The Stacks project

Remark 111.52.4. Freely use the following facts on dimension theory (and add more if you need more).

  1. The dimension of a scheme is the supremum of the length of chains of irreducible closed subsets.

  2. The dimension of a finite type scheme over a field is the maximum of the dimensions of its affine opens.

  3. The dimension of a Noetherian scheme is the maximum of the dimensions of its irreducible components.

  4. The dimension of an affine scheme coincides with the dimension of the corresponding ring.

  5. Let $k$ be a field and let $A$ be a finite type $k$-algebra. If $A$ is a domain, and $x \not= 0$, then $\dim (A) = \dim (A/xA) + 1$.

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