Sheaves and varieties

If we compare the study of affine algebraic sets and projective algebraic sets, we find many similarities and a few fundamental differences, such as the role played by homogeneous polynomials and graded rings in projective geometry. The most important difference, however, is the functions. If V is an affine algebraic set, we have a lovely function algebra Γ(V) and an almost perfect dictionary translating properties of V into properties of Γ(V). One of the problems of projective geometry is that elements of Γ _h (V) do not define functions on V, even in the simplest case, namely a homogeneous polynomial, since if x ∈ Pⁿ and F is homogeneous of degree d, then the quantity F(x) depends on the choice of representative: F(λx) = λ^dF(x).