Lech's multiplicity conjecture asserts that a flat local ring homomorphism between nonzero Noetherian local rings 
satisfies
 |
where 
is the Hilbert-Samuel multiplicity
with respect to each ring's own maximal
ideal. Flatness means that 
is a flat module over 
, and locality means that the homomorphism
sends the maximal ideal of 
into that of 
.
The maximal ideals need not satisfy 
.
