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Sheaves (from Algebraic Geometry by Hartshorne, Section 2.1)

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Sheaves (from Algebraic Geometry by Hartshorne, Section 2.1)

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Let's start the study of the second chapter of book Algebraic Geometry by Robin Hartshorne. Knowledge of the first chapter is not required.

Let X be a topological space. A presheaf F of sets on X consists of the following data:

  • For each open set U⊆X, there exists a set F(U). This set is also denoted Γ(U,F). The elements in this set are called the sections of F over U. The sections of F over X are called the global sections of F.
  • For each inclusion of open sets V⊆U, a function resVU:F(U)→F(V). In view of many of the examples below, the morphisms resVU are called restriction morphisms. If s∈F(U), then its restriction resVU(s) is often denoted s|V by analogy with restriction of functions.

The restriction morphisms are required to satisfy two additional (functorial) properties:

  • For every open set U of X, the restriction morphism resUU:F(U)→F(U) is the identity morphism on F(U).
  • If we have three open sets W⊆V⊆U, then the composite resWV∘resVU=resWU.

Informally, the second axiom says it does not matter whether we restrict to W in one step or restrict first to V, then to W.

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