Presheaf

Let X be a topological space. Then a presheaf F of sets on X:

For each open set U of X, there exist F(U). Another notation for this can be Γ(F,U) also called the sections of F over U. The sections of X is called the global sections of F. Γ(F,U)

For each V ⊆ U, a subsubset of X, a function called restriction morphism is from F(U) ----> F(V), the space is getting smaller conveniently for property malleable porpuses. s ∈ F(U) its restriction function means s is sent to the restriction section V also called res_U,V = s|_V

This restriction function forces to have two extra properties.

For every open set U of X: res_U,U: F(U) ----> F(U) as the identity morphism.

For three opensubsets of X, W ⊆ V ⊆ U, then composition restriction functor is res_W,V o res_V,U = res_W,U

Sheaf, at last!

Sheaf is a presheaf under two properties.

(Locally) Suppose U is an open set of X, {U_i} is the open cover of U, so for F(U), a presheaf of U. For all s, t in F(U), s|_U_i = t|_U_i, then s=t.

(Gluing) Suppose s_i in F(U) then s_i is a family of sections over the open cover of U. If all pairs of sections agree on its domain, then there exists a section s in F(U) such that s|_U_i = s_i for all i in I.

Lets S be the set of polynomials and Z be the vanishing set of those polynomials, such set over an affine space forms an affine variety.

Z(S) := {x in X | f(x)=0 for all f in S}