$X\in\mathrm{Sm}_{B}$. This finishes the proof. ∎

By definition, a presheaf $\mathcal{F}\in\mathrm{Pre}^{\mathrm{fr}}(B)$ is a
_semi-Zariski sheaf_ if $\gamma_{*}\mathcal{F}$ is a semi-Zariski sheaf, while
the class of semi-Zariski local isomorphisms in
$\mathrm{Pre}^{\mathrm{fr}}(B)$ is generated by the image of the class of
semi-Zariski isomorphisms under $\gamma^{*}$.

######  Proposition 7.9.

For any $\mathcal{F}\in\mathrm{Pre}^{\mathrm{fr}}(B)$, there is an isomorphism

$\gamma_{*}L_{\mathrm{sZar}}^{\mathrm{fr}}\mathcal{F}\simeq
L_{\mathrm{sZar}}\gamma_{*}\mathcal{F}.$

The same holds for $\mathcal{F}\in\mathrm{Spt}^{\mathrm{fr}}(B)$.

###### Proof.

Consider the commutative diagram of $\infty$-categories

$\textstyle{\mathrm{Corr}^{\mathrm{fr}}(\mathrm{EssSm}^{sLoc}_{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Corr}^{\mathrm{fr}}(\mathrm{EssSm}_{B})}$$\textstyle{\mathrm{EssSm}^{sLoc}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{sloc}}$$\textstyle{\mathrm{EssSm}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\gamma}$

Let $\alpha$ be a semi-Zariski local isomorphism in $\mathrm{Pre}(B)$. Since
$\gamma^{*}_{sloc}$ preserves isomorphisms, it follows that
$\gamma^{*}(\alpha)$ is an isomorphism on semi-local schemes. So
$\gamma_{*}\gamma^{*}(\alpha)$ is a semi-Zariski local isomorphism. Thus since
the class of semi-Zariski local isomorphisms in
$\mathrm{Pre}^{\mathrm{fr}}(B)$ is generated by morphisms of the from
$\gamma^{*}(\alpha)$ as above, it follows that $\gamma_{*}$ preserves semi-
Zariski isomorphisms. Then since $\gamma_{*}$ preserves semi-Zariski sheaves,
it follows that $\gamma_{*}$ commutes with $L_{\mathrm{sZar}}$.

The claim for $\mathrm{Spt}^{\mathrm{fr}}(B)$ follows from the one for
$\mathrm{Pre}^{\mathrm{fr}}(B)$ because $L_{\mathrm{sZar}}^{\mathrm{fr}}$ and
$\gamma^{\mathrm{fr}}_{*}$ commute with
$\Omega\colon\mathrm{Spt}^{\mathrm{fr}}(B)\to\mathrm{Spt}^{\mathrm{fr}}(B)$
and
$\Omega^{\infty}\colon\mathrm{Spt}^{\mathrm{fr}}(B)\to\mathrm{Pre}^{\mathrm{fr}}(B)$,
and similarly for $L_{\mathrm{sZar}}$ and $\gamma_{*}$. ∎

The following result holds for an infinite field $k$ and a local scheme
$U=X_{(x)}$ according to [18, Theorem 3.15(3)], while for semi-local schemes
the claim is proven in [10, Theorem 7.0.1] for presheaves with cohomological
correspondence transfers.

######  Theorem 7.10.

For any semi-local essentially smooth scheme $U$ over a field $k$ and quasi-
stable framed radditive abelian presheaf $\mathcal{F}$ on $\mathrm{Sm}_{k}$,
the homomorphism $\mathcal{F}(U)\to\mathcal{F}(U^{(0)})$ is injective.

###### Proof.

It suffices to consider semi-local schemes of the form
$U=X_{(x_{0},\dots,x_{l})}$ for $X\in\mathrm{Sm}_{k}$, $x_{0},\dots,x_{l}\in
X$, by taking filtered limits. By the argument of [18, Theorem 3.15(3)] it
suffices to construct, for any closed subscheme $Y\not\hookrightarrow X$ of
positive codimension, and for $N\gg 0$, a framed correspondence

$c\in\mathbb{Z}\mathrm{F}_{N}(U,X-Y)$

such that the $\mathbb{A}^{1}$-homotopy class in $\mathbb{Z}\mathrm{F}(U,X)$
of the composite of $c$ with the open immersion $j\colon X-Y\to X$ equals the
class of the $\sigma$-suspension of the canonical morphism $\mathrm{can}\colon
U\to X$, i.e.,

$[j\circ c]=[\mathrm{can}]\in\overline{\mathbb{Z}}\mathrm{F}(U,X)$ (7.2)

where $\mathbb{Z}\mathrm{F}(U,X)=\varinjlim_{n}\mathbb{Z}\mathrm{F}_{n}(U,X)$
and
$\overline{\mathbb{Z}}\mathrm{F}(U,X)=\mathbb{Z}\mathrm{F}(U,X)/\mathbb{Z}\mathrm{F}(U\times\mathbb{A}^{1},X)$.
Suppose that $k$ is infinite. Let us start with the diagram

$X\xleftarrow{v}\mathcal{C}\xrightarrow{j}\overline{\mathcal{C}}\xrightarrow{p}U$

and an ample line bundle $\mathcal{O}_{\overline{\mathcal{C}}}(1)$ on
$\overline{\mathcal{C}}$ provided by [10, Lemma 7.0.5] that satisfies the
respective list of properties (1)-(6). So, according to the properties (1),(4)
$p$ is a projective morphism of dimension one that is smooth over
$\mathcal{C}$, and $D=\overline{\mathcal{C}}\setminus\mathcal{C}$ is finite
over $U$. The argument from [10, Proof of Lemma 7.0.3] gives us sections

$s\in\Gamma(\overline{\mathcal{C}},\mathcal{O}(n))$ | $\widetilde{s}\in\Gamma(\overline{\mathcal{C}}\times\stackrel{{\scriptstyle\lambda}}{{\mathbb{A}^{1}}},\mathcal{O}(l))$ | $s^{\prime}\in\Gamma(\overline{\mathcal{C}},\mathcal{O}(l)\otimes\mathscr{L}(\Delta)^{-1})$ | $\delta\in\Gamma(\overline{\mathcal{C}},\mathscr{L}(\Delta))$
---|---|---|---
$Z(s|_{\mathcal{Z}\amalg D})=\varnothing$ | $\widetilde{s}|_{\overline{\mathcal{C}}\times 0}=s$ | $Z(s^{\prime}|_{\mathcal{Z}\amalg D\amalg\Delta})=\varnothing$ | $Z(\delta)=\Delta$
| $\widetilde{s}|_{\overline{\mathcal{C}}\times 1}=s^{\prime}\otimes\delta$ |  |
| $\widetilde{s}|_{D\times\mathbb{A}^{1}}=s$ |  |

Here $\mathcal{Z}=v^{-1}(Y)$, $\Delta$ is the graph of $\mathrm{can}$, and
$\mathscr{L}(\Delta)$ is the line bundle on $\overline{\mathcal{C}}$
corresponding to the divisor $\Delta$. Consider a closed immersion
$\overline{\mathcal{C}}\not\hookrightarrow\mathbb{P}^{N}_{U}$ provided by the
ampleness of $\mathcal{O}_{\overline{\mathcal{C}}}(1)$, and a retraction
$r\colon V\to\mathcal{C}$ for an étale neighborhood $V$ of ${\mathcal{C}}$ in
$\mathbb{A}^{N}_{U}$ provided by smoothness of $\mathcal{C}$, see [22, Theorem
I.8]. Let $\phi\in\mathcal{O}_{V}(V)$,
$\widetilde{\phi}\in\mathcal{O}_{V\times\mathbb{A}^{1}}(V\times\mathbb{A}^{1})$
be such that

$\phi\big{|}_{\mathcal{C}}=s/d^{l},\quad\widetilde{\phi}\big{|}_{\mathcal{C}}=\widetilde{s}/d^{l},$

and define $\phi^{\prime}$ as the inverse image of $\widetilde{\phi}$ along
the composite
$\mathbb{A}^{N}_{U}\cong\mathbb{A}^{N}_{U}\times\\{1\\}\hookrightarrow\mathbb{A}^{N}_{U}\times\mathbb{A}^{1}$.
Denote by $\mathrm{pr}_{V}\colon{V\times\mathbb{A}^{1}}\to{V}$ and
$\mathrm{pr}_{U}\colon{U\times\mathbb{A}^{1}}\to{U}$ the canonical
projections. We write $e$ for $v\circ r$ and its restrictions on open
subschemes of $V$, and $\widetilde{e}$ for $v\circ r\circ\mathrm{pr}_{V}$ and
its restrictions on open subschemes of $V\times\mathbb{A}^{1}$. We define the
framed correspondences

$\displaystyle\Upsilon$
$\displaystyle=(Z(s^{\prime}),V-\Delta,\phi^{\prime},e)\in\mathbb{Z}\mathrm{F}_{N}(U,X-Y),$
$\displaystyle\Phi^{\prime}$
$\displaystyle=(Z(s),V,\phi,e)-\Upsilon\in\mathbb{Z}\mathrm{F}_{N}(U,X-Y),$
$\displaystyle\Theta^{\prime}$
$\displaystyle=(Z(\widetilde{s}),\widetilde{\phi},\widetilde{e})-\Upsilon\circ\mathrm{pr}_{U}\in\mathbb{Z}\mathrm{F}_{N}(U\times\mathbb{A}^{1},X).$

Similarly to [10, Proof of Lemma 7.0.3], the properties of the sections above
imply that

$\Theta^{\prime}\circ
i_{0}=j\circ\Phi^{\prime}\in\mathbb{Z}\mathrm{F}(U,X),\quad[\Theta^{\prime}\circ
i_{1}]=[\langle\nu\rangle^{*}(\mathrm{can})]\in\overline{\mathbb{Z}}\mathrm{F}(U,X)$

and for some $\nu\in k[U]^{\times}$ (see Example 2.2 for $\langle\nu\rangle$).
Letting $\Phi=\langle\nu^{-1}\rangle^{*}(\Phi^{\prime})$,
$\Theta=\langle\nu^{-1}\rangle^{*}(\Theta^{\prime})$, we obtain (7.2) since

$\Theta\circ i_{0}=j\circ\Phi,\quad[j]=[\Theta\circ i_{1}].$

For a finite field $k$, the claim follows from the result over infinite
extensions of $k$ by the argument in [9, Proof of Lemma 6.19]. ∎

######  Theorem 7.11.

There are equivalences of endofunctors

$\Theta_{\mathbb{A}^{1},\mathrm{Nis}}\simeq
L_{\mathrm{Nis}}L_{\mathbb{A}^{1}}\gamma_{*}\gamma^{*}\simeq
L_{\mathrm{sZar}}L_{\mathbb{A}^{1}}\gamma_{*}\gamma^{*}$ (7.3)

on the $\infty$-categories $\mathrm{Pre}(k)$ and $\mathrm{Spt}(k)$, and
equivalences of endofunctors

$\Theta^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\simeq
L_{\mathrm{Nis}}^{\mathrm{fr}}L_{\mathbb{A}^{1}}\simeq
L_{\mathrm{sZar}}^{\mathrm{fr}}L_{\mathbb{A}^{1}}$ (7.4)

on the $\infty$-categories $\mathrm{Pre}^{\mathrm{fr}}(k)$ and
$\mathrm{Spt}^{\mathrm{fr}}(k)$.

###### Proof.

The first equivalences in (7.3) and (7.4) are provided by [14, Thereoms
3.5.14, 3.5.16]. Hence

$\Theta^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\mathcal{F}(E)\simeq
L_{\mathbb{A}^{1}}\mathcal{F}(E)$

for any $\mathcal{F}\in\mathrm{Pre}^{\mathrm{fr}}(k)$ and $0$-dimensional
reduced schemes $E\in\mathrm{EssSm}_{k}$ because the Nisnevich topology on the
small étale site over $E$ is trivial. Since the functors
$\Theta^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}$ and $L_{\mathbb{A}^{1}}$
land in $\mathbb{A}^{1}$-invariant objects, Theorem 7.10 implies there is an
equivalence

$\Theta^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\mathcal{F}(U)\simeq
L_{\mathbb{A}^{1}}\mathcal{F}(U)$

for any semi-local $U\in\mathrm{EssSm}_{k}$. This implies the second
isomorphism in (7.3). The second isomorphism in (7.4) follows by Proposition
7.9. ∎

The $\mathrm{zf}$-topology is generated by the Zariski topology and the
$\mathrm{tf}$-topology introduced in [7]. We also consider the
$\mathrm{szf}$-topology generated by the semi-Zariski topology and the
$\mathrm{tf}$-topology. That is, we have

$\mathrm{zf}=\mathrm{Zar}\cup\mathrm{tf},\quad\mathrm{szf}=\mathrm{sZar}\cup\mathrm{tf}.$

Next, we recall the reconstruction theorem [14, Theorem 3.5.12] and extend to
the $\mathrm{szf}$-topology an equivalence shown for the
$\mathrm{zf}$-topology in [7].

######  Theorem 7.12 ([14, Theorem 3.5.12], [7, Theorem 4.42]).

There are equivalences

$\mathbf{SH}_{\mathbb{A}^{1},\mathrm{Nis}}(B)\simeq\mathbf{SH}^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}(B)\simeq\mathbf{SH}^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{zf}}(B)\simeq\mathbf{SH}^{\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{szf}}(B)$

###### Proof.

The first equivalence is [14, Theorem 3.5.12], and the second one is [7,
Theorem 4.42]. To prove the third equivalence, we use induction on the Krull
dimension of $B$. Applying the localization theorem [7, Theorem 3.18] to the
$\mathrm{szf}$-topology, we may assume that $B$ is a field and conclude by
appealing to Theorem 7.11. ∎

Consider the subcategories of $\mathrm{tf}$-local objects
$\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)$ and
$\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)$ in the $\infty$-categories of
presheaves of spaces $\mathrm{Pre}^{\mathrm{fr}}(B)$ and of spectra
$\mathrm{Spt}^{\mathrm{fr}}(B)$ on the $\infty$-category
$\mathrm{Corr}^{\mathrm{fr}}(B)$. Consider the $\infty$-category of respective
bispectra
$\mathrm{Spt}^{\mathrm{fr},s,t}_{\mathrm{tf}}(B)=\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)[\mathbb{G}_{m}^{\wedge
1}]$, and the adjunctions of $\infty$-categories

$\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)\rightleftarrows\mathrm{Spt}^{\mathrm{fr},s,t}_{\mathrm{tf}}(B)\rightleftarrows\mathrm{Spt}^{\mathrm{fr},s,t}_{\mathbb{A}^{1},\mathrm{Nis}}(B).$
(7.5)

We denote by $\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}$
the unit of the composite adjunction of (7.5) and use the same notation for
the adjunction

$\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)\rightleftarrows\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B).$

######  Proposition 7.13.

There is a natural isomorphism of endofunctors

$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\simeq
L_{\mathrm{szf}}^{\mathrm{fr}}\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}$
(7.6)

on the $\infty$-categories $\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)$ and
$\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)$.

For any $\mathcal{F}\in\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)$ (resp.
$\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)$) there is an equivalence

$\gamma_{*}\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\mathcal{F}\simeq
L_{\mathrm{sZar}}\gamma_{*}\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}\mathcal{F}$
(7.7)

in $\mathrm{Pre}_{\mathrm{tf}}(B)$ (resp. $\mathrm{Spt}_{\mathrm{tf}}(B)$).

###### Proof.

Consider the case of (7.6) for the $\infty$-category
$\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)$. By [11, Theorem 15.10] we have

$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\simeq
L_{\mathrm{Nis}}^{\mathrm{fr}}\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}.$

Then, by [11, Theorem 15.4],
$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}$ preserves
Nisnevich local equivalences. Therefore,
$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}$ preserves
Zariski local equivalences, and hence also $\mathrm{szf}$-local equivalences.
Using [11, Theorem 15.4] we conclude that

$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{szf}}\simeq
L_{\mathrm{szf}}^{\mathrm{fr}}\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{tf}}$

Finally, since
$\mathbf{SH}^{\mathrm{fr}}_{\mathrm{Nis}}(B)\simeq\mathbf{SH}^{\mathrm{fr}}_{\mathrm{szf}}(B)$
by Theorem 7.12, there is an equivalence

$\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{Nis}}\simeq\Theta^{\mathrm{tf},\mathrm{fr}}_{\mathbb{A}^{1},\mathrm{szf}}.$

This implies (7.6) for $\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)$. We use
this to prove our claim for $\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)$:
Since the $\infty$-category $\mathrm{Corr}^{\mathrm{fr}}(B)$ is semi-additive,
see [14, Lemma 3.2.5], $\mathrm{Spt}^{\mathrm{fr}}(B)$ is equivalent to the
subcategory $\mathrm{Pre}^{\mathrm{fr}}(B)^{\mathrm{gp}}$ of group-like
objects in $\mathrm{Pre}^{\mathrm{fr}}(B)$, and, consequently,
$\mathrm{Spt}^{\mathrm{fr}}_{\mathrm{tf}}(B)\simeq\mathrm{Pre}^{\mathrm{fr}}_{\mathrm{tf}}(B)^{\mathrm{gp}}$.

The equivalence (7.7) follows from (7.6) by Proposition 7.9. ∎

Denote by $\Theta_{\mathbb{A}^{1},\mathrm{Nis}}$ the unit of the composite
adjunction of the sequence

$\mathrm{Pre}(B)\rightleftarrows\mathrm{Spt}(B)\rightleftarrows\mathrm{Spt}^{s,t}(B)\rightleftarrows\mathrm{Spt}^{s,t}_{\mathbb{A}^{1},\mathrm{Nis}}(B).$

For a scheme $X\in\mathrm{Sm}_{B}$, we denote by the same symbol the
representable presheaf in $\mathrm{Pre}(B)$. The standard loop-suspension
adjoint functors $\Sigma^{\infty}_{\mathbb{P}^{1}}$ and
$\Omega^{\infty}_{\mathbb{P}^{1}}$ yield the equivalence

$\Theta_{\mathbb{A}^{1},\mathrm{Nis}}X\simeq\Omega^{\infty}_{\mathbb{P}^{1}}\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+}.$

######  Theorem 7.14.

For any $X\in\mathrm{Sm}_{B}$, there are natural equivalences in
$\mathrm{Pre}(B)$

$\begin{array}[]{lcl}\Theta_{\mathbb{A}^{1},\mathrm{Nis}}X&\simeq&L_{\mathrm{sZar}}\varinjlim_{l}\Omega^{l}_{\mathbb{G}_{m}}(L_{\mathbb{A}^{1},\mathrm{tf}}\mathrm{Fr}(-,X_{+}\wedge\mathbb{G}_{m}^{\wedge
l}))^{\mathrm{gp}}\\\
&\simeq&L_{\mathrm{Zar}}\varinjlim_{l}\Omega^{l}_{\mathbb{G}_{m}}(L_{\mathbb{A}^{1},\mathrm{tf}}\mathrm{Fr}(-,X_{+}\wedge\mathbb{G}_{m}^{\wedge
l}))^{\mathrm{gp}}.\end{array}$ (7.8)

For any semi-local scheme $U\in\mathrm{EssSm}_{B}$, there is an equivalence of
spaces

$(\Theta_{\mathbb{A}^{1},\mathrm{Nis}}X)(U)\simeq\varinjlim_{l}\Omega^{l}_{\mathbb{G}_{m}}(L_{\mathbb{A}^{1},\mathrm{tf}}\mathrm{Fr}(U,X_{+}\wedge\mathbb{G}_{m}^{\wedge
l}))^{\mathrm{gp}}.$ (7.9)

###### Proof.

Proposition 7.13 shows the first equivalence in (7.8), cf. [11, Theorem
16.6]), which in turn implies the equivalence in (7.9). Moreover, the second
equivalence in (7.8) follows from (7.9) for local schemes
$U\in\mathrm{EssSm}_{B}$. ∎

Recall that
$\pi^{\mathbb{A}^{1},\mathrm{Nis}}_{i,j}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+})$
denotes the presheaf of stable motivic homotopy groups on $\mathrm{Sm}_{B}$
for $X\in\mathrm{Sm}_{B}$, see (1.10).

######  Theorem 7.15.

Suppose that $X\in\mathrm{Sm}_{B}$, $i<-\dim B+j$, and $j\in\mathbb{Z}$. Then,
for any essentially smooth semi-local $B$-scheme $U$, we have

$\pi^{\mathbb{A}^{1},\mathrm{Nis}}_{i,j}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+})(U)=0.$

Equivalently, the presheaf
$\pi^{\mathbb{A}^{1},\mathrm{Nis}}_{i,j}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+})$
is semi-Zariski locally trivial.

###### Proof.

Since by [11, Proposition 16.10] the functor $L_{\mathbb{A}^{1},\mathrm{tf}}$
takes connective objects to $(-\dim B)$-connective ones, the claim follows
from Theorem 7.14. ∎

##  8\. A family of non-acyclic Cousin complexes

In this section, we aim to show that the Cousin complex over a positive
dimensional base is generally non-acyclic. Throughout, the base scheme $B$ has
dimension $d>0$, and except for in Theorem 8.6 we assume that $B$ is local
with closed point $z$. We will work with

$\mathcal{F}=\Omega^{\infty}_{\mathbb{G}_{m}}\Sigma^{\infty}_{\mathbb{P}^{1}}(B/(B-z))\in\mathbf{SH}_{\mathbb{A}^{1},\mathrm{Nis}}^{s}(B).$
(8.1)

We aim to show that the associated Cousin complex
$\pi_{l}C^{\bullet}(B,\mathcal{F})$ is non-acyclic for
$l=\operatorname{codim}_{B}z$.

Given $U\in\mathrm{EssSm}_{B}$, a non-empty closed immersion
$Z\not\hookrightarrow B$, with $U_{Z}:=U\times_{B}Z$, we consider the
canonical map of motivic spaces

$U/(U-U_{Z})\to B/(B-Z)\to B/(B-z).$

We denote by $c_{Z}$ the corresponding element in

$[\Sigma^{\infty}_{\mathbb{P}^{1}}U/(U-U_{Z}),\Sigma^{\infty}_{\mathbb{P}^{1}}(B/(B-z))]_{\mathbf{SH}_{\mathbb{A}^{1},\mathrm{Nis}}^{\mathbb{P}^{1}}(B)}\cong\pi_{0}\mathcal{F}_{U_{Z}}(U)=\pi_{0}\mathcal{F}_{Z}(U).$

The isomorphism holds by (8.1), and the equality holds because
$\mathcal{F}_{Z}=\mathcal{F}_{U_{Z}}$ by Definition 3.10.

######  Lemma 8.1.

For any $U\in\mathrm{EssSm}_{B}$ such that $U\times_{B}z\neq\emptyset$, and
non-empty closed subscheme $Z$ in $B$, the element
$c_{Z}\in\pi_{0}\mathcal{F}_{Z}(U)$ is non-trivial.

###### Proof.

Base change along the closed embedding $z\not\hookrightarrow B$ takes $c_{Z}$
to the class in

$[\Sigma^{\infty}_{\mathbb{P}^{1}}(U_{z})_{+},\Sigma^{\infty}_{\mathbb{P}^{1}}z_{+}]_{\mathbf{SH}_{\mathbb{A}^{1},\mathrm{Nis}}^{s}(z)}$

defined by the canonical morphism of schemes $U_{z}\to z$, which is non-
trivial because $U_{z}\neq\emptyset$ by assumption. ∎

######  Lemma 8.2.

For any $U\in\mathrm{EssSm}_{B}$ and non-empty closed subscheme $Z$ in $B$,
there are stable equivalences of spectra

$\mathcal{F}_{Z-z}(U-U_{z})\simeq
0,\quad\mathcal{F}_{z}(U)\simeq\mathcal{F}_{Z}(U).$

###### Proof.

The first equivalence follows because

$\varinjlim_{\operatorname{codim}_{B}Z=d-1}\mathcal{F}_{Z-z}(U-U_{z})\simeq\varinjlim_{\operatorname{codim}_{B}Z=d-1}(u^{*}\mathcal{F})_{Z-z}(U-U_{z})\simeq
0,$

where $u\colon B-z\to B$, because $u^{*}\mathcal{F}\simeq
0\in\mathbf{SH}^{s}(B-z)$. The second equivalence follows from the first
because

$\mathcal{F}_{z}(U)\simeq\operatorname{fib}(\mathcal{F}_{Z}(U)\to\mathcal{F}_{Z-z}(U-U_{z})).$

∎

######  Proposition 8.3.

For any $U\in\mathrm{EssSm}_{B}$ such that $U\times_{B}z\neq\emptyset$,
consider the morphisms

$\partial\colon\bigoplus\limits_{\operatorname{codim}_{B}x=d-1}\pi_{l-(d-1)}\mathcal{F}_{x}(U)\to\bigoplus\limits_{\operatorname{codim}_{B}x=d}\pi_{l-d}\mathcal{F}_{x}(U)$
(8.2)

defined in (3.9). Then, for $l=d$, there is an isomorphism

$\operatorname{Coker}(\partial)\cong\pi_{0}\mathcal{F}_{z}(U).$ (8.3)

The element $c_{z}\in\pi_{0}\mathcal{F}_{z}(U)$ is non-trivial.

###### Proof.

Since $B$ is local, we have the isomorphisms

$\displaystyle\bigoplus\limits_{\operatorname{codim}_{B}x=d-1}\pi_{l-(d-1)}\mathcal{F}_{x}(U)$
$\displaystyle\cong\varinjlim_{\operatorname{codim}_{B}Z=d-1}\pi_{l-(d-1)}\mathcal{F}_{Z-z}(U-U_{z}),$
$\displaystyle\bigoplus\limits_{\operatorname{codim}_{B}x=d}\pi_{l-d}\mathcal{F}_{x}(U)$
$\displaystyle\cong\pi_{l-d}\mathcal{F}_{z}(U).$

Thus, the left-hand side of (8.2) is trivial by applying Lemma 8.2 to $Z=z$,
and (8.3) follows. The second claim follows by applying Lemma 8.1 to $Z=z$. ∎

######  Proposition 8.4.

For any $U\in\mathrm{EssSm}_{B}$ such that $U\times_{B}z\neq\emptyset$, the
morphism

$g\colon\pi_{l}\mathcal{F}(U)\to\bigoplus\limits_{\operatorname{codim}_{B}x=0}\pi_{l}\mathcal{F}_{x}(U)$
(8.4)

induces, for $l=0$, an isomorphism

$\operatorname{Ker}(g)\cong\pi_{0}\mathcal{F}(U).$ (8.5)

The element $c_{B}\in\pi_{0}\mathcal{F}(U)$ is non-trivial (under the
canonical equivalence $\mathcal{F}(U)\simeq\mathcal{F}_{B}(U)$).

###### Proof.

For $l=0$ the left-hand side of (8.4) is $\pi_{0}\mathcal{F}(U)$. The element
$c_{B}\in\pi_{0}\mathcal{F}(U)$ is non-trivial by Lemma 8.1 applied to $Z=B$.
The right-hand side of (8.4) is trivial by Lemma 8.2 applied to all closed
subschemes $Z$ in $B$ of codimension $1$. ∎

For any $U\in\mathrm{EssSm}_{B}$, we consider the $d+2$-term complex of
abelian groups

$\pi_{l}\mathrm{Cou}^{\bullet}_{\mathrm{tf}}(U,\mathcal{F})$

concentrated in degrees $-1,\dots,d$, see Definitions 3.11 and 3.13.

######  Theorem 8.5.

Suppose $B$ is a local scheme of Krull dimension $d>0$ with closed point $z$.
For any non-empty $U\in\mathrm{EssSm}_{B}$, the cohomology group

$H^{i}(\pi_{l}\mathrm{Cou}^{\bullet}_{\mathrm{tf}}(U,\mathcal{F}))\cong\begin{cases}\pi_{0}\mathcal{F}(U),&\text{if
}i=-1\text{ and }l=0,\\\ 0,&\text{if }i=0,\dots,d-1\text{ and
}l\in\mathbb{Z},\\\ \pi_{0}\mathcal{F}_{z}(U),&\text{if }i=d\text{ and
}l=d.\end{cases}$

The element $c_{B}\in\pi_{0}\mathcal{F}(U)$ defined by the canonical
equivalence $\mathcal{F}(U)\simeq\mathcal{F}_{B}(U)$ is non-trivial. Moreover,
$c_{z}\in\pi_{0}\mathcal{F}_{z}(U)$ is non-trivial.

###### Proof.

The claim for $i=0,\dots,d-1$ and $l\in\mathbb{Z}$ follows from Lemma 8.2. The
other claims follow from Propositions 8.4 and 8.3. ∎

######  Theorem 8.6.

Suppose $B$ is any base scheme of positive Krull dimension. Let
$U\in\mathrm{EssSm}_{B}$ be an essentially smooth local $B$-scheme. Assume the
image $z\in B$ of the closed point of $U$ has positive codimension in $B$.
Then

$H^{i}(\pi_{l}\mathrm{Cou}^{\bullet}(U,\mathcal{F}))\neq 0,\quad
i=\operatorname{codim}_{B}z,$

where $l=\operatorname{codim}_{B}z$,
$\mathcal{F}=\Omega^{\infty}_{\mathbb{G}_{m}}\Sigma^{\infty}_{\mathbb{P}^{1}}(B/(B-Z))$,
and $Z$ is the closure of $z$ in $B$.

###### Proof.

This follows by applying Theorem 8.5 to the local scheme $B_{(z)}$ of $B$ at
$z$. ∎

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to both the (finite) high-dimensional literature and the functional data
analysis literature open the door to many interesting avenues for future work,
which we outline in the following:

  * •

Inference for Covariance Operators: The techniques developed in this work, in
particular the dimension-free bounds for the sample covariance and variance
component functions open the door to lifting the finite high-dimensional
inference theory to the infinite dimensional setting. The review paper [10]
provides a detailed overview of the covariance inference literature in finite
dimensions. Our analysis can likely be used to extend this theory to the
covariance operator setting. To the best of our knowledge, the existing
literature on inference for covariance operators (see for example [34, 26])
does not account for potential sparse structure in the underlying operators,
though such structure can naturally be assumed in many cases of interest.

  * •

Estimating the Covariance Operator of a Multi-valued Gaussian Process: As
described in Section 1.1.2, the paper [21] considers covariance estimation for
multi-valued Gaussian processes under a sparsity assumption on the dependence
between the individual components of the process. In contrast to our approach
(see also Remark 2.2) their assumption does not impose any sparsity on each
component covariance function. It would be interesting therefore to extend our
analysis to the multi-valued functional data analysis setting where each
component satisfies a local-type sparsity constraint, such as belonging to
$\mathcal{K}^{*}_{q}$.

## Acknowledgments

The authors are grateful for the support of the NSF CAREER award DMS-2237628,
DOE DE-SC0022232, and the BBVA Foundation. The authors are also thankful to
Jiaheng Chen and Subhodh Kotekal for inspiring discussions.

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##  Appendix A Additional Numerical Simulations

This appendix shows random draws and results of operator estimation in $d=2$.
The experimental set-up is identical to the $d=1$ setting described in Section
4.2, however due to computational constraints in the higher dimensional
setting, we use different settings for the experimental parameters.
Specifically, our samples are generated by discretizing the domain
$D=[0,1]^{2}$ with a uniform mesh of $L=10,000$ points. We consider a total of
10 choices of $\lambda$ arranged uniformly in log-space and ranging from
$10^{-2.3}$ to $10^{-0.1}$. As in the $d=1$ case, the plots indicate that
adaptive thresholding is a significant improvement over both the sample
covariance and universal thresholding estimators in both the unweighted and
weighted settings.

Figure 4: Draws from a centered Gaussian process on $D=[0,1]^{2}$ with
covariance function SE in the first row, WSE($\alpha=0.1$) in the second and
WSE($\alpha=0.2$) in the third, with varying $\lambda$ parameter.  Figure 5:
Plots of the average relative errors and 95% confidence intervals achieved by
the sample ($\varepsilon$, dashed blue), universal thresholding
($\varepsilon^{U}_{\widehat{\rho}_{N}}$, red) and adaptive thresholding
($\varepsilon^{A}_{\widehat{\rho}_{N}}$, black) covariance estimators based on
a sample size ($N$, dotted green) for the (weighted) squared exponential
(left) and (weighted) Matérn (right) covariance functions in $d=2$ over 10
Monte-Carlo trials and 10 scale parameters $\lambda$ ranging from $10^{-2.3}$
to $10^{-0.1}$. The first row corresponds to the unweighted covariance
functions, the second and third rows correspond to the weighted variants with
$\alpha=0.1,0.2$ respectively.