The Stacks project

Proof. By More on Flatness, Lemma 38.20.9 the hypothesis on $\mathcal{F}$ is preserved by any base change. Also, formation of $[\mathcal{F}/X/S]_ r$ is compatible with any base change by Lemma 62.5.3. Since the condition of being compatible with specializations is checked after base change to the spectrum of a discrete valuation ring, this reduces us to the case where $S$ is the spectrum of a valuation ring. In this case the set $U = \{ x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\} $ is open in $X$ by More on Flatness, Lemma 38.13.11. Since the complement of $U$ in $X$ has fibres of dimension $< r$ over $S$ by assumption, we see that restriction along the inclusion $U \subset X$ induces an isomorphism on the groups of $r$-cycles on fibres after any base change, compatible with specialization maps and with formation of the relative cycle associated to $\mathcal{F}$. Thus it suffices to show compability with specializations for $[\mathcal{F}|_ U / U /S]_ r$. Since $\mathcal{F}|_ U$ is flat over $S$, this follows from Lemma 62.4.1 and the definitions. $\square$