The Stacks project

Remark 23.9.3. It appears difficult to define an good notion of “local complete intersection homomorphisms” for maps between general Noetherian rings. The reason is that, for a local Noetherian ring $A$, the fibres of $A \to A^\wedge $ are not local complete intersection rings. Thus, if $A \to B$ is a local homomorphism of local Noetherian rings, and the map of completions $A^\wedge \to B^\wedge $ is a complete intersection homomorphism in the sense defined above, then $(A_\mathfrak p)^\wedge \to (B_\mathfrak q)^\wedge $ is in general not a complete intersection homomorphism in the sense defined above. A solution can be had by working exclusively with excellent Noetherian rings. More generally, one could work with those Noetherian rings whose formal fibres are complete intersections, see [Rodicio-ci]. We will develop this theory in Dualizing Complexes, Section 47.23.