Lagmental Vicfred

Fibre Products of Affine Schemes Are Tensor Products by Vicfred

tags: mathematics, schemes, morphisms, functor of points

The geometric pullback over an affine base is represented by a tensor product of algebras. I want the notation, the mechanism, and the failure mode visible at the same time.

Objects and notation

An affine scheme \(X=\operatorname{Spec}A\) is determined by its ring together with its prime spectrum and structure sheaf. Maps \(X\to Y\) reverse the direction of ring homomorphisms.

$$ X=\operatorname{Spec}B,\quad Y=\operatorname{Spec}C,\quad S=\operatorname{Spec}A $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ X\times_SY\cong\operatorname{Spec}(B\otimes_AC) $$

Push the symbols

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{array}{ccc}\operatorname{Spec}(B\otimes_AC)&\longrightarrow&\operatorname{Spec}C\\\downarrow&&\downarrow\\\operatorname{Spec}B&\longrightarrow&\operatorname{Spec}A\end{array} $$

Structural reading

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad X=\operatorname{Spec}B,\quad Y=\operatorname{Spec}C,\quad S=\operatorname{Spec}A,\\[5pt] \mathsf{C}\;&:\quad X\times_SY\cong\operatorname{Spec}(B\otimes_AC). \end{aligned} $$

A hypothesis worth keeping

Field-valued points see only part of a scheme. Nilpotents, residue-field extensions, and families over nonfields require general test rings.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X\times_SY\cong\operatorname{Spec}(B\otimes_AC) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Tue 03 April 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.