Serre's condition 
for a commutative Noetherian
ring 
requires
 |
at every prime ideal 
. Here module depth
is the maximum length of a ring regular sequence
for 
in the maximal ideal of this local
ring, and 
denotes Krull dimension. A locally
Noetherian scheme satisfies 
when its local rings do. The
depth condition limits embedded and low-codimension
defects (The Stacks Project Authors 2026). It is required in the definition of a
semi-log canonical pair, alongside conditions
on codimension-one singularities.
