Algebraic Geometry ed

Sheaves ed

Topological space \(X\)

presheaf \(\mathfrak F\)

• for every open subset \(U \subset X\) giving an abelien group \(\mathfrak F(U)\)
• for inclusions \(V \subset U\), a morphism \(\rho_{UV} : \mathfrak F(U) \rightarrow \mathfrak F(V)\)

with

• \(\mathfrak F(\emptyset) = 0\)
• \(\rho_{UU} = 1\)
• \(W \subset V \subset U \quad \Rightarrow \quad \rho_{UW} = \rho_{VW} \circ \rho_{UV} \)

(standard) example

regular functions on \(X\)

A sheaf has additional "global from local" conditions for an open cover \(\{U_i\}\) of \(U\):

sheaf

• if an element \(s \in \mathfrak F(U)\) has \(s_{U_i}=0\) on each part, then also \(s=0\)
• if there are elements \(\{s_{U_i}\}\) on each part, that coincide on the overlap (\({s_{U_i}}_{|U_i \cap U_k} = {s_{U_k}}_{|U_i \cap U_k}\)), then there exists a global \(s\)