Proof. By [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.2 (1)], the statement is Zariski-local on $\mathrm {Spec}(R) \setminus V(I)$ , so it suffices to prove that $\dim _v(R[\tfrac {1}{f}]) + 1 \leqslant \dim _v(R)$ for $f \in I$ . Let $\mathfrak {p}$ be a prime ideal of $R[\tfrac {1}{f}]$ , which restricts to a prime ideal of R, which we also denote $\mathfrak {p}$ . Given a valuation ring V of rank r such that $R[\tfrac {1}{f}]/\mathfrak {p} \subseteq V \subseteq \mathrm {Frac}(R[\tfrac {1}{f}]/\mathfrak {p})$ , there exists a valuation ring extension $V'$ of V of rank $r+1$ such that $R/\mathfrak {p} \subseteq V' \subseteq \mathrm {Frac}(R/\mathfrak {p})$ . Consequently,

$$ \begin{align*}\dim_v(R[\tfrac{1}{f}]) + 1 = \sup_{\mathfrak{p} \subseteq R[\tfrac{1}{f}]} \dim_v(R[\tfrac{1}{f}]/\mathfrak{p}) + 1 \leqslant \sup_{\mathfrak{p} \subseteq R[\tfrac{1}{f}]} \dim_v(R/\mathfrak{p}) \leqslant \sup_{\mathfrak{p} \subseteq R} \dim_v(R/\mathfrak{p}) = \dim_v(R).\end{align*} $$

Proof. First note that, replacing valuative dimension by Krull dimension everywhere, this statement is a result of Clausen–Mathew [Reference Clausen and MathewCM21, Theorem 3.30], where we use that $\dim (\overline {\{x\}}) \leqslant d - \dim (X_x)$ for every point $x \in X$ . We adapt the argument of [Reference Clausen and MathewCM21, Theorem 3.30], and prove the desired statement by induction on the valuative dimension of X.

If the scheme X is local, then the argument of [Reference Clausen and MathewCM21, Theorem 3.30] proves the desired claim, by using Lemma 2.3 to implement the induction step.

In general, the underlying topological space of the qcqs scheme X is a spectral space [Sta25, Tag 094L], which is of Krull dimension at most d by [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.2 (8)]. The proof of [Reference Clausen and MathewCM21, Theorem 3.14] then implies, using the same argument as in Lemma 2.3 for the induction step, that the anima $E(X)$ is $(n-d)$ -connective if, for every point $x \in X$ , the anima $E(X_x)$ is $(n-\dim _v(X_x))$ -connective. The desired result is then a consequence of the previous paragraph.

Proof. A classical reference, although stated only in the case where S is the spectrum of a field, is [Reference MorelMor12, Proposition $5.20$ ]. The more general case where S is an arbitrary qcqs scheme is [Reference Antieau and ElmantoAE17, Theorem 4.27].

Proof. By Proposition 2.5, it suffices to prove that both the Nisnevich sheafification $L_{\mathrm {Nis}}$ and the singular construction $\mathrm {Sing}^{\mathbb {A}^1}$ preserve perverse n-connective objects. For the Nisnevich sheafification, this is Remark 2.2. For the singular construction, let ${E \in \mathcal {P}(\mathrm {Sm}_S)}$ be a presheaf of anima, and X be a henselian local essentially smooth S-scheme of valuative dimension d. By [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.2 (7)], for every integer $m \geqslant 0$ , the scheme $X \times \Delta ^m$ is of valuative dimension $d+m$ . By Theorem 2.4, this implies that, for every integer $m \geqslant 0$ , the anima $E(X \times \Delta ^m)$ is $(n-d-m)$ -connective. By [Reference AyoubAyo24, Lemma 2.4.6] (see also [Reference Bachmann, Elmanto and MorrowBEM25, Page 110, Footnote 41] for a more direct argument in the stable setting), this in turn implies that the geometric realisation $(\mathrm {Sing}^{\mathbb {A}^1} E)(X) := \vert E(X \times \Delta ^\bullet )\vert $ is $(n-d)$ -connective.

Proof. The proof is the same as in the first paragraph of the proof of [Reference Bouis and KunduBK25, Theorem 3.2], where we first reduce to S being the spectrum of a henselian local ring instead of a henselian valuation ring.

Proof. The main geometric ingredient here is the Nisnevich-local presentation lemma [Reference Bouis and KunduBK25, Corollary 2.25]. The proof that this implies the desired Gersten injectivity statement is classical and goes back to [Reference Colliot-Thélène, Hoobler and KahnCTHK97]; we reproduce the proof here for convenience, following [Reference Bouis and KunduBK25, Theorem 3.2].

Assume that the henselian local essentially smooth S-scheme X is the henselian local scheme $X_x^h$ of a smooth S-scheme X at some point $x \in X$ . Let $[\sigma ] \in \pi _i(F(X_x^h))$ be a class whose image in $\pi _i(F((X_x^h)_\eta ))$ vanishes. We want to prove that $[\sigma ]$ itself vanishes. The presheaf $\pi _i(F)$ commutes with filtered colimits of rings and the statement is Nisnevich local on X, so we can assume that the class $[\sigma ]$ comes from a class $[\sigma ] \in \pi _i(F(X))$ which vanishes in $\pi _i(F(X \setminus Z))$ , for $Z \hookrightarrow X$ a closed immersion of S-schemes satisfying $\dim (Z_s) < \dim (X_s)$ .

It now suffices to produce a Nisnevich neighbourhood $x \in X' \rightarrow X$ such that the pullback $[\sigma ']$ of the class $[\sigma]$ vanishes in $\pi _i(F(X'))$ . By the presentation lemma [Reference Bouis and KunduBK25, Corollary 2.25], there exist a smooth S-scheme T, a Nisnevich neighbourhood $X' \rightarrow X$ of x, and a Nisnevich square

such that the induced morphism $Z' \xrightarrow {\cong } \psi (Z') \rightarrow T$ is finite. By Nisnevich excision, the class $[\sigma '] \in \pi _i(F(X'))$ then lifts to a class $[\sigma '] \in \pi _i(F(\mathbb {A}^1_T))$ which vanishes in $\pi _i(F(\mathbb {A}^1_T \setminus \psi (Z')))$ . Again by Nisnevich excision (where we use that $\psi (Z')$ is finite over T), this class $[\sigma '] \in \pi _i(F(\mathbb {A}^1_T))$ further lifts to a class $[\sigma '] \in \pi _i(F(\mathbb {P}^1_T))$ which vanishes in $\pi _i(F(\mathbb {P}^1_T \setminus \psi (Z')))$ . By construction, the map $\infty ^\ast _T : \pi _i(F(\mathbb {P}^1_T)) \rightarrow \pi _i(F(T))$ factors through the map $\pi _i(F(\mathbb {P}^1_T)) \rightarrow \pi _i(F(\mathbb {P}^1_T \setminus \psi (Z')))$ , so the pullback class $\infty ^\ast _T [\sigma '] \in \pi _i(F(T))$ vanishes. By deflatability (Definition 3.3), this implies that $[\sigma '] = j^\ast _T [\sigma '] \in \pi _i(F(\mathbb {A}^1_T))$ also vanishes, which concludes the proof.

Proof. Let F be the fraction field of R, $F':=F(X)$ be the fraction field of $R'$ , and let $V'$ be a valuation ring satisfying that $R' \subseteq V' \subseteq F'$ . We want to prove that the rank of $V'$ is at most equal to d. Let V be the valuation ring defined as the intersection of $V'$ and F. By construction, we know that $R \subseteq V \subseteq F$ . In particular, by the assumption that R is of valuative dimension d, this implies that the valuation ring V is of rank at most d. The condition that $R' = R[X]_{\mathfrak {m}\cdot R[X]} \subseteq V'$ then implies that the valuation on $V'$ is the Gauss valuation induced by V: $\vert \sum _i a_i X^i \vert _{V'} = \min _i \vert a_i \vert _V$ (see for instance [Reference KunduKun23, proof of Lemma 3.10]). The value groups of V and $V'$ are then naturally isomorphic, and in particular the rank of $V'$ is equal to that of V, which is at most equal to d.

Proof. The statement is Zariski-local around $s \in S$ , so we can assume that the scheme S is local with closed point s [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.2 (3)]. By construction, there exists a smooth S-scheme Y and a generic point $y \in Y_s$ such that $X_\eta ^h$ is the henselisation of the local scheme of Y at y. By [Sta25, Tag 02G8] and [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.5], if $Y' \rightarrow Y$ is an unramified morphism of schemes, then $\dim _v(Y') \leqslant \dim _v(Y)$ . In particular, it suffices to prove that the localisation of Y at y is of valuative dimension at most that of S. Zariski-locally around $y \in Y$ , the map $Y \rightarrow S$ factors as a composite $Y \rightarrow \mathbb {A}^m_S \rightarrow S$ for some integer $m \geqslant 0$ and where the map $Y \rightarrow \mathbb {A}^m_S$ is étale. Using again [Sta25, Tag 02G8] and [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.5], we can assume that $Y=\mathbb {A}^m_S$ and y is the generic point of the special fibre of Y. By induction on the integer $m\geqslant 1$ , we can further assume that $m=1$ . Finally, in this case, by definition of the valuative dimension, the desired claim is a consequence of Lemma 3.7.

Proof. By Remark 3.2, the space $E \in \mathcal {P}_{\mathrm {Nis}}(\mathrm {Sm}_S)$ is perverse n-connective in the sense of Definition 2.1. By Corollary 2.6, this implies that the motivic space $L_{\mathrm {mot}} E \in \mathrm {H}(S)$ is perverse n-connective. In particular, any henselian local essentially smooth S-scheme X of valuative dimension e satisfies that the anima $(L_{\mathrm {mot}} E)(X)$ is $(n-e)$ -connective. By Example 3.4, the $\mathbb {A}^1$ -invariant Nisnevich sheaf $L_{\mathrm {mot}} E$ is deflatable. By Proposition 3.6 (applied to the deflatable Nisnevich sheaf $L_{\mathrm {mot}} E$ ) and Proposition 3.8, this implies that any henselian local essentially smooth S-scheme X satisfies that the anima $(L_{\mathrm {mot}}E)(X)$ is $(n-d)$ -connective, i.e., that the motivic space $L_{\mathrm {mot}} E \in \mathrm {H}(S)$ is $(n-d)$ -connective.

Proof. This is a consequence of Theorem 3.9 and of the fact that the valuative and Krull dimensions agree on locally noetherian schemes [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Proposition 2.3.2 (9)].

Proof. This is a consequence of the definition of n-connectivity for a motivic spectrum (see for instance [Reference DruzhininDru22, Section 2.1]) and of the shifted unstable $\mathbb {A}^1$ -connectivity theorem (Theorem 3.9). Note that if one rather starts from a spectra-valued sheaf $E \in \mathcal {P}_{\mathrm {Nis}}(\mathrm {Sm}_S;\mathrm {Sp})$ , then the same result still holds: either as a consequence of Theorem 3.9 if $n \geqslant d$ , or by using Proposition 3.6 to reproduce the classical argument of [Reference MorelMor04] (see also [Reference Schmidt and StrunkSS18, Reference Deshmukh, Hogadi, Kulkarni and YadavDHKY21]).

Proof. The proof is exactly as in [Reference Bachmann, Elmanto and MorrowBEM25, Proposition 3.44 and Lemma 4.8] (where the proof is stated for regular noetherian S of dimension at most one), using the shifted stable $\mathbb {A}^1$ -connectivity theorem (Corollary 3.12).