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].