k(x) is the residue field at x) yields the skyscraper sheaf δx(k(x)) = i{x*} k(x). These objects are frequently used to probe the local structure of X, to form short exact sequences that isolate data at x, and to model simple modules concentrated at a point. They also facilitate calculations of local cohomology and the behavior of pushforwards and pullbacks along closed immersions Closed immersion and Pushforward (sheaf theory).

Applications and connections

  • Local-to-global analysis: δ_x(A) serves as a compact way to study how global objects restrict to a single point, aiding in the computation of Hom and Ext groups via stalks Stalk (topology).
  • Derived categories: In derived categories of sheaves, skyscraper objects often act as building blocks for more complex objects, and they appear in discussions of t-structures, perverse sheaves, and related constructs when focusing on point-supported phenomena.
  • Local cohomology and supports: Skyscraper sheaves are tied to notions of local cohomology with support at a point, enabling a clean way to encode information concentrated at x Local cohomology.
  • Relation to other basic objects: They contrast with more global sheaves such as constant sheaves or locally constant sheaves, highlighting the spectrum from global to pointwise data.

Controversies and debates (from a typical mathematical perspective)

Within mathematics, discussions around skyscraper sheaves tend to revolve around their role as fundamental, highly local objects versus more global or categorical approaches. Some viewpoints emphasize that:

  • Skyscraper sheaves are essential building blocks for understanding local behavior and for explicit computations, and they adapt readily to both topological and algebro-geometric contexts.
  • In more advanced frameworks (for example, in the study of constructible or perverse sheaves), there is a preference to work with longer chains of objects that capture richer geometric information, with skyscraper sheaves serving as the simplest point-supported cases or as testing objects.
  • The use of point-supported objects can be contrasted with global sections and sheaves of higher rank, but in many theories the skyscraper construction remains a clean, concrete tool for isolating and transporting local data.

See also