The Gersten conjecture asserts exactness of the augmented residue chain complex for algebraic K-theory over a regular local ring 
. If 
has Krull dimension 
and field
of fractions 
,
its first terms are
 |
where the direct sum runs over prime ideals of ideal height 1 and 
is the residue field
of the localization 
. In particular, the conjecture requires 
to be injective
(Weibel 2013).
