An “Elementary” Calculation of the Krull Dimension of a Polynomial Ring over a Field

Problemy Matematyczne

An “Elementary” Calculation of the

Krull Dimension of a Polynomial Ring

over a Field

Andrzej Prószyński, M ałgorzata Rybińska

1. Introduction It is well known that the Krull dimension o f a domain R , which is finitely generated over a field K, is equal to the transcendence degree over К o f its field o f fractions; that is

d im (.ff) = t r .d e g A-(Tüo)

(see, for example, [1]). In the proof, the central role plays Noether normalization lemma (see [3]). Namely, if tr.degA-(jRo) = r, then R is an integral extension o f a subring S = K [ x i , . . . , ay], where X i , . . . , x r are algebraically independent over K. In particular, S is isomorphic to K [ T i , . . . ,T r], the polynomial ring in r variables over K. The rest o f the proof is a consequence o f the following two facts:

(1) d im ( A -[ T „ ...,r r]) = r,

(2 ) if R is an integral extension of S then dim (i?) = dim(5').

The proof o f (2) is not direct but, in fact, quite elementary ( “go­ ing up” theorem, see [1]). In contrast, the proof o f (1) involves some

44. A . Prószyński, M. Rybińska

non-element ary methods, like principal ideal theorem, systems of para­ meters, or Hilbert-Samuel polynomials (see [2], [1]). Our proof is based only on well known facts including the above normalization lemma, and was first published in the manuscript [4].

2. The proof of (1) We prove by induction on n that d im (S ) = n, where S = K [T \, . . . , Tn\. The chain o f prime ideals

0 С ( Г 1) С ( Г 1,Т2) С . . . С ( Г 1, . . . , Т„ )

(

1

)

P -i

С P0 С ... С

o f the ring S. Since Po is non-zero, it contains a non-zero and non - invertible polynomial g. One of its indecomposable factors, say / , does not belong to the prime ideal Pq. Since S' is a UFD, / generates

a prime ideal, and hence R = S / f is a domain. It follows from the choice o f / that d im (S ) > n. On the other hand, S = K [ t i , . . . ,f„], where t, denotes the coset o f 71,- for i = 1 , . . . , n , and then the relation / ( t x, = 0 gives us tr. degk(Ro) = r < n. Consequently, R is an integral extension o f K [ x i , . . . , x r\ for some algebraically independent elements x i , . . . , x r. By induction, dim ( K [ T i , . . . , Tr]) = r, and hence (2) gives us d im (S ) = r < n, a contradiction.

3. Corollary It follows from (0.1) that h t ( Ti , . . . ,7)) > i. We give the proof o f the equality

h t ( 7 i , . . . ,T ,) = i for i = l , . . . , n omitting the principal ideal theorem.

Suppose that h t ( T i , . . . , 7*) > i for some i. Then there exists a chain o f prime ideals

An e l e m n t a r y c a l c u l a t i o n .45

what gives us di m( A' [ Ti , . . . , T„]) > n, a contradiction.

R E F E R E N C E S

[1] M. Atiyah, I. M acDonald, Introduction to commutative algebra, Addison-Wesley, Reading 1969,

[2] S. Balcerzyk, T. Józefiak, Pierścienie przem ienne, PW N War­ szawa, 1985,

[3] S. Lang, Algebra, Addison-Wesley, Reading 1965,

[4] M. Rybińska, W ysokość ideału i wymiar Krulla pierścienia prze­

miennego, (M. Sc. dissertation - in Polish), W SP Bydgoszcz,

1994.

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Chodkiewicza 30 85 06Ą Bydgoszcz, Poland