Let ( R , m ) {\displaystyle {}(R,{\mathfrak {m}})} denote a two-dimensional regular local ring, let I = ( f 1 , … , f n ) {\displaystyle {}I={\left(f_{1},\ldots ,f_{n}\right)}} be an m {\displaystyle {}{\mathfrak {m}}} -primary ideal and f ∈ R {\displaystyle {}f\in R} an element with f ∉ I {\displaystyle {}f\notin I} . Let
be the corresponding forcing algebra.
Then for the extended ideal m B {\displaystyle {}{\mathfrak {m}}B} we have
In particular, the open subset D ( m B ) {\displaystyle {}D({\mathfrak {m}}B)} is an affine scheme if and only if f ∉ I {\displaystyle {}f\notin I} .
denote a two-dimensional regular local ring, let
be an 
-primary ideal and
an element with
.
Let
![{\displaystyle {}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6129de6909daf254da620457c2f29b3721e6b29e)
we have
is an affine scheme if and only if
.
