The Stacks project

Lemma 29.30.13. Let $R$ be a ring. Let $R \to A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ be a relative global complete intersection. Set $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$. Consider the morphism $f : X \to S$ associated to the ring map $R \to A$. The function $x \mapsto \dim _ x(X_{f(x)})$ is constant with value $n - c$.

Proof. By Algebra, Definition 10.136.5 $R \to A$ being a relative global complete intersection means all nonzero fibre rings have dimension $n - c$. Thus for a prime $\mathfrak p$ of $R$ the fibre ring $\kappa (\mathfrak p)[x_1, \ldots , x_ n]/(\overline{f}_1, \ldots , \overline{f}_ c)$ is either zero or a global complete intersection ring of dimension $n - c$. By the discussion following Algebra, Definition 10.135.1 this implies it is equidimensional of dimension $n - c$. Whence the lemma. $\square$

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