In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction.
The older notion of an algebraic variety over a field k is equivalent to a scheme over k with certain properties.
As always with universal properties, this condition determines the scheme X ×Y Z up to a unique isomorphism, if it exists.
Some important properties P of morphisms of schemes are preserved under arbitrary base change.
Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.
[6] These results form part of Grothendieck's theory of faithfully flat descent.