Proof. As pointed out in [Reference BalmerBal20a], this follows from the tensor product property and the fact that $\operatorname {\mathrm {Supp}}^h(E_{\mathscr {B}})=\{\mathscr {B}\}$ . On the other hand, part $(b)$ of Lemma 2.6 below (applied to the weak ring $E_{\mathscr {B}}$ ) provides a more ‘formal’ proof of the result.

Proof. $(a)$ : Let be the counit of the weak coring structure and let $\Delta \colon C \to C\otimes C$ be a choice of ‘comultiplication’, that is, a section of the map $1\otimes \epsilon \colon {C \otimes C \to C}$ . One can check using dinaturality of coevaluation (with respect to $\Delta $ ) that the composite

$$\begin{align*}\mathsf{hom}(C,t) \xrightarrow{1 \otimes \mathrm{coev}} \mathsf{hom}(C,t)\otimes \mathsf{hom}(C,C) \xrightarrow{} \mathsf{hom}(C\otimes C, t\otimes C) \xrightarrow{\mathsf{hom}(\Delta,1\otimes\epsilon)} \mathsf{hom}(C,t) \end{align*}$$

is the identity.

$(b)$ : Let be the unit of the weak ring structure and let $\mu :R\otimes R \to R$ be a choice of ‘multiplication’, that is, a retraction of the map $1\otimes \eta \colon R \to R \otimes R$ . One can check using dinaturality of coevaluation (with respect to $1\otimes \eta \colon t \to t\otimes R$ ) that the composite

$$\begin{align*}\mathsf{hom}(t,R) \xrightarrow{1 \otimes \mathrm{coev}} \mathsf{hom}(t,R) \otimes \mathsf{hom}(R,R) \xrightarrow{} \mathsf{hom}(t\otimes R,R\otimes R) \xrightarrow{\mathsf{hom}(1\otimes\eta,\mu)} \mathsf{hom}(t,R) \end{align*}$$

is the identity.

Proof. Let be the counit. One can readily check using dinaturality of evaluation (with respect to $\epsilon $ ) that the composite

is . Thus, $\mathsf {hom}(C,t)=0$ implies $t \otimes \epsilon = 0$ . Since the identity of $t\otimes C$ factors through $t \otimes C \otimes C \xrightarrow {1 \otimes \epsilon \otimes 1} t \otimes C$ , we conclude that $t\otimes C=0$ .

Proof. The case when t is a weak ring is [Reference BalmerBal20a, Theorem 4.7] while the case when t is a weak coring follows from Lemma 2.7 and part $(a)$ of Lemma 2.6.

Proof. We always have the $\subseteq $ inclusion by Proposition 2.5. On the other hand, if $\mathscr {B} \in \operatorname {\mathrm {Supp}}^n(t)$ then $E_{\mathscr {B}} \otimes t \neq 0$ by definition. Hence the h-detection property implies that $\operatorname {\mathrm {Supp}}^h(E_{\mathscr {B}} \otimes t) \neq \varnothing $ . By the tensor product property this implies $\varnothing \neq \operatorname {\mathrm {Supp}}^h(E_{\mathscr {B}}) \cap \operatorname {\mathrm {Supp}}^h(t) = \{\mathscr {B}\} \cap \operatorname {\mathrm {Supp}}^h(t)$ so that $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t)$ .

Proof. The inclusion $\operatorname {\mathrm {Supp}}^h(t) \subseteq \operatorname {\mathrm {Supp}}^n(t)$ is Proposition 2.5, so it remains to prove the ‘only if’ direction. First we claim that $f^! E_{\mathscr {B}} \neq 0$ . Otherwise, we would have . But $E_{\mathscr {B}}$ is a direct summand of by [Reference Balmer and CameronBC21, Theorem 3.1]. Thus we would have $\mathsf {hom}(E_{\mathscr {B}},E_{\mathscr {B}}) =0$ which contradicts $E_{\mathscr {B}} \neq 0$ .

If $\mathsf {hom}(t,E_{\mathscr {B}}) = 0$ , then $\mathsf {hom}(f^* t,f^! E_{\mathscr {B}}) = f^!\mathsf {hom}(t,E_{\mathscr {B}}) =0$ . Since $\mathscr {F}$ is a tt-field, it follows that $f^* t = 0$ by [Reference Balmer, Krause and StevensonBKS19, Theorem 5.21]. Hence by the projection formula. Therefore $E_{\mathscr {B}} \otimes t = 0$ since $E_{\mathscr {B}}$ is a direct summand of .

Proof. This follows immediately from Lemma 2.15.

2.17. We conclude this section with one further result concerning the relationship between the naive and genuine notions of homological support.

Proof. The $(\Rightarrow )$ direction holds unconditionally by Proposition 2.5, so it suffices to prove the $(\Leftarrow )$ implication. Let $t \in \mathscr {T}$ with $t \otimes E_{\mathscr {B}} \neq 0$ . By our minimality assumption, this implies that $E_{\mathscr {B}} \in \operatorname {\mathrm {Locid}}\langle t \otimes E_{\mathscr {B}} \rangle $ . If we had $\mathsf {hom}(t \otimes E_{\mathscr {B}}, E_{\mathscr {B}}) = 0$ , then $\mathsf {hom}(E_{\mathscr {B}}, E_{\mathscr {B}}) = 0$ which contradicts $E_{\mathscr {B}} \neq 0$ . Therefore, we see

$$\begin{align*}0 \neq \mathsf{hom}(t \otimes E_{\mathscr{B}}, E_{\mathscr{B}}) \simeq \mathsf{hom}(E_{\mathscr{B}},\mathsf{hom}(t,E_{\mathscr{B}})). \end{align*}$$

In particular, this implies that $\mathsf {hom}(t,E_{\mathscr {B}}) \neq 0$ .

Proof. The equivalence of $(a)$ and $(d)$ follows verbatim from the argument in [Reference Barthel, Castellana, Heard and SandersBCHS23, Theorem 6.4] which establishes the equivalence of the classical LGP property and the classical codetection property. On the other hand, part $(a)$ is equivalent to part $(b)$ by [Reference Barthel, Castellana, Heard and SandersBCHS23, (2.6)]. Finally, the equivalence of $(b)$ and $(c)$ is immediate from the definition of the naive h-support.

Proof. Suppose that $(a)$ holds, so that in particular h-detection holds. Indeed, ${\operatorname {\mathrm {Supp}}^h(t) = \varnothing }$ implies $\operatorname {\mathrm {Locid}}\langle t \rangle = 0$ by the h-LGP, hence $t = 0$ . It then follows from Lemma 2.11 that $(b)$ holds. Conversely, the equality in $(b)$ implies that

$$\begin{align*}\operatorname{\mathrm{Locid}}\langle t\otimes E_{\mathscr{B}} \mid \mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t) \rangle = \operatorname{\mathrm{Locid}}\langle t\otimes E_{\mathscr{B}} \mid \mathscr{B} \in \operatorname{\mathrm{Supp}}^n(t) \rangle, \end{align*}$$

so the h-LGP follows from h-codetection by Proposition 3.2.

Proof. $(a) \Rightarrow (b)$ : Consider $t \in \mathscr {T}$ . The homological local-to-global principle implies

$$ \begin{align*} \operatorname{\mathrm{Locid}}\langle t \rangle & = \operatorname{\mathrm{Locid}}\langle t \otimes E_{\mathscr{B}} \mid\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t) \rangle \\ & \subseteq \operatorname{\mathrm{Locid}}\langle E_{\mathscr{B}} \mid\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t) \rangle. \end{align*} $$

On the other hand, minimality implies $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle = \operatorname {\mathrm {Locid}}\langle t\otimes E_{\mathscr {B}} \rangle $ for all $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t)$ , which establishes the reverse inclusion.

$(b) \Rightarrow (c)$ : On the one hand, since $\operatorname {\mathrm {Supp}}^h(E_{\mathscr {B}}) = \{\mathscr {B}\}$ for any $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ , the map is always surjective. On the other hand, injectivity of the map is equivalent to the following statement:

$$\begin{align*}\forall t_1,t_2 \in \mathscr{T}\colon \operatorname{\mathrm{Supp}}^h(t_1) = \operatorname{\mathrm{Supp}}^h(t_2) \implies \operatorname{\mathrm{Locid}}\langle t_1 \rangle = \operatorname{\mathrm{Locid}}\langle t_2 \rangle. \end{align*}$$

This is a direct consequence of $(b)$ .

$(c) \Rightarrow (a)$ : We first check the minimality condition. To this end, let $\mathscr {B}$ be some homological prime and consider a nonzero $\mathscr {L} \subseteq \operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ . It follows that $\varnothing \subseteq \operatorname {\mathrm {Supp}}^h(\mathscr {L}) \subseteq \operatorname {\mathrm {Supp}}^h(E_{\mathscr {B}})$ . Injectivity of the map in $(c)$ guarantees that $\operatorname {\mathrm {Supp}}^h(\mathscr {L})$ is non-empty, for otherwise $\mathscr {L} = (0)$ . Therefore, $\operatorname {\mathrm {Supp}}^h(\mathscr {L}) = \{\mathscr {B}\}$ , so applying $(c)$ again gives $\mathscr {L} = \operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ . To prove the homological local-to-global principle, we compute

$$ \begin{align*} \operatorname{\mathrm{Supp}}^h(\operatorname{\mathrm{Locid}}\langle t \otimes E_{\mathscr{B}}\mid \mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t) \rangle) & = \bigcup_{\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t)}\operatorname{\mathrm{Supp}}^h(t) \cap \operatorname{\mathrm{Supp}}^h(E_{\mathscr{B}}) \\ & = \operatorname{\mathrm{Supp}}^h(t) \\ & = \operatorname{\mathrm{Supp}}^h(\operatorname{\mathrm{Locid}}\langle t \rangle). \end{align*} $$

Using the injectivity in $(c)$ once more, we see that $\operatorname {\mathrm {Locid}}\langle t \otimes E_{\mathscr {B}}\mid \mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t) \rangle = \operatorname {\mathrm {Locid}}\langle t \rangle $ , as desired.

Proof. Balmer and Cameron [Reference Balmer and CameronBC21, Theorem 3.1] establish that $E_{\mathscr {B}}$ is a direct summand of . Thus it suffices to prove that is a minimal localizing ideal. Consider . If $f^*(t) = 0$ then so that ${t \otimes t = 0}$ . This contradicts $t \neq 0$ since we have assumed that there are no tensor-nilpotent objects in $\mathscr {T}$ . Thus, $f^*(t) \neq 0$ and therefore by Example 4.4. Hence

by [Reference Barthel, Castellana, Heard and SandersBCHS23, (13.4)]. This proves that is minimal.

Proof. Certainly $(b)$ implies $(a)$ by Theorem 4.1. Conversely, h-LGP implies the h-detection property so, since we have enough tt-fields by assumption, we automatically get h-minimality by Corollary 4.6.

4.12. We give two examples where homological minimality fails:

Proof. $(a) \Rightarrow (b)$ : The first statement follows from Lemma 2.11. For the second, note that the well-defined order-preserving map $A(t)\mapsto \operatorname {\mathrm {Supp}}^n(t)=\operatorname {\mathrm {Supp}}^h(t)$ is a surjection since every subset S of $\operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ can be realized as the h-support of some object, say $\bigsqcup _{\mathscr {B} \in S}E_{\mathscr {B}}$ , thanks to [Reference BalmerBal20a, Proposition 4.3(b)] and Example 2.2. We claim that

$$\begin{align*}A(t) \le A(s) \iff \operatorname{\mathrm{Supp}}^h(t)\subseteq\operatorname{\mathrm{Supp}}^h(s). \end{align*}$$

Given the claim, the map sending $A(t)$ to $\operatorname {\mathrm {Supp}}^h(t)$ is an order-preserving bijection with order-preserving inverse $\operatorname {\mathrm {Supp}}^h(t)\mapsto A(t)$ and therefore an isomorphism of lattices. To prove the claim, it suffices to show that

$$\begin{align*}A(t) \le A(s) \impliedby \operatorname{\mathrm{Supp}}^h(t)\subseteq\operatorname{\mathrm{Supp}}^h(s). \end{align*}$$

This follows immediately from the equality

$$\begin{align*}A(t)=\big\{\,s\in\mathscr{T}\,\big|\,\operatorname{\mathrm{Supp}}^h(s)\cap\operatorname{\mathrm{Supp}}^h(t)=\varnothing\,\big\}, \end{align*}$$

which holds by h-detection and the tensor product formula.

$(b) \Rightarrow (a)$ : If $\operatorname {\mathrm {Supp}}^h(t)=\varnothing $ then $A(t)=A(0)$ and hence $t=0$ .

Proof. $(a) \Rightarrow (b)$ : The inclusion $\operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(t_1,t_2)) \subseteq \operatorname {\mathrm {Supp}}^n(t_1) \cap \operatorname {\mathrm {Cosupp}}^h(t_2)$ holds unconditionally. Since $(a)$ guarantees $\operatorname {\mathrm {Supp}}^h(t_1) = \operatorname {\mathrm {Supp}}^n(t_1)$ by Lemma 2.18, we obtain $\operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(t_1,t_2)) \subseteq \operatorname {\mathrm {Supp}}^h(t_1) \cap \operatorname {\mathrm {Cosupp}}^h(t_2)$ and it remains to verify the reverse inclusion. Consider some $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t_1) \cap \operatorname {\mathrm {Cosupp}}^h(t_2)$ . In particular, $E_{\mathscr {B}} \otimes t_1 \neq 0$ , hence $E_{\mathscr {B}} \in \operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}}\otimes t_1 \rangle $ since $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ is minimal by assumption. Since $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(t_2)$ , we have

$$\begin{align*}0 \neq \mathsf{hom}(E_{\mathscr{B}},t_2) \in \operatorname{\mathrm{Colocid}}\langle \mathsf{hom}(E_{\mathscr{B}}\otimes t_1,t_2) \rangle. \end{align*}$$

By adjunction, we get $0 \neq \mathsf {hom}(E_{\mathscr {B}}, \mathsf {hom}(t_1,t_2))$ , so $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(t_1,t_2))$ .

$(b) \Rightarrow (c)$ : This is a direct consequence of $\operatorname {\mathrm {Cosupp}}^h(0) = \varnothing $ .

$(c) \Rightarrow (a)$ : Consider two non-zero $t_1,t_2 \in \operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ . We first claim that $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(t_2)$ . Indeed, if $\mathscr {B} \notin \operatorname {\mathrm {Cosupp}}^h(t_2)$ then $\mathsf {hom}(E_{\mathscr {B}},t_2)=0$ , which would imply $\mathsf {hom}(t_2,t_2)=0$ , i.e., $t_2 = 0$ , giving the desired contradiction. Moreover, $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t_1)$ by the h-detection property. It then follows that

$$\begin{align*}\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t_1) \cap \operatorname{\mathrm{Cosupp}}^h(t_2), \end{align*}$$

so $\mathsf {hom}(t_1,t_2) \neq 0$ by $(c)$ . The minimality criterion [Reference Barthel, Castellana, Heard and SandersBCHS23, Lemma 7.12] then shows that $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ is minimal.

Proof. The h-LGP implies the h-detection property (Remark 3.6), so the equivalence follows from Theorem 4.1 and Proposition 5.1.

Proof. Throughout this proof, we will refer to (5.7) as the cosupport formula.

$(a) \Rightarrow (b)$ : By Corollary 5.2, h-stratification implies the h-LGP for $\mathscr {T}$ as well as the cosupport formula. But the h-LGP implies the h-codetection property for $\mathscr {T}$ (see Proposition 3.2 and Remark 3.3), so we get the statement of $(b)$ .

$(b) \Rightarrow (c)$ : The only if direction in $(c)$ follows directly from the cosupport formula. For the converse, suppose that $\operatorname {\mathrm {Supp}}^h(t_1) \cap \operatorname {\mathrm {Cosupp}}^h(t_2) = \varnothing $ . The cosupport formula then shows that $\operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(t_1,t_2)) = \varnothing $ , so h-codetection gives $\mathsf {hom}(t_1,t_2) = 0$ .

$(c) \Rightarrow (a)$ : Taking in $(c)$ and using Remark 5.4, we have if and only if $\operatorname {\mathrm {Supp}}^h(t_1)=\varnothing $ . Since $t_1 = 0$ if and only if , this establishes h-detection for $\mathscr {T}$ . We may thus appeal to the implication $(c) \Rightarrow (a)$ in Proposition 5.1 to deduce the minimality of $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle $ for all $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ .

Specializing $(c)$ to instead and using that , we get the h-codetection property for $\mathscr {T}$ . Lemma 2.18 together with Corollary 3.4 then imply that $\mathscr {T}$ has the h-LGP. Finally, the characterization of h-stratification in Theorem 4.1 shows that $\mathscr {T}$ is h-stratified.

Notation 6.1. We will write $\pi \colon \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c) \to \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ for the surjective comparison map between the two spectra.

Proof. For any Thomason subset $Y\subseteq \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ , [Reference Barthel, Heard and SandersBHS23a, Lemmma 3.8] establishes that $\operatorname {\mathrm {Supp}}^h(e_Y)=\pi ^{-1}(Y)$ and $\operatorname {\mathrm {Supp}}^h(f_Y)=\pi ^{-1}(Y^c)$ . Moreover, we have $\operatorname {\mathrm {Supp}}^h(e_Y)=\operatorname {\mathrm {Supp}}^n(e_Y)$ and $\operatorname {\mathrm {Supp}}^h(f_Y)=\operatorname {\mathrm {Supp}}^n(f_Y)$ since $e_Y$ is a (weak) coring and $f_Y$ is a (weak) ring (Proposition 2.8). Finally, we claim that if $a,b \in \mathscr {T}$ both have the property that their naive and genuine h-supports coincide, then so does their tensor product $a \otimes b$ . Indeed,

$$\begin{align*}\operatorname{\mathrm{Supp}}^n(a\otimes b) \subseteq \operatorname{\mathrm{Supp}}^n(a) \cap \operatorname{\mathrm{Supp}}^n(b) = \operatorname{\mathrm{Supp}}^h(a) \cap \operatorname{\mathrm{Supp}}^h(b) = \operatorname{\mathrm{Supp}}^h(a\otimes b), \end{align*}$$

while the reverse inclusion holds unconditionally. It follows that naive and genuine h-support coincide for $g_W$ for any weakly visible subset W.

Proof. We always have the inclusion

$$\begin{align*}\operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(a,t))\subseteq \operatorname{\mathrm{Supp}}^n(a) \cap \operatorname{\mathrm{Cosupp}}^h(t). \end{align*}$$

For the objects a under consideration, the naive and genuine h-support coincide by Remark 6.2 and Lemma 6.4, so we actually have

(6.7) $$ \begin{align} \operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(a,t))\subseteq \operatorname{\mathrm{Supp}}^h(a) \cap \operatorname{\mathrm{Cosupp}}^h(t). \end{align} $$

Now suppose a is compact and let $\mathscr {B}$ be in the right-hand side. We claim

$$\begin{align*}\mathsf{hom}(E_{\mathscr{B}},\mathsf{hom}(a,t)) \neq 0. \end{align*}$$

Otherwise, $\mathsf {hom}(a,\mathsf {hom}(E_{\mathscr {B}},t)) = 0$ so that $\mathsf {hom}(e_{\operatorname {\mathrm {supp}}(a)},\mathsf {hom}(E_{\mathscr {B}},t)) = 0$ . Hence $\mathsf {hom}(E_{\mathscr {B}},t) \simeq \mathsf {hom}(f_{\operatorname {\mathrm {supp}}(a)},\mathsf {hom}(E_{\mathscr {B}},t))$ . But $\mathsf {hom}(E_{\mathscr {B}},t) \neq 0$ by hypothesis. It follows that $E_{\mathscr {B}}\otimes f_{\operatorname {\mathrm {supp}}(a)}\neq 0$ . That is, $\mathscr {B} \in \operatorname {\mathrm {Supp}}^n(f_{\operatorname {\mathrm {supp}}(a)}) = \pi ^{-1}(\operatorname {\mathrm {supp}}(a)^c) = \pi ^{-1}(\operatorname {\mathrm {supp}}(a))^c = \operatorname {\mathrm {Supp}}^h(a)^c$ by Lemma 6.4, which contradicts the hypothesis. This proves the claim when a is compact.

Now suppose $a=e_Y$ . Again, consider $\mathscr {B}$ in the right-hand side of (6.7). If it is not contained in the left-hand side then $\mathsf {hom}(e_Y,\mathsf {hom}(E_{\mathscr {B}},t)) =0$ so that ${\mathsf {hom}(x,\mathsf {hom}(E_{\mathscr {B}},t)) = 0}$ for all compact x with $\operatorname {\mathrm {supp}}(x) \subseteq Y$ . That is, $\mathscr {B} \not \in \operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(x,t))$ so that $\mathscr {B} \not \in \operatorname {\mathrm {Supp}}^h(x)$ . Hence

$$\begin{align*}\mathscr{B} \not\in \bigcup_{x \in \mathscr{T}^c_Y} \operatorname{\mathrm{Supp}}^h(x) = \pi^{-1}(\bigcup_{x \in \mathscr{T}^c_Y} \operatorname{\mathrm{supp}}(x)) = \pi^{-1}(Y) = \operatorname{\mathrm{Supp}}^h(e_Y) \end{align*}$$

which contradicts the hypothesis.

Now suppose $a=f_Y$ . Consider any $\mathscr {B}$ contained in the right-hand side of (6.7). Note that $\mathscr {B} \not \in \operatorname {\mathrm {Supp}}^h(e_Y)$ since the latter is the complement of $\operatorname {\mathrm {Supp}}^h(f_Y)$ . Hence $\mathscr {B} \not \in \operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(e_Y,t))$ . That is, $\mathsf {hom}(E_{\mathscr {B}},\mathsf {hom}(e_Y,t)) = 0$ . Therefore,

$$\begin{align*}0 \neq \mathsf{hom}(E_{\mathscr{B}},t) \simeq \mathsf{hom}(f_Y,\mathsf{hom}(E_B,t)). \end{align*}$$

Hence $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(\mathsf {hom}(f_Y,t))$ .

It follows that we have the equality for a tensor product like $g_W = e_{Y_1} \otimes f_{Y_2}$ .

Proof. We proved in [Reference Barthel, Heard and SandersBHS23a, Proposition 3.10] that the inclusion $\pi (\operatorname {\mathrm {Supp}}^h(t)) \subseteq \operatorname {\mathrm {Supp}}(t)$ always holds. On the other hand, consider the purported inclusion

$$\begin{align*}\pi(\operatorname{\mathrm{Cosupp}}^h(t)) \subseteq \operatorname{\mathrm{Cosupp}}(t). \end{align*}$$

If $\mathscr {P}$ is not contained in the right-hand side, that is, $\mathsf {hom}(g_{\mathscr {P}},t) = 0$ , then

$$\begin{align*}\varnothing = \operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(g_{\mathscr{P}},t)) = \operatorname{\mathrm{Supp}}^h(g_{\mathscr{P}})\cap \operatorname{\mathrm{Cosupp}}^h(t) = \pi^{-1}(\{ \mathscr{P}\}) \cap \operatorname{\mathrm{Cosupp}}^h(t) \end{align*}$$

by Lemma 6.6 and Lemma 6.4. Thus, $\mathscr {P} \not \in \pi (\operatorname {\mathrm {Cosupp}}^h(t))$ .

Now for the equality in $(a)$ : If $\mathscr {P} \in \operatorname {\mathrm {Supp}}(t)$ then $g_{\mathscr {P}} \otimes t \neq 0$ . Hence, the h-detection property implies

$$\begin{align*}\varnothing \neq \operatorname{\mathrm{Supp}}^h(g_{\mathscr{P}} \otimes t) = \operatorname{\mathrm{Supp}}^h(g_{\mathscr{P}}) \cap \operatorname{\mathrm{Supp}}^h(t) = \pi^{-1}(\{\mathscr{P}\})\cap\operatorname{\mathrm{Supp}}^h(t) \end{align*}$$

so that $\mathscr {P} \in \pi (\operatorname {\mathrm {Supp}}^h(t))$ .

For the equality in $(b)$ : If $\mathscr {P} \in \operatorname {\mathrm {Cosupp}}(t)$ then $\mathsf {hom}(g_{\mathscr {P}},t) \neq 0$ . Hence, the h-codetection property implies

$$\begin{align*}\varnothing \neq \operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(g_{\mathscr{P}},t)) = \operatorname{\mathrm{Supp}}^h(g_{\mathscr{P}})\cap \operatorname{\mathrm{Cosupp}}^h(t) = \pi^{-1}(\{\mathscr{P}\}) \cap \operatorname{\mathrm{Cosupp}}^h(t) \end{align*}$$

by Lemma 6.6 and Lemma 6.4 so that $\mathscr {P} \in \pi (\operatorname {\mathrm {Cosupp}}^h(t))$ .

Proof. This immediately follows from Proposition 6.8.

6.11. For weak rings, it is possible to prove an unconditional comparison result between the homological and Balmer–Favi support:

Proof. The $\subseteq $ inclusion always holds (Proposition 6.8), so it suffices to establish $\pi (\operatorname {\mathrm {Supp}}^h(w)) \supseteq \operatorname {\mathrm {Supp}}(w)$ . Let $\mathscr {P} \in \operatorname {\mathrm {Supp}}(w)$ . We have $\{\mathscr {P}\} = \operatorname {\mathrm {supp}}(x) \cap \operatorname {\mathrm {gen}}(\mathscr {P})$ for some $x \in \mathscr {T}^c$ by [Reference Barthel, Heard and SandersBHS23b, Remark 2.8]. Moreover, replacing x by $x \otimes x^{\vee }$ we may assume that x is a (weak) ring. Then $w \otimes g_{\mathscr {P}} \neq 0$ implies that $w \otimes x \otimes f_{\operatorname {\mathrm {gen}}(\mathscr {P})^c} \neq 0$ is a nonzero weak ring. The homological support has the detection property for weak rings by [Reference BalmerBal20a, Theorem 4.7]. Hence

$$ \begin{align*} \varnothing \neq \operatorname{\mathrm{Supp}}^h(w \otimes x \otimes f_{\operatorname{\mathrm{gen}}(\mathscr{P})^c}) &= \operatorname{\mathrm{Supp}}^h(w)\cap \operatorname{\mathrm{Supp}}^h(x) \cap \operatorname{\mathrm{Supp}}^h(f_{\operatorname{\mathrm{gen}}(\mathscr{P})^c}) \\ &= \operatorname{\mathrm{Supp}}^h(w) \cap \pi^{-1}(\{\mathscr{P}\}) \end{align*} $$

by the tensor product property (2.4), Remark 6.2 and Lemma 6.4. Therefore $\mathscr {P} \in \pi (\operatorname {\mathrm {Supp}}^h(w))$ , as desired.

7.1. The following result, due to Balmer, is crucial:

Proof. This is established in [Reference BalmerBal20a, Lemma 5.6].

Proof. Let $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ be such that $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Supp}}^h(t)$ . By definition, this means $\mathsf {hom}(t,E_{\varphi ^h(\mathscr {B})}) \neq 0$ . Hence $\mathsf {hom}(t,f_*E_{\mathscr {B}}) \neq 0$ by Lemma 7.2. That is, $f_*\mathsf {hom}(f^*(t),E_{\mathscr {B}}) \neq 0$ . Therefore, $\mathsf {hom}(f^*(t),E_{\mathscr {B}}) \neq 0$ , i.e., $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f^*(t))$ . This establishes the first inequality.

Now suppose $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f^*(t))$ . Then by the first inequality applied to $E_{\varphi ^h(\mathscr {B})}$ , we have

(7.4) $$ \begin{align} \mathscr{B} \in (\varphi^h)^{-1}(\{\varphi^h(\mathscr{B})\}) \subseteq \operatorname{\mathrm{Supp}}^h(f^*(E_{\varphi^h(\mathscr{B})})) \end{align} $$

and thus $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f^*(t)) \cap \operatorname {\mathrm {Supp}}^h(f^*(E_{\varphi ^h(\mathscr {B})}))=\operatorname {\mathrm {Supp}}^h(f^*(t \otimes E_{\varphi ^h(\mathscr {B})}))$ . Therefore, $f^*(t \otimes E_{\varphi ^h(\mathscr {B})}) \neq 0$ . Hence, $t \otimes E_{\varphi ^h(\mathscr {B})} \neq 0$ . That is, $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Supp}}^n(t)$ . This establishes the second inequality.

Finally, suppose $\mathscr {B} \in (\varphi ^h)^{-1}(\operatorname {\mathrm {Supp}}^n(t))$ . That is, $E_{\varphi ^h(\mathscr {B})}\otimes t \neq 0$ . Lemma 7.2 then implies $f_*(E_{\mathscr {B}}) \otimes t \neq 0$ . By the projection formula, this means $f_*(E_{\mathscr {B}} \otimes f^*(t)) \neq 0$ so $E_{\mathscr {B}} \otimes f^*(t) \neq 0$ . Hence $\mathscr {B} \in \operatorname {\mathrm {Supp}}^n(f^*(t))$ . This establishes the third inequality.

Proof. In cases $(a)$ , $(b)$ and $(c)$ , we have $\operatorname {\mathrm {Supp}}^h(t)=\operatorname {\mathrm {Supp}}^n(t)$ by Proposition 2.8 and Lemma 2.11, hence the result follows from Proposition 7.3.

Proof. The equivalence of $(b)$ and $(c)$ is [Reference Barthel, Castellana, Heard and SandersBCHS24, Theorem 1.8] and $(a)$ certainly implies $(b)$ . The implication $(c) \Rightarrow (a)$ follows from Corollary 7.5.

Proof. Let $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ be such that $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Supp}}^h(t)$ , i.e., $0 \neq \mathsf {hom}(t,E_{\varphi ^h(\mathscr {B})})$ . By adjunction and using Lemma 7.2, this implies

$$\begin{align*}0 \neq \mathsf{hom}(t,f_*E_{\mathscr{B}}) \simeq f_*\mathsf{hom}(f^*t, E_{\mathscr{B}}). \end{align*}$$

In particular, we see that $0\neq \mathsf {hom}(f^*t, E_{\mathscr {B}})$ , that is $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f^*t)$ .

For the reverse inclusion, take $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f^*t)$ . We first claim that $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h (f^!E_{\varphi ^h(\mathscr {B})})$ . Indeed, since $\mathsf {hom}(E_{\varphi ^h(\mathscr {B})},E_{\varphi ^h(\mathscr {B})})\neq 0$ , Lemma 7.2 gives

$$\begin{align*}0 \neq \mathsf{hom}(f_*E_{\mathscr{B}},E_{\varphi^h(\mathscr{B})}) \simeq f_*\mathsf{hom}(E_{\mathscr{B}},f^!E_{\varphi^h(\mathscr{B})}), \end{align*}$$

so in particular $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(f^!E_{\varphi ^h(\mathscr {B})})$ . By assumption, $\mathscr {S}$ satisfies h-minimality at $\mathscr {B}$ , so Proposition 5.1 implies

$$\begin{align*}\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(f^*t) \cap \operatorname{\mathrm{Cosupp}}^h(f^!E_{\varphi^h(\mathscr{B})}) =\operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(f^*t,f^!E_{\varphi^h(\mathscr{B})})). \end{align*}$$

Hence $0 \neq \mathsf {hom}(f^*t,f^!E_{\varphi ^h(\mathscr {B})}) \simeq f^!\mathsf {hom}(t,E_{\varphi ^h(\mathscr {B})})$ , so $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Supp}}^h(t)$ .

Proof. If $\operatorname {\mathrm {Supp}}^h(t)=\varnothing $ then $\operatorname {\mathrm {Supp}}^h(f^*(t))=\varnothing $ by Proposition 7.9 and hence $f^*(t)=0$ since h-stratified categories have the h-detection property. Since $f^*$ is conservative, we conclude that $t=0$ .

7.11. While this paper does not systematically develop the theory of homological cosupport, we do include the following cosupport-version of Proposition 7.9, which will be useful in Section 8 below.

Proof. The proof is similar to the one of Proposition 7.9. Let $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ be such that $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Cosupp}}^h(t)$ , i.e., $0 \neq \mathsf {hom}(E_{\varphi ^h(\mathscr {B})},t)$ . By adjunction and using Lemma 7.2, this implies

$$\begin{align*}0 \neq \mathsf{hom}(f_*E_{\mathscr{B}},t) \simeq f_*\mathsf{hom}(E_{\mathscr{B}},f^!t). \end{align*}$$

In particular, we see that $0\neq \mathsf {hom}(E_{\mathscr {B}},f^!t)$ , that is $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(f^!t)$ .

To prove the reverse inclusion, consider some $\mathscr {B} \in \operatorname {\mathrm {Cosupp}}^h(f^!t)$ . Since $\mathscr {S}$ satisfies h-minimality at $\mathscr {B}$ , (7.4) and Proposition 5.1 give

$$\begin{align*}\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(f^*E_{\varphi^h(\mathscr{B})}) \cap \operatorname{\mathrm{Cosupp}}^h(f^!t) =\operatorname{\mathrm{Cosupp}}^h(\mathsf{hom}(f^*E_{\varphi^h(\mathscr{B})},f^!t)). \end{align*}$$

Hence $0 \neq \mathsf {hom}(f^*E_{\varphi ^h(\mathscr {B})},f^!t) \simeq f^!\mathsf {hom}(E_{\varphi ^h(\mathscr {B})}, t)$ , so $\varphi ^h(\mathscr {B}) \in \operatorname {\mathrm {Cosupp}}^h(t)$ .

Proof. For the first statement, observe that

$$\begin{align*}\mathscr{B} \in \varphi^h(\operatorname{\mathrm{Supp}}^h(R)) &\Longleftrightarrow (\varphi^h)^{-1}(\{\mathscr{B}\}) \cap \operatorname{\mathrm{Supp}}^h(R) \neq \varnothing \\ &\Longleftrightarrow \operatorname{\mathrm{Supp}}^h(f^*(E_{\mathscr{B}})\otimes R) \neq \varnothing &&(\textrm{{{C}orollary~7.5}})\\ &\Longleftrightarrow f^*(E_{\mathscr{B}}) \otimes R \neq 0 &&(\textrm{{{R}emark~2.9}}) \\ &\Longleftrightarrow f_*(f^*(E_{\mathscr{B}}) \otimes R)\neq 0 &&([\mathrm{BCHS23},\ \textrm{Remark~13.12}])\\ &\Longleftrightarrow E_{\mathscr{B}}\otimes f_*(R) \neq 0 \\ &\Longleftrightarrow \mathscr{B} \in \operatorname{\mathrm{Supp}}^h(f_*(R)). \end{align*}$$

For the second statement, suppose $\mathscr {B} \in \operatorname {\mathrm {Supp}}^h(f_*(s))$ . This implies $E_{\mathscr {B}} \otimes f_*(s) \neq 0$ . That is, $f_*(f^*(E_{\mathscr {B}})\otimes t)\neq 0$ . Hence $f^*(E_{\mathscr {B}}) \otimes s\neq 0$ . The h-detection property for $\mathscr {S}$ then implies $\operatorname {\mathrm {Supp}}^h(f^*(E_{\mathscr {B}})\otimes s) \neq \varnothing $ . It then follows that $(\varphi ^h)^{-1}(\{\mathscr {B}\}) \cap \operatorname {\mathrm {Supp}}^h(s) \neq \varnothing $ by using Example 7.6. Hence $\mathscr {B} \in \varphi ^h(\operatorname {\mathrm {Supp}}^h(s))$ .

Proof. First we establish injectivity. Let $\mathscr {C}_1, \mathscr {C}_2 \in \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ and suppose $\varphi ^h(\mathscr {C}_1) = \varphi ^h(\mathscr {C}_2)$ . Then $E_{\varphi ^h(\mathscr {C}_1)} \otimes E_{\varphi ^h(\mathscr {C}_2)} \neq 0$ . Hence $f_*(E_{\mathscr {C}_1}) \otimes f_*(E_{\mathscr {C}_2}) \neq 0$ by Lemma 7.2. Moreover, since $f^*$ is a finite localization, it is smashing, so $f_*$ preserves the tensor product (but not necessarily the unit). Therefore, $f_*(E_{\mathscr {C}_1}) \otimes f_*(E_{\mathscr {C}_2}) \simeq f_*(E_{\mathscr {C}_1} \otimes E_{\mathscr {C}_2})$ and hence we conclude that $E_{\mathscr {C}_1} \otimes E_{\mathscr {C}_2} \neq 0$ . Thus $\mathscr {C}_1 = \mathscr {C}_2$ .

Recall from [Reference BalmerBal20b, Remark 3.4] and [Reference Barthel, Heard and SandersBHS23a, Example 2.8] that a basis of closed sets for the topology on $\operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ is given by the $\operatorname {\mathrm {Supp}}^h(x)$ for $x \in \mathscr {S}^c$ . Since $f^*$ is a finite localization, $x\oplus \Sigma x = f^*(c)$ for some compact $c \in \mathscr {T}^c$ and (replacing c with $c \otimes c^\vee $ ) we can assume c is a (weak) ring. Then by Corollary 7.5 we have

$$\begin{align*}\varphi^h(\operatorname{\mathrm{Supp}}^h(x))=\varphi^h(\operatorname{\mathrm{Supp}}^h(f^*(c))) = \varphi^h((\varphi^h)^{-1}(\operatorname{\mathrm{Supp}}^h(c))) = \operatorname{\mathrm{Supp}}^h(c) \cap \operatorname{\mathrm{im}}(\varphi^h). \end{align*}$$

Moreover since $\varphi ^h$ is injective, it preserves intersections. It follows that for an arbitrary closed subset $Z \subseteq \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ , we similarly have $\varphi ^h(Z) = Z' \cap \operatorname {\mathrm {im}}(\varphi ^h)$ for some closed subset $Z' \subseteq \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ . This establishes that $\varphi ^h$ is an embedding.

To describe the image of $\varphi ^h$ , note that the object is the right idempotent of the finite localization so by Lemma 6.4 and by [Reference BalmerBal20a, Theorem 5.12].

Proof. Let $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ . We claim that $\mathscr {B}$ is in the image of $\varphi ^h$ if and only if $\pi (\mathscr {B})$ is in the image of $\varphi $ , but this is precisely what $\operatorname {\mathrm {im}}(\varphi ^h)=\pi ^{-1}(\operatorname {\mathrm {im}}(\varphi ))$ says.

Proof. This follows from Proposition 7.14 and the definitions.

Terminology 8.1. Let $(f_i^*\colon \mathscr {T} \to \mathscr {S}_i)_{i \in I}$ be a family of geometric functors between rigidly-compactly generated tt-categories, indexed on a (not necessarily finite) set I. We refer to $(f_i^*)_{i \in I}$ as a geometric family.

Proof. We first prove $(a) \Leftrightarrow (b)$ . Write and consider its right orthogonal $\mathscr {L}^{\perp }$ in $\mathscr {T}$ . Note that

The first equivalence follows from [Reference Barthel, Castellana, Heard and SandersBCHS23, Definition 2.9], the second equivalence follows from the internal adjunction [Reference Balmer, Dell’Ambrogio and SandersBDS16, (2.18)], and the final equivalence follows from the observation that a left adjoint is conservative on the essential image of its right adjoint, by the unit-counit identity.

$(a) \Rightarrow (c)$ : If $t_1 \in \operatorname {\mathrm {Locid}}\langle t_2 \rangle $ then $f_i^*(t_1) \in \operatorname {\mathrm {Locid}}\langle f_i^*(t_2) \rangle $ for all $i \in I$ , since $f_i^*$ are tt-functors, so it remains to prove the reverse implication in $(c)$ . Tensoring (8.3) with $t_1$ and using the projection formula gives

Assuming $f_i^*(t_1) \in \operatorname {\mathrm {Locid}}\langle f_i^*(t_2) \rangle $ and using the projection formula again, we deduce $(f_i)_*f_i^*(t_1) \in \operatorname {\mathrm {Locid}}\langle (f_i)_*f_i^*(t_2) \rangle $ . Now varying over $i \in I$ yields

as claimed.

$(c) \Rightarrow (a)$ : By the triangle identity for the $f_i^* \dashv (f_i)_*$ adjunction, is a retract of , hence for all $i \in I$ . Therefore, condition $(c)$ implies , so $(f_i^*)_{i \in I}$ is weakly descendable.

Proof. $(a)\Rightarrow (b)$ : This follows from Lemma 8.7 and the proof of [Reference Barthel, Castellana, Heard and SandersBCHS23, Proposition 13.21]. $(b)\Rightarrow (c)$ : This is immediate from the definitions. $(c)\Leftrightarrow (d)$ : This is [Reference Barthel, Castellana, Heard and SandersBCHS24, Theorem 1.8]. $(d) \Rightarrow (e)$ : This holds because $\pi $ is surjective. $(e)\Rightarrow (a)$ : Since $\varphi $ is surjective, we have

$$\begin{align*}\operatorname{\mathrm{Spc}}(\mathscr{T}^c)=\operatorname{\mathrm{im}}\varphi=\bigcup_{i\in I}\operatorname{\mathrm{im}}\varphi_i. \end{align*}$$

If the $f_i^*$ are weakly closed then, by definition, each is compact. We then have by the argument given at the beginning of the proof of [Reference BalmerBal18, Theorem 1.7]. Hence

so that by the classification of thick ideals.

Proof. The implications $(a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (d)$ are part of Proposition 8.11, so it remains to show $(d) \Rightarrow (a)$ . Indeed, if $\varphi ^h$ is surjective, then

where the second equality uses [Reference BalmerBal20a, Theorem 5.12]. Since $\mathscr {T}$ is h-stratified, this implies that , so $(f_i^*)_{i\in I}$ is weakly descendable.

Proof. By hypothesis and Remark 3.3, . Hence it suffices to prove that for . This follows from the fact that ; see [Reference ZouZou25, Lemma 8.6].

Proof. Recall that $g_{\mathscr {P}} = e_{Y} \otimes f_{\operatorname {\mathrm {gen}}(\mathscr {P})^c}$ for some Thomason subset Y and . Thus

The local-to-global principle states that is contained in the left-hand side, while the family is weakly descendable if is contained in the right-hand side.

Proof. If $\mathscr {T}$ has the h-detection property then for every $\mathscr {B}$ by Proposition 4.5, which implies the desired equivalence. Certainly h-LGP implies h-detection. To finish the proof, we claim that $(a)$ also implies h-detection. Indeed, since tt-fields are h-stratified (Example 4.4), we have $(\varphi ^h_{\mathscr {B}})^{-1}(\operatorname {\mathrm {Supp}}^h(t)) = \operatorname {\mathrm {Supp}}^h(f_{\mathscr {B}}^*(t))$ for every $\mathscr {B}$ and every object $t \in \mathscr {T}$ by Proposition 7.9. Therefore, the h-detection property is equivalent to the family $\{f_{\mathscr {B}}^*\colon \mathscr {T} \to \mathscr {F}_{\mathscr {B}}\}_{\mathscr {B}}$ being jointly conservative. The claim thus follows from Proposition 8.11.

Proof. We will verify condition $(b)$ of Theorem 4.1. To this end, let $t \in \mathscr {T}$ and first fix some $i \in I$ . Using Proposition 7.9 twice, we obtain equalities

$$\begin{align*}\operatorname{\mathrm{Supp}}^h(f_i^*(t)) = (\varphi_i^h)^{-1}(\operatorname{\mathrm{Supp}}^h(t)) = \operatorname{\mathrm{Supp}}^h(\big\{\,f_i^*(E_{\mathscr{B}})\,\big|\,\mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t)\,\big\}). \end{align*}$$

Theorem 4.1 $(b)$ applied to $\mathscr {S}_i$ then shows that

$$\begin{align*}\operatorname{\mathrm{Locid}}\langle f_i^*(t) \rangle = \operatorname{\mathrm{Locid}}\langle f_i^*(E_{\mathscr{B}}) \mid \mathscr{B} \in \operatorname{\mathrm{Supp}}^h(t) \rangle. \end{align*}$$

Keeping in mind that (8.3) holds by assumption, Lemma 8.7 implies that $\operatorname {\mathrm {Locid}}\langle t \rangle = \operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}}\mid \mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t) \rangle $ , as desired.

8.25. We also include an alternative proof of Theorem 8.24 based on cosupport.

Proof Alternative proof of Theorem 8.24

We will verify condition (c) of Theorem 5.6. Note that the collection of maps $(\varphi _i^h)$ is jointly surjective by Proposition 8.11. Since $(f_i^*)$ is weakly descendable and $\mathscr {S}_i$ is h-stratified for all i, for any objects $t_1,t_2 \in \mathscr {T}$ we have equivalences:

$$ \begin{align*} 0 = \mathsf{hom}(t_1,t_2) & \iff \forall i \in I\colon 0 = f_i^!\mathsf{hom}(t_1,t_2) \simeq \mathsf{hom}(f_i^*t_1,f_i^!t_2) & & (8.7) \\ & \iff \forall i \in I\colon \varnothing = \operatorname{\mathrm{Supp}}^h(f_i^*t_1) \cap \operatorname{\mathrm{Cosupp}}^h(f_i^!t_2) & & (5.6) \\ & \iff \varnothing = \bigcup_{i \in I} (\varphi_i^h)^{-1}(\operatorname{\mathrm{Supp}}^h(t_1) \cap \operatorname{\mathrm{Cosupp}}^h(t_2)) && (7.9,7.12) \\ & \iff \varnothing = \operatorname{\mathrm{Supp}}^h(t_1) \cap \operatorname{\mathrm{Cosupp}}^h(t_2) & & (8.11), \end{align*} $$

as desired.

Proof. The only if part follows immediately from the diagram in Corollary 7.15. Now suppose each $\mathscr {T}(V_i)$ satisfies the Nerves of Steel conjecture. For any $\mathscr {P} \in \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ there exists some $V_i$ containing $\mathscr {P}$ . Applying Proposition 7.14 to the finite localization $\mathscr {T} \to \mathscr {T}(V_i)$ we obtain the commutative diagram

which implies that $\pi ^{-1}(\{\mathscr {P}\})$ consists of exactly one point.

Proof. $(a)\Rightarrow (b)$ : By [Reference Barthel, Heard and SandersBHS23a, Theorem 4.7] if $\mathscr {T}$ is tt-stratified, then the Nerves of Steel conjecture holds for $\mathscr {T}$ , so it remains to show that $\mathscr {T}$ is h-stratified. It was proved in [Reference Barthel, Heard and SandersBHS23a, Lemma 3.7] that $\operatorname {\mathrm {Supp}}(E_{\mathscr {B}})=\{ \pi (\mathscr {B})\}$ for any $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ . Hence $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \rangle = \operatorname {\mathrm {Locid}}\langle g_{\pi (\mathscr {B})} \rangle $ when $\mathscr {T}$ is tt-stratified. Therefore,

$$\begin{align*}\operatorname{\mathrm{Locid}}\langle t \rangle = \operatorname{\mathrm{Locid}}\langle g_{\mathscr{P}} \mid \mathscr{P} \in \operatorname{\mathrm{Supp}}(t) \rangle = \operatorname{\mathrm{Locid}}\langle E_{\mathscr{B}} \mid \mathscr{B} \in \pi^{-1}(\operatorname{\mathrm{Supp}}(t)) \rangle \end{align*}$$

for any $t \in \mathscr {T}$ . This coincides with $\operatorname {\mathrm {Locid}}\langle E_{\mathscr {B}} \mid \mathscr {B} \in \operatorname {\mathrm {Supp}}^h(t) \rangle $ since [Reference Barthel, Heard and SandersBHS23a, Theorem 4.7] establishes that $\pi ^{-1}(\operatorname {\mathrm {Supp}}(t))=\operatorname {\mathrm {Supp}}^h(t)$ when $\mathscr {T}$ is tt-stratified. This establishes that $\mathscr {T}$ is h-stratified by Theorem 4.1.

$(b)\Rightarrow (a)$ : If $\mathscr {T}$ is h-stratified then the h-detection property holds. Thus, Proposition 6.8 implies that $\pi (\operatorname {\mathrm {Supp}}^h(t))=\operatorname {\mathrm {Supp}}(t))$ for each $t \in \mathscr {T}$ . It follows that the diagram

(9.7)

commutes. By hypothesis, the horizontal and vertical maps are injective (and thus bijective), hence so is the diagonal map; that is, $\mathscr {T}$ is tt-stratified.

Proof. This immediately follows from Theorem 9.6.

Proof. For each $i \in I$ , the tt-category $\mathscr {S}_i$ is tt-stratified by assumption, hence h-stratified by Theorem 9.6. Descent for h-stratification as established in Theorem 8.24 then implies that $\mathscr {T}$ is h-stratified. Therefore, Theorem 9.6 implies that $\mathscr {T}$ is tt-stratified, due to our assumption that $\mathscr {T}$ satisfies Nerves of Steel.

9.10. We conclude this section with a proof of Theorem A from the introduction.

Proof of Theorem A

The equivalence between $(a)$ and $(b)$ is Corollary 9.8. Corollary 8.14 gives $(b) \Rightarrow (c)$ , while the reverse implication is Theorem 8.24.

Proof. Let and consider the commutative square

where the vertical arrows are the finite localizations and the bottom arrow is the induced functor. This diagram satisfies the Beck–Chevalley condition $h^* f_* \simeq g_* k^*$ ; cf. [Reference SandersSan22, Lemma 5.5]. Indeed, this may be checked after applying the conservative $h_*$ . For any $s \in \mathscr {S}$ , we have where we have invoked [Reference Balmer and SandersBS17, Proposition 5.11]. With this in hand, consider a compact object $x \in \mathscr {S}(\varphi ^{-1}(V))^c$ . Then $x \oplus \Sigma x = k^*(c)$ for some $c \in \mathscr {S}^c$ and it suffices to show that $g_* k^*(c)$ is compact. This follows from our hypothesis, since $g_* k^*(c) \simeq h^*f_*(c)$ .

Proof. Consider the commutative diagram

Recall from Proposition 8.11 that $(f_i^*)$ being jointly nil-conservative implies that $\varphi $ is also surjective. If $\varphi $ is in addition injective as assumed in $(a)$ , then it is bijective. Since $\sqcup \pi _{\mathscr {S}_i}$ is bijective, $\varphi ^h$ is injective and hence bijective. Therefore $\pi _{\mathscr {T}}$ is bijective.

Now assume either condition $(b)$ or $(c)$ . By Corollary 7.15, the Nerves of Steel conjecture holds also for any finite localization of $\mathscr {S}_i$ . Moreover, the property of being strongly closed is local in the target by Lemma 10.1, while the property of being nil-conservative is local in the target by Proposition 8.11[ $(c) \Leftrightarrow (d)$ ] and Corollary 7.16. Therefore, we can reduce to the case when $\mathscr {T}$ is local and check that the fibre of $\pi _{\mathscr {T}}\colon \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)\to \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ over the unique closed point $\mathfrak m \in \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ is a singleton. Since $\varphi $ is surjective, we may choose $i \in I$ and a closed point $\mathscr {M} \in \operatorname {\mathrm {Spc}}(\mathscr {S}_i^c)$ lying over $\mathfrak m$ . For the remainder of this proof, we set $\mathscr {S} = \mathscr {S}_i$ and simplify the notation accordingly.

Observe that $\{\mathscr {M}\}$ is Thomason closed if $\operatorname {\mathrm {Spc}}(\mathscr {S}^c)$ is noetherian or $\varphi $ has discrete fibres. In either case, by [Reference BalmerBal05, Proposition 2.14] there exists some compact object $z \in \mathscr {S}^c$ with $\operatorname {\mathrm {supp}}(z)=\{\mathscr {M}\}$ . Replacing z by $z \otimes z^{\vee }$ we may assume that z is a (weak) ring. Let $\mathscr {A}\in \operatorname {\mathrm {Spc}}^h(\mathscr {S}^c)$ be the unique homological prime such that $\pi _{\mathscr {S}}(\mathscr {A}) = \mathscr {M}$ , so that $\operatorname {\mathrm {Supp}}^h(z) = \{\mathscr {A}\}$ .

Suppose for a contradiction that there exists a point $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ lying over $\mathfrak m$ such that $\mathscr {B}\neq \varphi ^h(\mathscr {A})$ . Recall from Example 7.6 that

$$\begin{align*}\operatorname{\mathrm{Supp}}^h(f^*(E_{\mathscr{B}})) = (\varphi^h)^{-1}(\{\mathscr{B}\}). \end{align*}$$

It follows that

$$\begin{align*}\operatorname{\mathrm{Supp}}^h(z\otimes f^*(E_{\mathscr{B}})) = \operatorname{\mathrm{Supp}}^h(z) \cap \operatorname{\mathrm{Supp}}^h(f^*(E_{\mathscr{B}})) = \{\mathscr{A}\} \cap (\varphi^h)^{-1}(\{\mathscr{B}\}) = \varnothing \end{align*}$$

and hence $f^*(E_{\mathscr {B}})\otimes z=0$ because homological support detects weak rings (Remark 2.9). Therefore, $E_{\mathscr {B}} \otimes f_*(z)=0$ by the projection formula. That is $\mathscr {B} \not \in \operatorname {\mathrm {Supp}}^h(f_*(z))$ . By the hypothesis that f is strongly closed, $f_*(z)$ is compact and hence ${\operatorname {\mathrm {Supp}}^h(f_*(z)) = \pi _{\mathscr {T}}^{-1}(\operatorname {\mathrm {supp}}(f_*(z)))}$ by Remark 6.2. Therefore we have $\mathscr {B} \not \in \pi _{\mathscr {T}}^{-1}(\operatorname {\mathrm {supp}}(f_*(z)))$ so $\mathfrak m = \pi _{\mathscr {T}}(\mathscr {B}) \not \in \operatorname {\mathrm {supp}}(f_*(z))=\varphi (\operatorname {\mathrm {supp}}(z))$ where the last equality is by [Reference Barthel, Castellana, Heard and SandersBCHS23, Theorem 13.13] since z is a compact weak ring. But $\varphi (\operatorname {\mathrm {supp}}(z))$ contains $\varphi (\{\mathscr {M}\}) = \mathfrak m$ yielding the desired contradiction.

Proof. Recall from [Reference Balmer and CameronBC21, Lemma 2.2] that every tt-field induces a homological residue field. More precisely, for each $\mathscr {P} \in \operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ , there exists a homological prime $\mathscr {B} \in \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ lying over $\mathscr {P}$ and a commutative diagram

where the functor $\overline {F}$ is faithful; see [Reference Balmer and CameronBC21, (2.3)]. It follows from this diagram and the fact that the tt-field $\mathscr {F}_{\mathscr {P}}$ is phantomless that a morphism in $\mathscr {T}$ is killed by $f_{\mathscr {P}}^*$ if and only if it is killed by the homological residue field . Thus, the family $(f_{\mathscr {P}}^*)_{\mathscr {P} \in \mathscr {\operatorname {\mathrm {Spc}}}(\mathscr {T}^c)}$ jointly detects $\otimes $ -nilpotence if and only if the corresponding family of homological residue fields (also indexed on $\operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ ) jointly detects $\otimes $ -nilpotence. The latter is equivalent to the Nerves of Steel conjecture by [Reference BalmerBal20b, Corollary 4.7 and Theorem 5.4]. This establishes the equivalence of $(a)$ and $(b)$ . On the other hand, contemplating the commutative diagram

we see that $(a)$ implies that the map $\varphi ^h\colon \bigsqcup _{\mathscr {P} \in \operatorname {\mathrm {Spc}}(\mathscr {T}^c)} \operatorname {\mathrm {Spc}}^h(\mathscr {F}_{\mathscr {P}}^c)\to \operatorname {\mathrm {Spc}}^h(\mathscr {T}^c)$ is surjective. This is equivalent to the family being nil-conservative by [Reference Barthel, Castellana, Heard and SandersBCHS24, Theorem 1.9]. Thus $(a)\Rightarrow (c)$ . Finally, the implication $(c)\Rightarrow (a)$ follows from Proposition 10.2(a).

Proof. Assume first that the Nerves of Steel conjecture holds for $\mathscr {T}$ . Using Proposition 6.12 twice as well as the tensor product formula for homological support (2.4), we compute

$$ \begin{align*} \operatorname{\mathrm{Supp}}(w_1 \otimes w_2) & = \pi \operatorname{\mathrm{Supp}}^h(w_1 \otimes w_2) \\ & = \pi(\operatorname{\mathrm{Supp}}^h(w_1) \cap \operatorname{\mathrm{Supp}}^h(w_2)) \\ & = \pi\operatorname{\mathrm{Supp}}^h(w_1) \cap \pi\operatorname{\mathrm{Supp}}^h(w_2) \\ & = \operatorname{\mathrm{Supp}}(w_1) \cap \operatorname{\mathrm{Supp}}(w_2), \end{align*} $$

where the third equality uses the hypothesis that $\pi $ is a bijection.

Conversely, suppose that the tensor product formula holds for all weak rings. The argument in the proof of [Reference Barthel, Heard and SandersBHS23a, Proposition 3.13] then implies the Nerves of Steel conjecture for $\mathscr {T}$ . Note that the result there assumed the full tensor product formula, but it is only used for the weak rings $E_{\mathscr {B}}$ .

Proof. By Theorem 9.6 each $\mathscr {S}_i$ is h-stratified and satisfies the Nerves of Steel conjecture. The result then follows from Proposition 10.2 and Corollary 9.9 since $(f_i^*)_{i \in I}$ is jointly nil-conservative by Proposition 8.11.

Proof. Since A is descendable, the base-change functor $A\otimes -\colon \mathscr {C} \to \operatorname {\mathrm {Mod}}_{\mathscr {C}}(A)$ is conservative and hence induces a surjection $\operatorname {\mathrm {Spc}}(\operatorname {\mathrm {Mod}}_{\mathscr {C}}(A)^c) \to \operatorname {\mathrm {Spc}}(\mathscr {C}^c)$ ; see [Reference Barthel, Castellana, Heard and SandersBCHS24, Theorem 1.4] or [Reference Barthel, Castellana, Heard, Naumann, Pol and SandersBCH⁺24, Proposition 7.8]. Hence $\operatorname {\mathrm {Spc}}(\mathscr {C}^c)$ is also noetherian. We may then invoke Proposition 11.1 $(c)$ .

11.5. We now turn to a cohomological criterion for stratification.

Proof. The various comparison maps fit into a commutative diagram

in which the three indicated vertical maps are surjections. The top and middle horizontal maps are also surjections by Proposition 8.11. The bottom map is a bijection by assumption, hence $\rho _{\mathscr {T}}$ is surjective as well. Since the tt-categories $\mathscr {S}_i$ are cohomologically stratified, the left vertical composite is a bijection, which then implies that all maps in this diagram are bijections. The result thus follows from Corollary 9.9.

Notation 12.3. We let $\operatorname {\mathrm {Sub}}(G)$ denote the set of closed subgroups of G and $\operatorname {\mathrm {Sub}}(G)/G$ the set of closed subgroups of G up to conjugation.

Recollection 12.4. We will use a number of facts about the equivariant homotopy category in the following result. In particular, recall that for $H \le G$ we have a restriction functor $\mathrm {res}^G_H \colon \operatorname {\mathrm {Sp}}_G \to \operatorname {\mathrm {Sp}}_H$ that admits left and right adjoints $\operatorname {\mathrm {ind}}_H^G$ and $\operatorname {\mathrm {coind}}_H^G$ , respectively. If G is finite, then these left and right adjoints are naturally isomorphic. This is not true in general, but rather there is a natural equivalence

$$\begin{align*}\operatorname{\mathrm{coind}}_H^G(Y) \simeq \operatorname{\mathrm{ind}}_H^G(Y \otimes S^L), \end{align*}$$

where L is the tangent H-representation of $G/H$ . Moreover, we have

$$\begin{align*}\operatorname{\mathrm{ind}}_H^G(S^0_H) \simeq G/H_+. \end{align*}$$

Finally, the projection formula holds in this context: there is a natural equivalence

$$\begin{align*}\operatorname{\mathrm{coind}}_H^G(\mathrm{res}^G_H(X) \otimes Y) \simeq X\otimes \operatorname{\mathrm{coind}}_H^G(Y). \end{align*}$$

Proof. Let us write $\Psi _G^H$ for the right adjoint of $\Phi _G^H$ (which sometimes goes by the name of geometric inflation). We denote the collection of conjugacy classes of proper closed subgroups of G by $\mathcal {P}$ . We will establish that

(12.6)

Note that we can define an ordering on compact Lie groups by the pair $(\dim H,\pi _0H)$ ordered lexicographically. Specifically, $H \le G$ if and only if $\dim (H) \leq \dim (G)$ or $\dim (H) = \dim (G)$ and the number of connected components of H is less than or equal to the number of connected components of G. We will prove (12.6) by induction, with the base case $G = e$ being tautological. We then assume that (12.6) holds for all $H \lneq G$ , and let $M \in \operatorname {\mathrm {Mod}}_G(R)$ .

If $G = H$ , then $\Phi ^G_G$ is a finite localization away from $\operatorname {\mathrm {Locid}}\langle G/H_+ \otimes R \mid H \in \mathcal {P} \rangle $ , factoring as

$$\begin{align*}\Phi_G^G\colon \operatorname{\mathrm{Mod}}_G(R) \to \operatorname{\mathrm{Mod}}_G(R \otimes \tilde{E}\mathcal{P}) \simeq \operatorname{\mathrm{Mod}}(\Phi_G^GR), \end{align*}$$

where the first map is base-change and the second map is the symmetric monoidal equivalence induced by taking categorical G-fixed points. This implies that

(12.7)

This also follows from [Reference HillHil12, Definition 4.1 and Proposition 4.4]. We claim that

(12.8)

It suffices to check this for the generators of $\operatorname {\mathrm {Mod}}_G(R)$ , i.e., for M of the form $G/K_+ \otimes R$ . The case $K=G$ is (12.7), while for $K \lneq G$ the claim holds since $G/K_+ \otimes \tilde {E}\mathcal {P} = 0$ .

Now let H be a proper closed subgroup of G and consider subgroups $K \le H$ . The corresponding geometric fixed-point functors then factor as

$$\begin{align*}\Phi_G^K\colon \operatorname{\mathrm{Mod}}_G(R) \xrightarrow{\operatorname{\mathrm{res}}} \operatorname{\mathrm{Mod}}_H(R) \xrightarrow{\Phi_H^K} \operatorname{\mathrm{Mod}}(\Phi_H^KR) \simeq \operatorname{\mathrm{Mod}}(\Phi_G^KR). \end{align*}$$

By the inductive hypothesis

in $\operatorname {\mathrm {Mod}}_H(R)$ . In particular, , where L is the tangent H-representation of $G/H$ . Applying coinduction, we get

By Recollection 12.4 we have

$$ \begin{align*} \operatorname{\mathrm{coind}}_H^G(\operatorname{\mathrm{res}}_H(M) \otimes S^L) & \simeq \operatorname{\mathrm{coind}}_H^G(S^L) \otimes M \\ & \simeq \operatorname{\mathrm{ind}}_H^G(S_H^0) \otimes M \\ & \simeq G/H_+ \otimes M. \end{align*} $$

Since $\operatorname {\mathrm {coind}}_H^G\Psi _H^K \simeq \Psi _G^K$ , this implies that

(12.9)

The isotropy separation sequence takes the form

$$\begin{align*}M \otimes E\mathcal{P}_+ \to M \to M \otimes \tilde{E}\mathcal{P}. \end{align*}$$

We claim that $M \otimes E\mathcal {P}_+ \in \operatorname {\mathrm {Loc}}\langle G/H_+\otimes M \mid H \in \mathcal {P} \rangle $ . Indeed, as a consequence of work of Illman [Reference IllmanIll83] on the existence of G-triangulations (see [Reference SandersSan19, Remark 3.6]), we have

$$\begin{align*}\operatorname{\mathrm{Thick}}\langle G/H_+ \mid H \in \mathcal{P} \rangle = \operatorname{\mathrm{Thickid}}\langle G/H_+ \mid H \in \mathcal{P} \rangle. \end{align*}$$

It follows that

$$\begin{align*}E\mathcal{P}_+ \in \operatorname{\mathrm{Locid}}\langle G/H_+ \mid H \in \mathcal{P} \rangle = \operatorname{\mathrm{Loc}}\langle G/H_+ \mid H \in \mathcal{P} \rangle, \end{align*}$$

as desired. Consequently, from isotropy separation and the previous claim we get

$$\begin{align*}M \in \operatorname{\mathrm{Loc}}\langle M \otimes \tilde{E}\mathcal{P}, M \otimes E\mathcal{P}_+ \rangle \subseteq \operatorname{\mathrm{Loc}}\langle M \otimes \tilde{E}\mathcal{P} , G/H_+\otimes M \mid H \in \mathcal{P} \rangle. \end{align*}$$

Substituting (12.8) and (12.9) thus gives

which finishes the proof.

Proof. First we show that the claim is true for $R=S_G^0$ . Indeed, by [Reference Barthel, Greenlees and HausmannBGH20, Theorem 3.14], the analogous result is true for the Balmer spectrum. Because the Nerves of Steel conjecture holds for $\operatorname {\mathrm {Sp}}_G$ by [Reference BalmerBal20b, Corollary 5.10] (which in turn relies on [Reference Barthel, Greenlees and HausmannBGH20, Corollary 3.18]), the claim follows.

For the general case, consider the commutative diagram

induced by base-change $S_G^0\to R$ and the H-geometric fixed point functors for $\operatorname {\mathrm {Mod}}_G(R)$ and $\operatorname {\mathrm {Sp}}_G$ . This diagram shows that the images of the maps $\varphi ^h_H$ are disjoint in $\operatorname {\mathrm {Spc}}^h(\operatorname {\mathrm {Mod}}_G(R)^c)$ for non-conjugate subgroups, since this is true for the $R=S^0_G$ case just mentioned. As a consequence of Lemma 12.5 and Proposition 8.11, these images also cover the homological spectrum, which gives the desired decomposition of the homological spectrum. The analogous statement for the tt-spectrum is proved similarly.

Proof. The first part follows from Theorem 8.24, which applies due to Lemma 12.5. The claimed bijection is then Theorem 4.1 combined with Lemma 12.10.

Proof. Since the geometric fixed points for $R_G$ are split by inflation,

$$\begin{align*}\varphi_H\colon \operatorname{\mathrm{Spc}}(\operatorname{\mathrm{Mod}}(R)^c) \to \operatorname{\mathrm{Spc}}(\operatorname{\mathrm{Mod}}_G(R_G)^c) \end{align*}$$

is a homeomorphism onto its image for any subgroup H in G. In particular, using Lemma 12.10, these maps induce a bijection

$$\begin{align*}\operatorname{\mathrm{Spc}}(\operatorname{\mathrm{Mod}}_G(R_G)^c) \simeq \bigsqcup_{H \in \operatorname{\mathrm{Sub}}(G)/G} \operatorname{\mathrm{Spc}}(\operatorname{\mathrm{Mod}}(R)^c). \end{align*}$$

By Proposition 10.2(a), the Nerves of Steel conjecture descends to $\operatorname {\mathrm {Mod}}_G(R_G)$ , i.e., $\pi \colon \operatorname {\mathrm {Spc}}^h(\operatorname {\mathrm {Mod}}_G(R_G)^c) \to \operatorname {\mathrm {Spc}}(\operatorname {\mathrm {Mod}}_G(R_G)^c)$ is a bijection. Proposition 12.11 then implies that $\operatorname {\mathrm {Mod}}_G(R_G)$ is h-stratified, as well as the claimed parametrization of localizing ideals.

Question 13.2. Suppose $\mathscr {T}$ is a rigidly-compactly generated tt-category. If $\mathscr {T}$ is h-stratified and satisfies the Nerves of Steel conjecture, does it follow that $\operatorname {\mathrm {Spc}}(\mathscr {T}^c)$ is weakly noetherian?

Question 13.5. If (∗) holds, does it follow that $\operatorname {\mathrm {Spec}}(R)$ is weakly noetherian?

Proof. We always have $\pi (\operatorname {\mathrm {Supp}}^h(t)) \subseteq \operatorname {\mathrm {Supp}}(t)$ by [Reference ZouZou25, Lemma 8.5] and the latter is localizing closed, hence $ \overline {\pi (\operatorname {\mathrm {Supp}}^h(t))}^{\mathrm {loc}} \subseteq \operatorname {\mathrm {Supp}}(t)$ . For the reverse inclusion, recall that weakly visible subsets form a basis of open subsets for the localizing topology. If ${\mathscr {P} \notin \overline {\pi (\operatorname {\mathrm {Supp}}^h(t))}^{\mathrm {loc}}}$ then there exists a weakly visible subset $W \ni \mathscr {P}$ such that $\pi (\operatorname {\mathrm {Supp}}^h (t))\cap W = \varnothing $ and hence $\pi ^{-1}(W) \cap \operatorname {\mathrm {Supp}}^h (t) = \varnothing $ . By the tensor product formula and Lemma 6.4, we obtain $\operatorname {\mathrm {Supp}}^h(t \otimes g_W) = \varnothing $ . It then follows from h-detection that $t \otimes g_W = 0$ . Therefore, $\mathscr {P} \not \in \operatorname {\mathrm {Supp}}(t)$ by the definition of the Balmer–Favi–Sanders support; see [Reference ZouZou25, Definition 5.17].