Dimensions of monomial varieties

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Jul 23, 2017 - are Atiyah and Macdonald [1, Chapter 11], Cox, Little, and O'Shea [2, ..... aℓ,jim+j and b′. ℓ = iℓ + k+1. ∑ j=1. aℓ,jim+j = bℓ + aℓ,k+1im+k+1.

arXiv:1612.06841v1 [math.AG] 20 Dec 2016

ELEMENTARY CALCULATION OF THE DIMENSIONS OF SOME MONOMIAL VARIETIES MELVYN B. NATHANSON

Abstract. The dimensions of certain varieties defined by monomials are computed using only high school algebra.

1. Krull dimension and varieties In this paper, a ring R is a commutative ring with a multiplicative identity, and a field F is an infinite field of any characteristic. Let S be a nonempty set of polynomials in F[t1 , . . . , tn ]. The variety (also called the algebraic set ) V determined by S is the set of points in Fn that are common zeros of the polynomials in S, that is, V = {(x1 , . . . , xn ) ∈ Fn : f (x1 , . . . , xn ) = 0 for all f ∈ S} . The set I(V ) of polynomials that vanish on the variety V , that is, I(V ) = {f ∈ F[t1 , . . . , tn ] : f (x1 , . . . , xn ) = 0 for all (x1 , . . . , xn ) ∈ V } is an ideal in the polynomial ring F[t1 , . . . , tn ]. The quotient ring F(V ) = F[t1 , . . . , tn ]/I(V ) is called the coordinate ring of V . Note that I(V ) contains S, and so I(V ) contains the ideal generated by S. A prime ideal chain of length n in the ring R is a strictly increasing sequence of n + 1 prime ideals of R. The Krull dimension of R is the supremum of the lengths of prime ideal chains in R. For example, the polynomial ring F[t1 , . . . , tn ] has Krull dimension n (Atiyah and Macdonald [1, Chapter 11], Cox, Little, and O’Shea [2, Chapter 9], Kunz [3, Chapter 2], Nathanson [4] ). We define the dimension of the variety V as the Krull dimension of its coordinate ring F(V ). We adopt standard polynomial notation. Let N0 denote the set of nonnegative integers. Associated to every n-tuple I = (i1 , . . . , in ) ∈ Nn0 is the monomial tI = ti11 · · · tinn . Every polynomial f ∈ R[t1 , . . . , tn ] can be represented uniquely in the form X c I tI f= I∈Nn 0

where ci ∈ R and cI 6= 0 for only finitely many I ∈ Nn0 . Date: December 21, 2016. 2010 Mathematics Subject Classification. Primary 13C15, 12D99, 12-01,13-01. Key words and phrases. Dimension of varieties, elementary methods. 1

2

MELVYN B. NATHANSON

In this paper, we need only two results about polynomials from high school algebra to compute the dimensions of certain varieties defined by monomials. The first is the factorization formula xi − y i = (x − y)∆i (x, y).

(1) where (2)

∆i (x, y) =

i−1 X

xi−1−j y j .

j=0

The second result, which follows from the fact that of degree n has P a polynomial cI tI ∈ F[t1 , . . . , tn ] and if at most n roots in a field, states that if f = I∈Nn 0 f (x1 , . . . , xn ) = 0 for all (x1 , . . . , xn ) ∈ Fn , then (because the field F is infinite) cI = 0 for all I ∈ Nn0 . 2. An m-dimensional variety (k = 1) Let m and k be positive integers, and let n = m + k. We begin with the case k = 1 and n = m + 1. Lemma 1. For λ ∈ F and (a1 , . . . , am ) ∈ Nm 0 , let f ∗ = tm+1 − λta1 1 · · · tamm ∈ F[t1 , . . . , tm+1 ]. For I = (i1 , . . . , im , im+1 ) ∈ Nm+1 , let 0 bℓ = iℓ + aℓ im+1

for ℓ = 1, . . . , m.

There exists a polynomial gI ∈ F[t1 , . . . , tm+1 ] such that tI − λim+1 tb11 · · · tbmm = gI f ∗ . Proof. Recall the polynomial ∆i (u, v) defined by (2), and let gI = ti11 · · · timm ∆im+1 (tm+1 , λta1 1 · · · tamm ) ∈ F[t1 , . . . , tm+1 ]. We have tI −λim+1 tb11 · · · tbmm i

i +a i

m+1 im +am im+1 = ti11 · · · timm tm+1 − λim+1 t11 1 m+1 · · · tm   im+1 a i = ti11 · · · timm tm+1 − λim+1 t1 1 m+1 · · · tamm im+1   im+1 i = ti11 · · · timm tm+1 − (λta1 1 · · · tamm ) m+1

= ti11 · · · timm ∆im+1 (tm+1 , λta1 1 · · · tamm ) (tm+1 − λta1 1 · · · tamm ) = gI f ∗ .

This completes the proof.



Theorem 1. Let F be an infinite field. For λ ∈ F and (a1 , . . . , am ) ∈ Nm 0 , consider the polynomial f ∗ = tm+1 − λta1 1 ta2 2 · · · tamm ∈ F[t1 , . . . , tm , tm+1 ] and the corresponding variety  V = (x1 , . . . , xm , xm+1 ) ∈ Fm+1 : f ∗ (x1 , . . . , xm , xm+1 ) = 0 .  = (x1 , . . . , xm , λxa1 1 xa2 2 · · · xamm ) ∈ Fm+1 : (x1 , . . . , xm ) ∈ Fm .

DIMENSION OF SOME VARIETIES

3

The vanishing ideal I(V ) is the principal ideal generated by f ∗ . Proof. Because the vanishing ideal I(V ) contains the principal ideal generated by f ∗ , it suffices to prove that I(V ) is contained in the principal ideal generated by f ∗. Every polynomial f ∈ F[t1 , . . . , tm , tm+1 ] can be represented uniquely in the form X X X f= c I tI = c I tI (b1 ,...,bm )∈Nm 0

I∈Nm+1 0

=

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m i

X

X

(b1 ,...,bm )∈Nm 0

m+1 cI ti11 · · · timm tm+1 .

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m

A polynomial f ∈ F[t1 , . . . , tm , tm+1 ] is in the vanishing ideal I(V ) if and only if, for all (x1 , . . . , xm ) ∈ Fm , a

a

0 = f (x1 , . . . , xm , λx1 1,j x2 2,j · · · xamm,j ) X X = =

cI xi11 · · · ximm (λxa1 1 xa2 2 · · · xamm )im+1

(b1 ,...,bm )∈Nm 0

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m

X

X

(b1 ,...,bm )∈Nm 0

i +a1 im+1

cI λim+1 x11

· · · ximm +am im+1

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m





    X  im+1  b1 cI λ =   x1 · · · xbmm .   m+1 (b1 ,...,bm )∈Nm  0 I=(i1 ,...,im ,im+1 )∈N0 X

iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m

Because F is an infinite field, all of the coefficients of this polynomial are zero, and so X (3) cI λim+1 = 0 I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m

for all (b1 , . . . , bm ) ∈ Nm 0 . The (m + 1)-tuple I = (b1 , . . . , bm , 0) is one of the terms in the sum (3), and so −c(b1 ,...,bm ,0) =

X

I=(i1 ,...,im ,im+1 )∈Nm+1 , 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m, I6=(b1 ,...,bm ,0)

cI λim+1 .

4

MELVYN B. NATHANSON

Therefore, f ∈ I(V ) implies X X f= (b1 ,...,bm )∈Nm 0

c I tI

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m





      X X   b1 I bm   = cI t + c(b1 ,...,bm ,0) t1 · · · tm     m+1 (b1 ,...,bm )∈Nm 0 I=(i1 ,...,im ,im+1 )∈N0  iℓ +aℓ im+1 =bℓ   for ℓ = 1, . . . , m I6=(b1 ,...,bm ,0)





      X X X   I im+1 b1 bm   = cI t − t1 · · · tm  cI λ    m+1 (b1 ,...,bm )∈Nm I=(i1 ,...,im ,im+1 )∈Nm+1 0 I=(i1 ,...,im ,im+1 )∈N0  0 iℓ +aℓ im+1 =bℓ iℓ +aℓ im+1 =bℓ   for ℓ = 1, . . . , m I6=(b1 ,...,bm ,0)

=

=

for ℓ = 1, . . . , m I6=(b1 ,...,bm ,0)

(b1 ,...,bm )∈Nm 0

X

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m I6=(b1 ,...,bm ,0)

  cI tI − λim+1 tb11 · · · tbmm

X

X

cI g I f ∗

X

(b1 ,...,bm )∈Nm 0

I=(i1 ,...,im ,im+1 )∈Nm+1 0 iℓ +aℓ im+1 =bℓ for ℓ = 1, . . . , m I6=(b1 ,...,bm ,0)

by Lemma 1. It follows that the vanishing ideal I(V ) is contained in the principal ideal generated by f ∗ . This completes the proof.  Theorem 2. For λ ∈ F and (a1 , . . . , am ) ∈ Nm 0 , the variety  V = (x1 , . . . , xm , λxa1 1 xa2 2 · · · xamm ) ∈ Fm+1 : (x1 , . . . , xm ) ∈ Fm has dimension m.

Proof. The function ϕ : F[t1 , . . . , tm+1 ] → F[t1 , . . . , tm ] defined by ϕ(tℓ ) = tℓ

for ℓ = 1, . . . , m

and ϕ(tm+1 ) = λta1 1 · · · tamm is a ring isomorphism with  kernel(ϕ) = f ∈ F[t1 , . . . , tm ] : f (t1 , . . . , tm , λim+1 ta1 1 · · · tamm = 0 = I(V ).

Therefore,

F[t1 , . . . , tm , tm+1 ]/I(V ) ∼ = F[t1 , . . . , tm ]

DIMENSION OF SOME VARIETIES

5

and the coordinate ring of I(V ) has Krull dimension m. This completes the proof.  3. An m-dimensional variety (k ≥ 1) Let m and k be positive integers, and let n = m + k. For j = 1, 2, . . . , k, let λj ∈ F and (a1,j , a2,j , . . . , am,j ) ∈ Nm 0 . Consider the polynomials (4)

a

a

fj∗ = tm+j − λj t1 1,j t2 2,j · · · tamm,j ∈ F[t1 , . . . , tn ].

Let V be the variety in Fn determined by the set of polynomials S = {fj∗ : j = 1, . . . , k} and let I(V ) be the vanishing ideal of V . We shall prove that the coordinate ring F[t1 , . . . , tn ]/I(V ) is isomorphic to the polynomial ring F[t1 , . . . , tm ], and so V has dimension m. Notation: For (b1 , . . . , bm ) ∈ Nm 0 , let X X = . I=(i1 ,...,im ,im+1 ,...,im+k )∈Nn 0 P iℓ + k j=1 aℓ,j im+j =bℓ for ℓ = 1, . . . , m

I(b1 ,...,bm )

Lemma 2. Let R be a ring. For j = 1, 2, . . . , k, let λj ∈ R and (a1,j , a2,j , . . . , am,j ) ∈ ∗ Nm 0 , and let fj be the polynomial defined by (4). For I = (i1 , . . . , im , im+1 , . . . , im+k ) ∈ m+k N0 , let k X aℓ,j im+j for ℓ = 1, . . . , m. bℓ = iℓ + j=1

There exists polynomials gI,1 , . . . , gI,k ∈ F[t1 , . . . , tm+k ] such that (5)

tI −

k Y

i

λjm+j tb11 · · · tbmm =

k X

gI,j fj∗ .

j=1

j=1

Proof. The proof is by induction on k. The case k = 1 is Lemma 1. Assume that the Lemma 2 is true for some positive integer k. We shall prove that the Lemma is true for k + 1. Let I = (i1 , . . . , im+k ) ∈ N0m+k tI =

m Y

ℓ=1

and

tiℓℓ

k Y

i

m+j tm+j

j=1

I ′ = (i1 , . . . , im+k+1 ) ∈ Nm+k+1 0 ′

tI =

m Y

ℓ=1

For ℓ = 1, . . . , m, let

tiℓℓ

k+1 Y

i

i

m+j m+k+1 = tI tm+k+1 . tm+j

j=1

bℓ = iℓ +

k X j=1

aℓ,j im+j

6

MELVYN B. NATHANSON

and b′ℓ = iℓ +

k+1 X

aℓ,j im+j = bℓ + aℓ,k+1 im+k+1 .

j=1

We have ′

tI −

k+1 Y

i

b′

b′



k Y

λjm+j t11 · · · tmm

j=1

i

m+k+1  I = tm+k+1 t −



i

m+k+1  + tm+k+1

j=1 k Y

j=1

i



i



λjm+j tb11 · · · tbmm  λjm+j tb11 · · · tbmm  −

k+1 Y

b′

i

b′

λjm+j t11 · · · tmm

j=1

By the induction hypothesis, there exist polynomials gI,1 , . . . , gI,k ∈ F[t1 , . . . , tm+k ] that satisfy (5). Using the function ∆i defined by (2), we define the polynomial   ! m k Y Y aℓ,k+1 im+j b1 bm   tℓ . t1 · · · tm ∆im+k+1 tm+k+1 , λk+1 gI,k+1 = λj j=1

ℓ=1

Applying the factorization formula (1), we obtain   k+1 k Y i Y b′ b′ im+j b1 im+k+1  bm  λjm+j t11 · · · tmm t1 · · · tm − tm+k+1 λj j=1

j=1



=

=



= 

=

=

k Y

i λjm+j

tb11

j=1

k Y

j=1 k Y



· · · tbmm 

im+k+1 tm+k+1



i



i λjm+j

tb11

j=1

∗ gI,k+1 fk+1 .

m Y

a i tℓ ℓ,k+1 m+k+1

ℓ=1

i

m+k+1 λjm+j tb11 · · · tbmm  tm+k+1 −



im+k+1 λk+1

· · · tbmm  ∆im+k+1

λk+1

m Y

ℓ=1

!

!im+k+1  a  t ℓ,k+1

tm+k+1 , λk+1



m Y

a tℓ ℓ,k+1

ℓ=1

This completes the proof.

!

tm+k+1 −

m Y

ℓ=1



Theorem 3. Let n = m + k. Let F be an infinite field. For j = 1, 2, . . . , k, let ∗ λj ∈ F and (a1,j , a2,j , . . . , am,j ) ∈ Nm 0 , and let fj be the polynomial defined by (4). n Let V ⊆ F be the variety determined by the set S = {f1∗ , . . . , fk∗ } ⊆ F[t1 , . . . , tn ]. I(V ) ∼ = F[t1 , . . . , tm ]. The vanishing ideal I(V ) is the principal ideal generated by S. Proof. The variety determined by S is   a a a V = x1 , . . . , xm , λ1 x1 1,1 · · · xamm,1 , . . . , λk x1 1,k · · · xmm,k : (x1 , . . . , xm ) ∈ Fm .

a tℓ ℓ,k+1

!

DIMENSION OF SOME VARIETIES

7

Every polynomial f ∈ F[t1 , . . . , tn ] has a unique representation in the form X X X c I tI c I tI = f= I∈Nn 0

(b1 ,...,bm )∈Nm 0 I(b1 ,...,bm )

where cI ∈ F and cI 6= 0 for only finitely many n-tuples I. If f ∈ I(V ), then  a a a 0 = f x1 , . . . , xm , λ1 x1 1,1 · · · xamm,1 , . . . , λk x1 1,k · · · xmm,k X X im+1 i a a a · · · λk x1 1,k · · · xmm,k m+k cI xi11 · · · ximm λ1 x1 1,1 · · · xamm,1 = (b1 ,...,bm )∈Nm 0 I(b1 ,...,bm )

X

=

X

cI

(b1 ,...,bm )∈Nm 0 I(b1 ,...,bm )

=

X

(b1 ,...,bm )∈Nm 0

 

k Y

j=1

X

cI

I(b1 ,...,bm )

m Y

i

λjm+j

iℓ +

xℓ

Pk

j=1

aℓ,j im+j

ℓ=1

k Y



i

j=1

λjm+j  xb11 · · · xbmm

for all (x1 , . . . , xm ) ∈ Fm . It follows that 0=

X

I(b1 ,...,bm )

cI

k Y

i

λjm+j

j=1

= c(b1 ,...,bm ,0,...,0) +

X

cI

I(b1 ,...,bm ) I6=(b1 ,...,bm ,0,...,0)

k Y

i

λjm+j

j=1

for all (b1 , . . . , bm ) ∈ Nm 0 , and so X X c I tI f= (b1 ,...,bm )∈Nm 0 I(b1 ,...,bm )

=

X

(b1 ,...,bm )∈Nm 0

=

X

(b1 ,...,bm )∈Nm 0

=

X

(b1 ,...,bm )∈Nm 0

   

   



X

I(b1 ,...,bm ) I6=(b1 ,...,bm ,0,...,0)

X

 cI tI + c(b1 ,...,bm ,0,...,0) tb11 · · · tbmm  

I(b1 ,...,bm ) I6=(b1 ,...,bm ,0,...,0)

X

I(b1 ,...,bm ) I6=(b1 ,...,bm ,0,...,0)

X

c I tI −

I(b1 ,...,bm ) I6=(b1 ,...,bm ,0,...,0)



cI tI −

k Y

j=1

i

cI

k Y

j=1





 i λjm+j tb11 · · · tbmm  

λjm+j tb11 · · · tbmm  .

Lemma 2 immediately implies that f is in the ideal generated by S. This completes the proof.  Theorem 4. The variety V has dimension m. Proof. The function ϕ : F[t1 , . . . , tm+1 , . . . , tm+k ] → F[t1 , . . . , tm ]

8

MELVYN B. NATHANSON

defined by ϕ(tℓ ) = tℓ and

for ℓ = 1, . . . , m

a

ϕ(tm+j ) = λj t1 1,j · · · tamm,j is a ring isomorphism with

for j = 1, . . . , k

kernel(ϕ)   a a a = f ∈ F[t1 , . . . , tn ] : f t1 , . . . , tm , λ1 t1 1,1 · · · tamm,1 , . . . , λk t1 1,k · · · tmm,k = 0 = I(V ).

Therefore,

F[t1 , . . . , tn ]/I(V ) ∼ = F[t1 , . . . , tm ] and the coordinate ring of I(V ) has Krull dimension m. This completes the proof.  References [1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969. [2] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, third ed., Springer, New York, 2007. [3] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Modern Birkh¨ auser Classics, Birkh¨ auser/Springer, New York, 2013. [4] M. B. Nathanson, An elementary proof for the Krull dimension of a polynomial ring, arXiv:1612.05670, 2017. Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468 E-mail address: [email protected]