Continuous closure, axes closure, and natural closure
Authors: Epstein Neil , Hochster Melvin 
Published: August 2016

Let

R
R

be a reduced affine

C

\mathbb {C}

-algebra with corresponding affine algebraic set

X
X

. Let

C

(
X
)

\mathcal {C}(X)

be the ring of continuous (Euclidean topology)

C

\mathbb {C}

-valued functions on

X
X

. Brenner defined the continuous closure

I

c
o
n
t

I^{\mathrm {cont}}

of an ideal

I
I

as

I

C

(
X
)
∩
R

I\mathcal {C}(X) \cap R

. He also introduced an algebraic notion of axes closure

I

a
x

I^{\mathrm {ax}}

that always contains

I

c
o
n
t

I^{\mathrm {cont}}

, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining

f
∈

I

a
x

f \in I^{\mathrm {ax}}

if its image is in

I
S

IS

for every homomorphism

R
→
S

R \to S

, where

S
S

is a one-dimensional complete seminormal local ring. We also introduce the natural closure

I

♮

I^{\natural }

of

I
I

. One of many characterizations is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript normal ♮ Baseline equals upper I plus StartSet f element-of upper R colon there-exists n greater-than 0 normal w normal i normal t normal h f Superscript n Baseline element-of upper I Superscript n plus 1 Baseline EndSet">

I

♮

=
I
+
{
f
∈
R
:
∃
n
>
0

 
w
i
t
h
 

f
n

∈

I

n
+
1

}

I^{\natural } = I + \{f \in R: \exists n >0 \mathrm {\ with\ } f^n \in I^{n+1}\}

. We show that

I

♮

⊆

I

a
x

I^{\natural } \subseteq I^{\mathrm {ax}}

and that when continuous closure is defined,

I

♮

⊆

I

c
o
n
t

⊆

I

a
x

I^{\natural } \subseteq I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}

. Under mild hypotheses on the ring, we show that

I

♮

=

I

a
x

I^{\natural } = I^{\mathrm {ax}}

when

I
I

is primary to a maximal ideal and that if

I
I

has no embedded primes, then

I
=

I

♮

I = I^{\natural }

if and only if

I
=

I

a
x

I = I^{\mathrm {ax}}

, so that

I

c
o
n
t

I^{\mathrm {cont}}

agrees as well. We deduce that in the polynomial ring

C

[

x
1

,
…
,

x
n

]

\mathbb {C} \lbrack x_1, \ldots , x_n \rbrack

, if

f
=
0

f = 0

at all points where all of the

∂
f

∂

x
i

{\partial f \over \partial x_i}

are 0, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of left-parenthesis StartFraction partial-differential f Over partial-differential x 1 EndFraction comma ellipsis comma StartFraction partial-differential f Over partial-differential x Subscript n Baseline EndFraction right-parenthesis upper R">

f
∈
(

∂
f

∂

x
1

,

…
,

∂
f

∂

x
n

)
R

f \in ( {\partial f \over \partial x_1}, \, \ldots , \, {\partial f \over \partial x_n})R

. We characterize

I

c
o
n
t

I^{\mathrm {cont}}

for monomial ideals in polynomial rings over

C

\mathbb {C}

, but we show that the inequalities

I

♮

⊆

I

c
o
n
t

I^{\natural } \subseteq I^{\mathrm {cont}}

and

I

c
o
n
t

⊆

I

a
x

I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}

can be strict for monomial ideals even in dimension 3. Thus,

I

c
o
n
t

I^{\mathrm {cont}}

and

I

a
x

I^{\mathrm {ax}}

need not agree, although we prove they are equal in

C

[

x
1

,

x
2

]

\mathbb {C}[x_1, x_2]

.