Dimension of the quotient of a local ring

Question

Let ($R$ , $$) be a local Noetherian ring. Suppose that $I=(x_1,..., x_k) \subseteq $. Is it true that $\dim R/I=\dim R-k$ ?

Thank you.

Answer 1

As referenced in the comments, this is not true in general. In fact, it happens exactly when $(x_1,...,x_k)$ forms part of a system of parameters.

Answer 2

It is not true in general. You can prove this one:

• $\dim( R/I )\ge \dim( R)-k$

• $(x_{1},...,x_{k})$ is part of a system of parameters iff $\dim (R/I)= \dim( R) - k$