Following our discussion of glueing schemes: Categorical descriptions for glueing sheaves and schemes. We now discuss the construction of the fibre product of schemes by glueing.
Given arbitrary schemes , let and be the given morphisms. Let be an open affine cover of . Let , , choose an affine open cover for and an affine open cover for . The fibre product is constructed by glueing various together.
We rewrite the glueing construction of fibre product in a more categorical way as follows. Note that the colimit here is glueing construction and the consequences of the two pullback squares should be clear thinking in terms of schemes.
View as pdf: Construction of fibre product
Theorem[Thm 3.3, / Thm 9.1.1, ] For any two schemes and over a scheme , the fibre product exists and is unique up to unique isomorphism.