2. Krull dimension of a ring 5
Theorem 1.17. Let (i?, m) be a local ring, and d a nonnegative integer.
The following conditions are equivalent:
(1) dimR^d;
(2) there exists an m-primary ideal generated by d elements;
(3) for t ^ 0; the length function
£(R/mt)
agrees with a polynomial in t
of degree at most d.
Definition 1.18. Let (i?,m) be a local ring of dimension d. Elements
x i , . . . , Xd are a system of parameters for R if rad(xi,..., x^) m.
Theorem 1.17 implies that every local ring has a system of parameters.
Example 1.19. Let K be a field, and take
R = K[x, y, z}{X}V^/(xy, xz).
Then R has a chain of prime ideals (x) C (x, y) C (x, y, z), so dim R ^ 2. On
the other hand, the maximal ideal (x,y,z) is the radical of the 2-generated
ideal (y,x z), implying that dimi? ^ 2. It follows that dimi? = 2 and
that y, x z is a system of parameters for i?.
Exercise 1.20. Let K be a field. For the following local rings (i?,m),
compute dimi? by examining
^(R/m1)
for t » 0. In each case, find a system
of parameters for R and a chain of prime ideals
Po £ Pi £ * £ Pd ? where i = dim i?.
(1) i? =
K[x2,x3](x2^3).
(2) jR = K[x2,xy,j/2](a.2a.yy2).
(3) i? = K[^, x, y, ^ ( ^ ^ ^ ^ / ( ^ x - yz).
(4) R = Z(p) where p is a prime integer.
If a finitely generated algebra over a field is a domain, then its dimension
may be computed as the transcendence degree of a field extension:
Theorem 1.21. // a finitely generated K-algebra R is a domain, then
dim R = tr. degK Prac(i?),
where Frac(i?) is the fraction field of R. Moreover, any chain of primes in
Speci? can be extended to a chain of length dimi? which has no repeated
terms. Hence dim Rm dim R for every maximal ideal m of R, and
height p + dim R/p = dim R for all p G Spec R.
When K is algebraically closed, this is [6, Corollary 11.27]; for the general
case see [115, Theorem 5.6, Exercise 5.1].
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