DOI: 10.1017/s0010437x26103133 ISSN: 0010-437X
Zariski–Nagata theorems for singularities and the Uniform Izumi–Rees Property
Thomas Polstra Abstract
We introduce and explore the Uniform Izumi–Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if
R
is a normal domain essentially of finite type over a field, there exists a constant
C
such that for all prime ideals
German p subset of or equal to German q element of normal upper S normal p normal e normal c left parenthesis upper R right parenthesis
p
⊆
q
∈
S
p
e
c
(
R
)
$\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$
, if
German p subset of or equal to German q Superscript left parenthesis t right parenthesis
p
⊆
q
(
t
)
$\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$
, then for all
n element of double struck upper N
n
∈
N
$n\in\mathbb{N}$
, there is a containment of symbolic powers
German p Superscript left parenthesis upper C n right parenthesis Baseline subset of or equal to German q Superscript left parenthesis t n right parenthesis
p
(
C
n
)
⊆
q
(
t
n
)
$\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$
.