Algebraic Subset of N-Dimensional Vector Space on Affine Scheme
Jiaming Luo
School of Mathematics and Statistics, Research Center of Modern Mathematics and Its Application, Kashi University,
Xinjiang, China
Keywords: Vector Space, Algebraic Subset, Affine Scheme.
Abstract: In this paper, we study the connection between the most basic Hodge theory in compact complex manifold
and the affine scheme in algebraic geometry. By introducing the definitions of algebraic subset and affine
scheme, the Hodge operator on-dimensional affine space is defined, the ringed space of algebraic subset
defined on affine space is constructed, and the proof the Nullstellensatz theorem is obtained.
1 INTRODUCTION
As we known, Hodge theory in compact complex
manifold is a very important theory in algebraic
manifold. It has become an important research topic
in algebraic geometry whether similar theories can be
established in the theory of algebraic variety, that is,
the famous Cheeger-Goresky-MacPherson
conjecture (Goresky M, 1980), (Goresky M, 1983). This
is still a very difficult job that is to solve at present.
2 ALGEBRAIC SUBSETS ON
HERMITIAN VECTOR SPACE
The polynomial ring
 
1
,,
n
k x x
defined on
n
-
dimensional vector space is a commutative ring. By
using the definition of algebraic subset, the Hodge
operator defined on
n
-dimensional affine space is
obtained, which is equivalent to Hermitian exterior
algebra. What is more important is to construct a
ringed space on this basis, establish the relationship
between the algebraic subset of the vector space and
the affine scheme, and further prove the
Nullstellensatz theorem satisfied by the algebraic
subset defined in the vector space.
2.1 Hermitian Exterior Algebra and
Hodge Operator on Algebraic
Subset
Definition 2.1.1:
Let be a
polynomial with coeffcients in a field
k
. The
function defined by
f
is called a polynomial
function on
n
-dimensional vector space
n
k
over
k
,
with values in
k
.
If
k
is infinite, then all polynomials on
n
-
dimensional vector space
n
k
over
k
can form a
commutative ring
 
1
,,
n
k x x
. Thus distinct
polynomials define distinct polynomial functions by
the “Definition 2.1.1([
David Eisenbud, 2008
])”. Since
no polynomial function other than 0 can vanish
identically on
n
k
, then
n
k
is usually called affine
n
-space over
k
, written
()
n
Ak
or
n
A
. Hence we
have the following definition:
Definition 2.1.2:
Given a subset
 
1
,,
n
I k x x
,
then we define a corresponding algebraic subset of
n
k
to be
11
( ) ( , , ) ( , , ) 0
n
nn
Z I a a k f a a for all f I

   


. (1)
The “Definition 2.1.2” gives an isomorphism
between algebraic subsets and subsets of a affine
space. Consider
 
1
,,
n
k x x
as a polynomial ring,
if
 
1
,,
n
I k x x
is an ideal, then we have an
isomorphism:
()Z I I
.
Similarly, we also have another isomorphism about a
subset of an affine space. Given any set
n
Xk
, we
can define
 
1 1 1
( ) , , ( , , ) 0 ( , , )
n n n
I X g k x x g b b for all b b X

   


. (2)
It is clear that
()IX
is an ideal and
()I X X
is
isomorphic.
 
1
,,
n
f k x x
538
Luo, J.
Algebraic Subset of n-dimensional Vector Space on Affine Scheme.
DOI: 10.5220/0012287200003807
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 2nd International Seminar on Artificial Intelligence, Networking and Information Technology (ANIT 2023), pages 538-541
ISBN: 978-989-758-677-4
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

Let
V
be a real finite-dimensional vector space
of dimension
m
which is equipped with an inner
product
,
, a Euclidean vector space. Actually,
V
is an affine
m
-space by the previous discussion.
Namely, if
 
1
,,
m
ee
is an orthonormal basis for
V
, then we have
 
 
 
, 1,2, ,
ii
I e e i n
are isomorphic by the (2). Thus for the exterior
algebra
s
V
, we can obtain the corresponding
orthonormal basis
 
 
12
1
:1
p
is
ps
Z I e i i i m




    






, (3)
moreover the Hodge operator can be defined. The
Hodge operator is a mapping
 
 
 
 
1 2 1 2
11
: :1 :1
pq
i s j m s
p s q m s
H Z I e i i i m Z I e j j j m


    
   
   
   
          
   
   
   
   
   

by the (3).
In order to better define the form of the Hodge
operator
H

, we need to prove the proposition
introduced below. It will be beneficial to establish a
relationship with the scheme theory.
Proposition 2.1.3:
The intersection of collection of
algebraic subsets is algebraic subset, i.e.
 
 
ii
ii
Z J Z J
. (4)
Furthermore, if we define
1
n
i
i
J


to be the set
consisting of all products of one function from each
i
J
, then we have
 
1
1
n
n
ii
i
i
Z J Z J







. (5)
Proof. By the “Definition 2.1.1”,

   
1
,,
ri
i
ZJ  
we have
 
1
, , 0
ir
f  
for
all
ii
fJ
, this implies that

i
i
fJ
such that
 
1
, , 0
r
f  
. Then
 
 
 
 
1
,,
r i i i
i i i
Z J Z J Z J    
; (6)
Conversely,

 
 
1
,,
ri
i
ZJ  
, it satisfies
 
1
, , 0
r
f  
for any
i
i
fJ
. Hence

ii
fJ
we have
 
1
, , 0
ir
f  
such that
   
1
,,
ri
ZJ  
. Thus
     
 
1
,,
r i i i
i i i
Z J Z J Z J    
. (7)
Then (4) right by (6) and (7).
Consider
   
1
1
,,
n
ri
i
ZJ 


, then for all
ii
fJ
,
1,2, ,in
, we have
 
1
, , 0
ir
f  
.
Since
1
n
i
i
fJ



,
:
rn
f k k
defined, then
   
1
, , 0, ,0
r
f  
. Thus
   
1
11
1
,,
n
nn
r i i i
ii
i
Z J Z J Z J 


   
  
   
   

; (8)
Conversely,
 
1
1
,,
n
ri
i
ZJ 






and
1
n
i
i
gJ



,
there exists the corresponding
ii
gJ
such that
     
 
 
1 1 1 1
, , , , , , , , 0, ,0
r r n r
g g g       
,
where
1,2, ,in
. Then we have
       
11
1
11
, , 0 , ,
nn
n
i r r i i i
i
ii
g Z J Z J Z J    



    



(9)
for all
ii
gJ
, where
1,2, ,in
. Then (5) right
by (8) and (9).
Q.E.D
According to the “Proposition 2.1.3”, we can let
 
 


 
 
1
1
pp
s
i i i
ps
ps
Z I e Z I e



  



. (10)
Hence the Hodge operator become
:
s m s
ij
H

  
by the (10), and
s m s m
ij

   
is an
orthonormal basis for
V
. If we find that
,
s
a b V
, then we have
,
s m s m
I I J J
I s J s
a H b a b a b


   
        
   
   
   

. (11)
Therefore, after the above discussion, a
representation combining Hermitian exterior algebra
and Hodge operator with algebraic subsets theory has
been obtained. Moreover, the operation method
between one element and another element
transformed by Hodge operator
H

in exterior
algebra
s
V
is given through formulas (11).
2.2 Affine Scheme of Algebraic Subset
and Nullstellensatz Theory
Now we need to construct a ringed space based on the
affine spaces discussed in the previous section, and
introduce the scheme theory to study related issues.
Algebraic Subset of n-dimensional Vector Space on Affine Scheme
539

Consider polynomial commutative ring
 
1
,,
n
k x x
and
n
-dimensional vector space
n
k
,
the topological space here is obtained by
commutative ring from the “Definition 2.1.1”, that is
to say, the topological space is obtained by
 
 
1
,,
n
Spec k x x
, where the elements are Prime
ideals in
 
1
,,
n
k x x
. Thus we can construct such a
ringed space
 
 
 
1
, , ,
n
Spec k x x
, where the sheaf
of ring
 
 
 
 


 
 
 
11
, , : , ,
nn
kK
k
K K Spec k x x K Spec k x x

   
,
and
 
 
 
 
 
 
11
, , , , , : , ,
nn
kK
k
K K Spec k x x K Spec k x x 

   
.
Definition 2.2.1:
A ringed space
 
,
X
X
is an affine
scheme, which means that it is isomorphic to a ringed
space of the form
 
,SpecA A
, where
A
is a ring, and
then we call
 
,
X
X
the ring of affine scheme.
For the ringed space formed by
n
-dimensional
Vector space
n
k
, we can find
n
Yk
, there exists a
mapping
 
 
1
: ( ) , ,
n
I Y Spec k x x
by setting
()h I Y
,
:
h
hK
.
It is clear that

is isomorphic. By the “Definition
2.2.1 (
Alexander Grothendieck, 2018
)”,
 
 
 
,
IY
IY
is an
affine scheme. Next, we can obtain the
Nullstellensatz theorem satisfied by the base space of
affine scheme
 
 
 
,
IY
IY
.
Proposition 2.2.2:
Let
 
IO
be the base space of the
affine probability form constructed above, and
 
 
1
, , ,
n
n
O k I O k x x
. Then
 
 
 
 
 
I Z I O rad I O
. (12)
Thus, the correspondences
   
 
I O Z I O
and
 
O I O
induce a bijection between the collection
of algebraic subsets of
n
k
and radical ideals of
 
1
,,
n
k x x
.
Proof. Since
k
is an algebraically closed field,
then we can obtain that the affine space
n
k
over
k
and defined polynomial function
 
1
,,
n
f k x x
.
Thus for any polynomial function
 
 
 
 
     
 
 
1 1 1
, , , , 0 , ,
n n n
I Z I O f k x x f a a for all a a Z I O      
.
That is to say,
 
1
, , 0
n
aa  
for all
   
 
     
 
1 1 1
, , , , , , 0
n
n n n
a a Z I O b b k g b b for all g I O    
.
By the property of that
k
is an algebraically closed
field, then

has no multiple roots, it implies that
   
 
rad I O  
. Hence we have
 
 
 
 
 
I Z I O rad I O
. (13)
For any
 
 
rad I O  
, there exists a
m
such
that
 
m
IO  
. Thus
   
 
1
,,
n
b b Z I O
, there
exists
 
1
, , 0
m
n
bb  
. Hence
 
 
 
m
I Z I O  
implies
 
 
 
 
 
I Z I O rad I O
. (14)
 
 
 
 
 
I Z I O rad I O
right by the (12) and (13).
Consider the bijection
 
O I O
, then we have
 
 
 
 
 
~
I Z I O Z I O
by the (12), and
 
 
 
 
 
 
 
 
~ contain
Z I O I Z I O rad I O I O  
.
Consider the bijection
   
 
I O Z I O
, we have
   
 
~ contain
I O Z I O O 
. Therefore, we can
use Figure 1 below to describe the corresponding
relationship mentioned above.
,
Figure 1: The commutative graph of algebraic subset and
ideals.
where
, 
are surjective by that
 
 
contain
Z I O O
,
and
, 
are isomorphic. Q.E.D
3 CONCLUSION
The expression of Hodge operator is given by
defining affine space on
n
-dimensional vector space
and introducing algebraic subset. The process of
constructing prime spectrum based on polynomial
ring is given, and on this basis, the ringed space
satisfying the affine scheme condition is constructed.
Finally, the Nullstellensatz theorem on the base space
of affine scheme has been obtained. However, this
paper only addresses some of the most basic issues in
Hodge's theory and does not extend to the relevant
theories of scheme. Therefore, it is necessary to
further investigate the relationship between Hodge
ANIT 2023 - The International Seminar on Artificial Intelligence, Networking and Information Technology
540

decomposition theory (Fox J, 1994) and Hodge's
harmonic representation (Nagase M, 1988) and
scheme in future.
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