Categorical Construction of Fibre Product of Schemes

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 X,Y,S, let q:X\to S and r: Y\to S be the given morphisms. Let \{S_i\} be an open affine cover of S. Let X_i=q^{-1}(S_i), Y_i=r^{-1}(S_i), choose an affine open cover X_{ij} for X_i and an affine open cover Y_{jk} for Y_k. The fibre product is constructed by glueing various X_i\times_{S_i} Y_i  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, [1]/ Thm 9.1.1, [2]] For any two schemes X and Y over a scheme S, the fibre product X\times _S Y exists and is unique up to unique isomorphism.