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.

construction1construction2

reference

One thought on “Categorical Construction of Fibre Product of Schemes

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s