concentrated in degrees $[0,r]$. The inclusion of the bottom term gives a map

(27) $\mathcal{O}_{\mathbb{P}_{X}V}\to\det\mathcal{V}(-r-1)[r].$

Since $p_{*}\Lambda^{i}\mathcal{V}(-i)=0$ for $i=1,\ldots,r$ this map induces
an equivalence

$p_{*}\mathcal{O}_{\mathbb{P}_{X}V}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}p_{*}\det\mathcal{V}(-r-1)[r]=p_{*}\mathcal{O}(-r-1)\otimes\det
V[r].$

Now recall that the unit map $\mathcal{O}_{X}\to
p_{*}p^{*}\mathcal{O}_{X}=p_{*}\mathcal{O}_{\mathbb{P}_{X}V}$ is an
equivalence by [Khan-blow-up]*Theorem 3.3 (i). Combining this with the above
equivalence we obtain an induced equivalence

(28) $\tau\colon
p_{*}\mathcal{O}(-r-1)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\det
V^{\vee}[-r].$

######  Lemma 6.1.3.

Suppose $n-r$ is odd. Then there is a split Poincaré-Verdier sequence

$\big{(}\mathcal{C},\Qoppa^{\mathrm{s}}_{p^{*}L(n)}|_{\mathcal{C}}\big{)}\to\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)}\xrightarrow{p_{*}(-\otimes\mathcal{O}(-\frac{n+r+1}{2}))}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}$

######  Remark 6.1.4.

When $n=-r-1$, the Poincaré-Verdier projection of Lemma 6.1.3 coincides with
the Poincaré functor

$(p_{*},\eta)\colon\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{!}L}\big{)}\to\big{(}\mathcal{D}^{\mathrm{p}}(X),L\big{)}$

of Lemma 5.1.5 under the identification

$p^{!}L=p^{*}L\otimes
p^{!}\mathcal{O}_{X}=p^{*}L\otimes\det\Omega_{\mathbb{P}_{X}V/X}=p^{*}L(-r-1),$

see Lemma A.6.2 and Remark 5.1.9.

###### Proof of Lemma 6.1.3.

Since $(\mathcal{A}(\frac{n+r+1}{2}),\mathcal{C})$ is a semi-orthogonal
decomposition we have a left split Verdier sequence

$\mathcal{C}\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)\to\mathcal{A}\big{(}\frac{n+r+1}{2}\big{)}\
.$

At the same time, the functor $p_{*}(-\otimes\mathcal{O}(-\frac{n+r+1}{2}))$
is trivial on $\mathcal{C}$ and restricts to an equivalence from
$\mathcal{A}(\frac{n+r+1}{2})$ to $\mathcal{D}^{\mathrm{p}}(X)$, and so
determines an equivalence between the above sequence and the sequence

$\mathcal{C}\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)\xrightarrow{p_{*}(-\otimes\mathcal{O}(-\frac{n+r+1}{2}))}\mathcal{D}^{\mathrm{p}}(X)\
.$

and so the latter is again a left-split Verdier sequence. As the first arrow
is a Poincaré-Verdier inclusion by definition, we can upgrade it to a left-
split Poincaré-Verdier sequence, of the form

$(\mathcal{C},\Qoppa^{\mathrm{s}}_{p^{*}L(n)}|_{\mathcal{C}})\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-\frac{n+r+1}{2}))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa)$

which is then automatically split, see [9-authors-II]*Observation 1.2.1. It
remains to compute the Poincaré structure $\Qoppa$. We know by [9-authors-
II]*Proposition 1.2.2 (i) that it is given by pre-composing
$\Qoppa^{\mathrm{s}}_{p^{*}L(n)}$ with the left adjoint
$p^{*}(-)\otimes\mathcal{O}(\frac{n+r+1}{2})$ of
$p_{*}(-\otimes\mathcal{O}(-\frac{n+r+1}{2}))$, and so we calculate

$\displaystyle\Qoppa(P)$
$\displaystyle=\mathrm{hom}_{\mathbb{P}_{X}V}(p^{*}P\otimes
p^{*}P\otimes\mathcal{O}(n+r+1),p^{*}L(n))^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle=\mathrm{hom}_{\mathbb{P}_{X}V}(p^{*}(P\otimes
P),p^{*}L(-r-1))^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle=\mathrm{hom}_{X}(P\otimes P,L\otimes
p_{*}\mathcal{O}(-r-1))^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle\stackrel{{\scriptstyle\tau}}{{=}}\mathrm{hom}_{X}(P\otimes
P,L\otimes\det V^{\vee}[-r])^{\mathrm{h}\mathrm{C_{2}}}$

where in last step we have used the equivalence (28) above. ∎

Let $\mathcal{L}\subseteq\mathcal{C}$ be the stable subcategory generated by
$\mathcal{A}(\frac{n+r^{\prime}}{2})\cup\cdots\cup\mathcal{A}(\lceil\frac{n+1}{2}\rceil)$.
Note that $\mathcal{L}$ has a semi-orthogonal decomposition of the form
$(\mathcal{A}(\frac{n+r^{\prime}}{2}),\dots,\mathcal{A}(\lceil\frac{n+1}{2}\rceil))$.
In particular, for every additive invariant $\mathcal{F}$ there’s an
equivalence

$\mathcal{F}^{\mathrm{hyp}}(\mathcal{L})\simeq\mathcal{F}^{\mathrm{hyp}}(\mathcal{D}^{\mathrm{p}}(X))^{\oplus\lfloor\frac{r^{\prime}+1}{2}\rfloor}$

induced by the functors
$\\{p^{*}(-)\otimes\mathcal{O}(i)\\}_{i=\frac{n+r^{\prime}}{2},\dots,\lceil\frac{n+1}{2}\rceil}$
(see [Khan-blow-up]*Lemma 2.8).

######  Lemma 6.1.5.

If $n$ is odd then the stable subcategory $\mathcal{L}\subseteq\mathcal{C}$ is
a Lagrangian with respect to the Poincaré structure
$\Qoppa^{\mathrm{s}}_{p^{*}L}|_{\mathcal{C}}$. If $n$ is even then
$\mathcal{L}$ is isotropic with respect to the Poincaré structure and its
homology inclusion can be identified with the Poincaré functor

$p^{*}(-)\otimes\mathcal{O}\big{(}\frac{n}{2}\big{)}\colon(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}).$

###### Proof.

Since
$(\mathcal{A}(\frac{n+r^{\prime}}{2}),\dots,\mathcal{A}(\frac{n-r^{\prime}}{2}))$
is a semi-orthogonal decomposition of $\mathcal{C}$ the inclusion
$\mathcal{L}\subseteq\mathcal{C}$ has a right adjoint
$r\colon\mathcal{C}\to\mathcal{L}$ whose kernel can be identified using Lemma
6.1.2 with either $\mathrm{D}_{p^{*}L(n)}\mathcal{L}$ if $n$ is odd or with
the stable subcategory generated by $\mathrm{D}_{p^{*}L(n)}\mathcal{L}$ and
$\mathcal{A}(\frac{n}{2})$ if $n$ is even. Hence by [9-authors-II]*Remark
3.2.3 and Lemma 6.1.2 the orthogonal complement $\mathcal{L}^{\perp}$ of
$\mathcal{L}$ is either $\mathcal{L}$ if $n$ is odd or the subcategory
generated by $\mathcal{L}$ and $\mathcal{A}(\frac{n}{2})$ if $n$ is even. In
particular, $\mathcal{L}\subseteq\mathcal{L}^{\perp}$, and since the Poincaré
structure is a symmetric one this suffices to prove that
$\Qoppa^{\mathrm{s}}_{p^{*}L(n)}|_{\mathcal{L}}=0$. Thus $\mathcal{L}$ is an
isotropic subcategory of $\mathcal{C}$ which is furthermore a Lagrangian when
$n$ is odd. Finally, when $n$ is even $\mathcal{L}^{\perp}$ admits a semi-
orthogonal decomposition of the form $(\mathcal{L},\mathcal{A}(\frac{n}{2})$
and so the duality invariant subcategory
$\mathcal{L}^{\perp}\cap\mathrm{D}_{p^{*}L(n)}\mathcal{L}^{\perp}=\mathcal{L}^{\perp}\cap\ker(r)$
coincides with $\mathcal{A}(\frac{n}{2})$. We may then identify the associated
homology inclusion with the resulting Poincaré functor

$\big{(}\mathcal{A}\big{(}\frac{n}{2}\big{)},\Qoppa^{\mathrm{s}}_{p^{*}L(n)}|_{\mathcal{A}(\frac{n}{2})}\big{)}\hookrightarrow\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)}.$

Now since the functor
$p^{*}(-)\otimes\mathcal{O}(\frac{n}{2})\colon\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
is fully-faithful with image $\mathcal{A}(\frac{n}{2})$ the last Poincaré
functor is equivalent to the Poincaré functor

$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa)\hookrightarrow\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)}$

where $\Qoppa$ is obtained by restricting $\Qoppa^{\mathrm{s}}_{p^{*}L(n)}$
along $p^{*}(-)\otimes\mathcal{O}(\frac{n}{2})$. We then compute

$\displaystyle\Qoppa(P)$
$\displaystyle=\mathrm{hom}_{\mathbb{P}_{X}V}(p^{*}P\otimes
p^{*}P\otimes\mathcal{O}(n),p^{*}L(n))^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle=\mathrm{hom}_{\mathbb{P}_{X}V}(p^{*}(P\otimes
P),p^{*}L)^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle=\mathrm{hom}_{X}(P\otimes P,L)^{\mathrm{h}\mathrm{C_{2}}}$
$\displaystyle=\Qoppa^{\mathrm{s}}_{L}(P)\ ,$

as desired. ∎

Putting the two lemmas together we obtain the projective bundle formula.

######  Theorem 6.1.6 (The projective bundle formula).

Let $X$ be a qcqs scheme, let $L$ be an invertible perfect complex over $X$
with $\mathrm{C_{2}}$-action and let $V$ be a vector bundle over $X$ of rank
$r+1$. Let $\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{E}$ be an
additive functor valued in some stable $\infty$-category $\mathcal{E}$. Then
the following holds

  1. (1)

If $n$ and $r$ are both odd then the exact functors
$p^{*}(-)\otimes\mathcal{O}(i)\colon\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
for $i=\frac{n+r}{2},\dots,\frac{n+1}{2}$ induce an equivalence

$\mathcal{F}^{\mathrm{hyp}}(\mathcal{D}^{\mathrm{p}}(X))^{\oplus(r+1)/2}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}).$

  2. (2)

If $n$ and $r$ are both even then the exact functors
$p^{*}(-)\otimes\mathcal{O}(i)\colon\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
for $i=\frac{n+r}{2},\dots,\frac{n+2}{2}$ together with the Poincaré functor
$p^{*}(-)\otimes\mathcal{O}(\frac{n}{2})\colon(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})$
induce an equivalence

$\mathcal{F}^{\mathrm{hyp}}(\mathcal{D}^{\mathrm{p}}(X))^{\oplus
r/2}\oplus\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}).$

  3. (3)

If $n$ is odd and $r$ is even then we have a fiber sequence

$\mathcal{F}^{\mathrm{hyp}}(\mathcal{D}^{\mathrm{p}}(X))^{\oplus
r/2}\to\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)}\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2}))}\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\ ,$

where the first arrow is induced by the functors
$p^{*}(-)\otimes\mathcal{O}(i)\colon\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
for $i=\frac{n+r-1}{2},\dots,\frac{n+1}{2}$.

  4. (4)

If $n$ is even and $r$ is odd then we have a fiber sequence

$\mathcal{F}^{\mathrm{hyp}}(\mathcal{D}^{\mathrm{p}}(X))^{\oplus(r-1)/2}\oplus\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\to\mathcal{F}(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2}))}\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\ ,$

where the first arrow is induced by the functors
$p^{*}(-)\otimes\mathcal{O}(i)\colon\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
for $i=\frac{n+r-1}{2},\dots,\frac{n+2}{2}$ together with the Poincaré functor
$p^{*}(-)\otimes\mathcal{O}(\frac{n}{2})\colon(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})$.

###### Proof.

Apply [9-authors-II]*Theorem 3.2.10 to the isotropic subcategory
$\mathcal{L}\subseteq\mathcal{C}$, together with Lemma 6.1.3, Lemma 6.1.5 and
[Khan-blow-up]*Lemma 2.8. ∎

Our goal now is to study the splitting of the fiber sequences ((4)) and ((3)).
In order to do so we will need to compute a special case of the boundary
operator. Suppose $r=\mathrm{rk}V-1$ is odd. Then the non-degenerate symmetric
bilinear form

$\wedge\colon\Lambda^{\frac{r+1}{2}}V[\frac{r+1}{2}]\otimes\Lambda^{\frac{r+1}{2}}V[\frac{r+1}{2}]\to\det
V[r+1]\,.$

determines a class $e(V)\in\operatorname{L}^{\mathrm{s}}_{0}(X,\det
V[r+1])=\operatorname{L}^{\mathrm{s}}_{-r-1}(X,\det V)$. We call it the _Euler
class_ of $V$. Note that $e(V)=0$ whenever $V$ has a quotient bundle of odd
rank [Walter]*Proposition 8.1, in particular whenever $V$ is trivial.

######  Lemma 6.1.7.

Suppose that $r$ is odd and $n$ is even. Then the boundary map

$\partial\colon\operatorname{L}^{\mathrm{s}}_{0}(X)\to\operatorname{L}^{\mathrm{s}}_{-1}(X,\det
V[r])=\operatorname{L}^{\mathrm{s}}_{-r-1}(X,\det V)$

of the fiber sequence of Theorem 6.1.6(4) for $\mathcal{F}=\operatorname{L}$
and $\operatorname{L}=\det V[r]$, sends 1 to $e(V)$.

###### Proof.

We first note that since $\operatorname{L}^{\mathrm{hyp}}=0$ the domain and
target of this boundary map indeed as indicated. Second, the parity conditions
on $n$ and $r$ imply that $n+r+1$ is even, and hence by Remark 6.1.1 we may
assume without loss of generality that $n=-r-1$ (this assumption will simplify
the notation in the following argument, though mathematically it makes no
essential difference). In particular, our Poincaré-Verdier projection takes
the form

$p_{*}\colon\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{!}\mathcal{O}_{X}}\big{)}=\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\mathrm{det}\mathcal{V}(-r-1)[r]}\big{)}\to(\mathcal{D}^{\mathrm{p}}(X),\mathcal{O}_{X}),$

see Remark 6.1.4. We now carry out the explicit procedure to produce the
boundary map as described in [9-authors-II]*Proposition 4.4.8. For this, we
need to find a hermitian object in
$\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\mathrm{det}\mathcal{V}(-r-1)[r]}\big{)}$
lifting the unit Poincaré object $\mathcal{O}_{X}$ in
$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{X})$. Since $p_{*}$ is a
Poincaré-Verdier projection there is a natural candidate for such a lift,
obtained by applying to $\mathcal{O}_{X}$ the left adjoint $p^{*}$, in which
case the hermitian structure map

$\Qoppa^{\mathrm{s}}_{\mathrm{det}\mathcal{V}(-r-1)[r]}(\mathcal{O}_{\mathbb{P}_{X}V})=\Qoppa^{\mathrm{s}}_{\mathrm{det}\mathcal{V}(-r-1)[r]}(p^{*}\mathcal{O}_{X})\to\Qoppa^{\mathrm{s}}_{X}(\mathcal{O}_{X})$

is an equivalence. Let
$q\in\Omega^{\infty}\Qoppa^{\mathrm{s}}_{\mathrm{det}\mathcal{V}(-r-1)[r]}(\mathcal{O}_{\mathbb{P}_{X}V})$
be the hermitian form corresponding to
$\mathrm{id}\in\Omega^{\infty}\Qoppa^{\mathrm{s}}_{X}(\mathcal{O}_{X})$ under
the above equivalence. To calculate the boundary of $q$, we first note that
the induced map of spaces

$\operatorname{Map}_{\mathbb{P}_{X}V}\big{(}\mathcal{O}_{\mathbb{P}_{X}V},\mathrm{D}_{\mathrm{det}\mathcal{V}(-r-1)[r]}\mathcal{O}_{\mathbb{P}_{X}V}\big{)}\to\operatorname{Map}_{X}(\mathcal{O}_{X},\mathrm{D}_{X}\mathcal{O}_{X})$

is already an equivalence before taking $\mathrm{C_{2}}$-homotopy fixed
points, and that the second, and hence also the first, are discrete spaces. In
particular, $q$ is determined up to homotopy by the associated map

$q_{\sharp}\colon
p^{*}\mathcal{O}_{X}=\mathcal{O}_{\mathbb{P}_{X}V}\to\mathrm{D}_{\mathrm{det}\mathcal{V}(-r-1)[r]}\mathcal{O}_{\mathbb{P}_{X}V}=\mathrm{det}\mathcal{V}(-r-1)[r],$

which in turn is determined up to homotopy by the condition that it maps to
the identity map $\mathcal{O}_{X}\to\mathcal{O}_{X}$ by $p_{*}$, under the
identifications $\mathcal{O}_{X}=p_{*}p^{*}\mathcal{O}_{X}$ and
$p_{*}\mathrm{det}\mathcal{V}(-r-1)[r]=\mathcal{O}_{X}$ determines by the unit
and the map $\tau$ of (28) used to define to Poincaré structure on $p_{*}$. By
the construction of $\tau$ the map $q_{\sharp}$ must then be homotopy
equivalent to the map

$\mathcal{O}_{\mathbb{P}_{X}V}\to\det\mathcal{V}(-r-1)[r]$

of (27), determined by the inclusion of the bottom term in the complex

$\Lambda^{r}\mathcal{V}(-r)\to\cdots\to\Lambda^{1}\mathcal{V}(-1)\to\mathcal{O}_{\mathbb{P}_{X}V}\,.$

Unwinding the definitions, we then see that the boundary of the hermitian
object $(\mathcal{O}_{\mathbb{P}_{X}V},q)$ is the object
$K\in\mathcal{C}\subseteq\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V)$
represented by the complex

$\Lambda^{r}\mathcal{V}(-r)\to\cdots\to\Lambda^{1}\mathcal{V}(-1)$

concentrated in degrees $[1,r]$, with the Poincaré form
$\beta\in\Qoppa^{\mathrm{s}}_{\det\mathcal{V}[r+1](-r-1)}(K)$ given by the
collection of wedge pairings

$\Lambda^{i}\mathcal{V}(-i)\otimes\Lambda^{r+1-i}\mathcal{V}(-r-1+i)\to\Lambda^{r+1}\mathcal{V}(-r-1).$

Note that the subcomplex of $K$ given by

$\Lambda^{\frac{r-1}{2}}\mathcal{V}\big{(}-\frac{r-1}{2}\big{)}\to\cdots\to\Lambda^{1}\mathcal{V}(-1)$

lives in $\mathcal{L}$ and so it forms canonically a surgery datum (since
$\mathcal{L}$ is isotropic). The result of the surgery is exactly
$\Lambda^{\frac{r+1}{2}}\mathcal{V}[\frac{r+1}{2}]\big{(}\frac{-r-1}{2}\big{)}=\Lambda^{\frac{r+1}{2}}\mathcal{V}[\frac{r+1}{2}]\big{(}\frac{n}{2}\big{)}$
with bilinear form given by the wedge pairing

$\Lambda^{\frac{r+1}{2}}\mathcal{V}[\frac{r+1}{2}]\big{(}\frac{n}{2}\big{)}\otimes\Lambda^{\frac{r+1}{2}}\mathcal{V}[\frac{r+1}{2}]\big{(}\frac{n}{2}\big{)}\to\Lambda^{r+1}\mathcal{V}[r+1](n).$

that is, it’s the pullback of $e(V)$ along
$p^{*}(-)\otimes\mathcal{O}\big{(}\frac{n}{2}\big{)}$, as required. ∎

######  Corollary 6.1.8.

Suppose $n+r$ is odd. If either $r$ is even or $r$ is odd and $e(V)=0$ then
for every $L\in{\mathcal{P}ic}(X)^{{\mathrm{h}\mathrm{C_{2}}}}$ the Poincaré
functor

$(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2}))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]})$

admits a Poincaré section. In particular, in these cases the fiber sequence in
Items (3) and (4) of Theorem 6.1.6 split.

###### Proof.

Assume that $n+r$ is odd. We claim that if either $r$ is even or $r$ is odd
and $e(V)=0$ then $1\in\operatorname{L}^{\mathrm{s}}_{0}(X)$ belongs to the
image of the map

$\operatorname{L}^{\mathrm{s}}_{0}(\mathbb{P}_{X}V,\det\mathcal{V}(n)[r])\xrightarrow{p_{*}((-)\otimes\mathcal{O}(\frac{n+r+1}{2}))}\operatorname{L}^{\mathrm{s}}_{0}(X)\
.$

Indeed when $r$ is even this map is an equivalence by Theorem 6.1.6(3) and
when $r$ is odd the result follows from Lemma 6.1.7. This means that there
exists a Poincaré object $(x,q)$ in
$(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\det\mathcal{V}(n)[r]})$
whose image under $p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2}))$ is
cobordant to $(\mathcal{O}_{x},\mathrm{id})$ (where
$\mathrm{id}\in\Omega^{\infty}\Qoppa^{\mathrm{s}}_{X}(\mathcal{O}_{X})$ is the
identity form on $\mathcal{O}_{X}$). Now since
$p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2}))$ is a split Poincaré-Verdier
projection [9-authors-II]*Theorem 2.5.3 tells that the induced functor on
cobordism categories is bicartesian fibration, and hence we can lift any
cobordism between $p_{*}(x\otimes\mathcal{O}(-\frac{n+r+1}{2}))$ (with its
induced form) and $(\mathcal{O}_{X},\mathrm{id})$ to a cobordism between
$(x,q)$ and some Poincaré object $(x^{\prime},q^{\prime})$ in
$\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\det\mathcal{V}(n)[r]}\big{)}$
lying above $(\mathcal{O}_{X},\mathrm{id})$. In particular,
$p_{*}(x^{\prime}\otimes\mathcal{O}(-\frac{n+r+1}{2}))\simeq\mathcal{O}_{X}$.

We are now going to use the Poincaré object $(x^{\prime},q^{\prime})$ in order
to construct the desired Poincaré section. For this, note that the symmetric
monoidal structure maps of the top horizontal functor in (12) determine in
particular a Poincaré functor

$(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\det\mathcal{V}(n)[r]})\otimes_{(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{X})}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)})$

which is an equivalence on underlying stable $\infty$-categories with duality.
In particular, the Poincaré object $(x^{\prime},q^{\prime})$ in
$(\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{\det\mathcal{V}(n)[r]})$
determines a $(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{X})$-linear
Poincaré functor

$F_{(x^{\prime},q^{\prime})}\colon\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\to\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)},$

whose underlying exact functor is $F_{x^{\prime}}(-)=p^{*}(-)\otimes
x^{\prime}$. Consider the composite Poincaré functor

$G_{(x^{\prime},q^{\prime})}\colon\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\xrightarrow{F_{(x^{\prime},q^{\prime})}}\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L(n)}\big{)}\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-\frac{n+r+1}{2})}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}.$

On the level of underlying exact functors this composite is given by

$z\mapsto p_{*}\big{(}p^{*}(z)\otimes
x^{\prime}\otimes\mathcal{O}\big{(}-\frac{n+r+1}{2}\big{)}\big{)}=z\otimes
p_{*}\big{(}x^{\prime}\otimes\mathcal{O}\big{(}-\frac{n+r+1}{2}\big{)}\big{)}=z,$

where the first equivalence is by the projection formula and the second by the
construction of $x^{\prime}$. We conclude that $G_{(x^{\prime},q^{\prime})}$
is an equivalence on underlying stable $\infty$-categories, and hence on the
level of underlying stable $\infty$-categories with duality. Since
$\Qoppa^{\mathrm{s}}_{L\otimes\det V^{\vee}[-r]}$ is the symmetric Poincaré
structure associated to the underlying duality we conclude that
$G_{(x^{\prime},q^{\prime})}$ is an equivalence of Poincaré
$\infty$-categories. In fact, one can show that $G_{(x^{\prime},q^{\prime})}$
is homotopic to the identity Poincaré functor on
$\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}$, so that $F_{(x^{\prime},q^{\prime})}$ gives the desired
section, though we will not have to prove this: even if
$G_{(x^{\prime},q^{\prime})}$ is merely some equivalence, we get a section by
taking $F_{(x^{\prime},q^{\prime})}\circ G_{(x^{\prime},q^{\prime})}^{-1}$.

∎

###  6.2. Genuine refinement for the projective line

In §6.1 we have proven a general projective bundle formula for the symmetric
Poincaré structure. In this section, we want to address the analogous question
for the genuine Poincaré structures $\Qoppa^{\geq m}$. For simplicity, and
since it is the only case we will need to use, we will consider only the case
$V=\mathcal{O}^{\oplus 1}$ and $m=0$, so that
$\mathbb{P}_{X}V=X\times\mathbb{P}^{1}$ and $r=1$. In this case, the
projective bundle formula of Theorem 6.1.6 (4) provides a Poincaré-Verdier
sequence

(29)
$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\xrightarrow{p^{*}}(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L[-1]})\,.$

Our goal in this subsection is to prove the following refinement of this
result:

######  Proposition 6.2.1.

For every $m\in\mathbb{Z}\cup\\{-\infty,\infty\\}$ the sequence (29) refines
to a split Poincaré-Verdier sequence

$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq
m}_{L})\xrightarrow{p^{*}}(\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}\times
X),\Qoppa^{\geq
m}_{p^{*}L})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq
m-1}_{L[-1]}).$

Before going into the proof of Proposition 6.2.1, we note that the underlying
Verdier sequence of (29) induces for every
$M\in\mathcal{D}^{\mathrm{qc}}(X\times\mathbb{P}^{1})$ a fiber sequence of the
form

(30) $p^{*}p_{*}M\to M\to p^{*}p_{*}(M(-1))\otimes\Sigma\mathcal{O}(-1)\ ,$

also known as the Beilinson sequence. The proof of Proposition 6.2.1 will then
be based on the following lemma:

######  Lemma 6.2.2.

  1. (1)

There are natural equivalences
$p_{*}\mathcal{E}_{p^{*}L}\simeq\mathcal{E}_{L}$ and
$p_{*}\mathcal{E}_{p^{*}L}(-1)\simeq\Sigma^{-1}\mathcal{E}_{L}$. In
particular, the Beilinson sequence (30) for $M=\mathcal{E}_{p^{*}L}$ has the
form

$p^{*}\mathcal{E}_{L}\to\mathcal{E}_{p^{*}L}\to
p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-1).$

  2. (2)

The above exact sequence splits Zariski locally on $\mathbb{P}^{1}\times X$.

  3. (3)

For every $m\in\mathbb{Z}$ the associated sequence of $m$-connective covers

$\tau_{\geq m}(p^{*}\mathcal{E}_{L})\to\tau_{\geq
m}(\mathcal{E}_{p^{*}L})\to\tau_{\geq
m}(p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-1))$

remains exact in $\mathcal{D}^{\mathrm{qc}}(\mathbb{P}^{1}\times X)$.

###### Proof.

We first prove (1). In the split Poincaré-Verdier sequence (29), the Verdier
inclusion $p^{*}$ has a right adjoint $p_{*}$ and a left adjoint
$p_{*}((-)\otimes\Sigma\mathcal{O}(-2))$, and the Poincaré projection
$p_{*}((-)\otimes\mathcal{O}(-1))$ has a left adjoint
$p^{*}(-)\otimes\mathcal{O}(1)$ and a right adjoint
$p^{*}(-)\otimes\Sigma\mathcal{O}(-1)$. For $M\in\mathcal{D}^{\mathrm{qc}}(X)$
we consequently have

$\displaystyle\mathrm{hom}_{X}(M,p_{*}\mathcal{E}_{p^{*}L})$
$\displaystyle=\mathrm{hom}_{\mathbb{P}^{1}\times
X}(p^{*}M,\mathcal{E}_{p^{*}L})$
$\displaystyle=\mathrm{hom}_{\mathbb{P}^{1}\times X}(p^{*}M\otimes
p^{*}M,p^{*}L)^{{\mathrm{t}\mathrm{C_{2}}}}$
$\displaystyle=\mathrm{hom}_{X}(M\otimes M,L)^{{\mathrm{t}\mathrm{C_{2}}}}$
$\displaystyle=\mathrm{hom}_{X}(M,\mathcal{E}_{L})\ ,$

and

$\displaystyle\mathrm{hom}_{X}(M,p_{*}\mathcal{E}_{p^{*}L}(-1))$
$\displaystyle=\mathrm{hom}_{\mathbb{P}^{1}\times
X}(p^{*}M\otimes\mathcal{O}(1),\mathcal{E}_{p^{*}L})$
$\displaystyle=\mathrm{hom}_{\mathbb{P}^{1}\times X}(p^{*}M\otimes
p^{*}M,p^{*}L\otimes\mathcal{O}(-2))^{{\mathrm{t}\mathrm{C_{2}}}}$
$\displaystyle=\mathrm{hom}_{X}(M\otimes M,L\otimes
p_{*}\mathcal{O}(-2))^{{\mathrm{t}\mathrm{C_{2}}}}$
$\displaystyle=\mathrm{hom}_{X}(M,\Sigma^{-1}\mathcal{E}_{L})\ .$

This establishes (1).

For Claim (2), the statement is local in $X$ and hence we may as well assume
that $X=\mathrm{spec}(R)$ is affine. We will prove that the sequence in
question splits when restricted to $\mathbb{A}^{1}_{R}=\mathbb{A}^{1}\times R$
for any embedding $j\colon\mathbb{A}^{1}_{R}\hookrightarrow\mathbb{P}^{1}_{R}$
over $\mathrm{spec}(R)$. Since $\mathbb{P}^{1}_{R}$ can be covered by (two)
such copies of $\mathbb{A}^{1}_{R}$ this implies the desired result. Now for
an open embedding $j\colon\mathbb{A}^{1}_{R}\hookrightarrow\mathbb{P}^{1}_{R}$
it follows from Corollary 4.2.12 (applied to the $\geq-\infty=\mathrm{s}$
decoration) and the fact that left Kan extensions commute with taking linear
parts that $\mathcal{E}_{j^{*}p^{*}L}=j^{*}\mathcal{E}_{p^{*}L}$. Wring
$f=pj\colon\mathbb{A}^{1}_{R}\to\mathrm{spec}(R)$ we are reduced to showing
that the comparison map

$f^{*}\mathcal{E}_{L}\to\mathcal{E}_{f^{*}T}$

admits a retraction in $\mathcal{D}^{\mathrm{qc}}(\mathbb{A}^{1}_{R})$. Making
this map explicit, we note that $L$ is an invertible $R$-module with an
$R$-linear $\mathrm{C_{2}}$-action and $\mathcal{E}_{L}$ is the object of
$\mathcal{D}(R)$ corresponding to the $\mathrm{H}(R)$-module spectrum
$\mathrm{H}(L)^{{\mathrm{t}\mathrm{C_{2}}}}$, with $\mathrm{H}(R)$ acting via
the Tate Frobenius
$\mathrm{H}(R)\to\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$ (the
$\mathrm{C_{2}}$ action on $\mathrm{H}(R)$ being the trivial one). On the
level of $\mathrm{H}(R[t])$-module spectra the above comparison map can then
be written as the $\mathrm{H}(R[t])$-linear map

(31)
$\mathrm{H}(L)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}(R)}\mathrm{H}(R[t])\to\mathrm{H}(L\otimes_{R}R[t])^{{\mathrm{t}\mathrm{C_{2}}}}.$

induced by the $\mathrm{H}(R)$-linear map
$\mathrm{H}(L)^{{\mathrm{t}\mathrm{C_{2}}}}\to\mathrm{H}(L\otimes_{R}R[t])^{{\mathrm{t}\mathrm{C_{2}}}}$.
We want to construct a retraction of (31). For this we first reduce to the
case where $L=R$ with constant $\mathrm{C_{2}}$-action. Consider the
commutative square of ${\mathrm{E}_{\infty}}$-ring spectra

${\mathrm{H}(R)}$${\mathrm{H}(R[t])}$${\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}}$${\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}}$

where the vertical maps are the corresponding Tate Frobenius maps. This square
determines a map of ${\mathrm{E}_{\infty}}$-ring spectra

(32)
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}(R)}\mathrm{H}(R[t])\to\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}.$

At the same time, the map (31) factors as a composite

$\mathrm{H}(L)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}(R)}\mathrm{H}(R[t])\to(\mathrm{H}L)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}}\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}\to\mathrm{H}(L\otimes_{R}R[t])^{{\mathrm{t}\mathrm{C_{2}}}},$

where the first map is induced by the various Tate Frobenius maps and the
second map by the lax monoidal structure of
$(-)^{{\mathrm{t}\mathrm{C_{2}}}}$. We then observe that the second map is an
equivalence: indeed, since the Tate construction commutes with infinite direct
sums of uniformly truncated objects we have that
$\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}$ is an infinite direct sum of
copies of $\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$ indexed by the
monomials $t^{i}$ and
$\mathrm{H}(L\otimes_{R}R[t])^{{\mathrm{t}\mathrm{C_{2}}}}$ is an infinite
direct sum of copies of $\mathrm{H}(L)^{{\mathrm{t}\mathrm{C_{2}}}}$ indexed
in a corresponding manner. We then see that the map (31) is obtained from (32)
by tensoring with $(\mathrm{H}L)^{{\mathrm{t}\mathrm{C_{2}}}}$ over
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$. It will hence suffice to show
that the map (32) admits a
$[\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}(R)}\mathrm{H}(R[t])]$-linear
retraction.

Explicitly,
$(\mathrm{H}R)^{{\mathrm{t}\mathrm{C_{2}}}}\otimes_{\mathrm{H}R}\mathrm{H}(R[t])$
is an infinite direct sum of copies of
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$, indexed by the monomials
$t^{i}$, where the multiplication is determines by the
${\mathrm{E}_{\infty}}$-ring structure of
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$ and the rule
$t^{i}t^{j}=t^{i+j}$. As above, since $(-)^{{\mathrm{t}\mathrm{C_{2}}}}$
commutes with uniformly truncated infinite direct sums the underlying spectrum
of $\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}$ has the exact same form,
and the exact same multiplication law. The map (32) hence corresponds to the
endomorphism

(33) $\bigoplus_{i\geq 0}\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle
t^{i}\rangle\to\bigoplus_{i\geq
0}\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{i}\rangle$

induced by the Tate Frobenius of $\mathrm{H}(R[t])$. We claim that this map
identifies the $\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle
t^{i}\rangle$ factor on the left with the
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{2i}\rangle$ factor on
the right. Indeed, since this map is
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$ linear it will suffice to check
that it sends the component of
$\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{i}\rangle$ corresponding
to the unit of $\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}$ to the component
of $\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{2i}\rangle$
corresponding to the same unit. Now since the $\mathrm{C_{2}}$-action on $R$
is trivial we have that
$\pi_{0}\mathrm{H}(R[t])^{{\mathrm{t}\mathrm{C_{2}}}}=\hat{H}^{0}(R[t])=R[t]/2=(R/2)[t]$,
and the Tate Frobenius is the map

$R[t]\to(R/2)[t]\quad\quad x\mapsto[x^{2}]$

on the level of $\pi_{0}$. In particular, it sends $t^{i}\in R[t]$ to
$t^{2i}\in(R/2)[t]$, as needed. We conclude that, as an $\bigoplus_{i\geq
0}\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{i}\rangle$-module, the
right hand side of (33) splits as a direct sum of two copies of
$\bigoplus_{i\geq 0}\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle
t^{i}\rangle$ (one spanned by the even powers of $t$ and one by the odd
powers), such that the map (32) is the inclusion of the even summand. It
follows that this map admits an $\bigoplus_{i\geq
0}\mathrm{H}(R)^{{\mathrm{t}\mathrm{C_{2}}}}\langle t^{i}\rangle$-linear
retraction, as desired.

Finally, to prove (3), we note that for a sequence of quasi-coherent sheaves
$M\to N\to P$ the comparison map $\eta_{m}\colon\operatorname{cof}[\tau_{\geq
m}M\to\tau_{\geq m}N]\to\tau_{\geq m}P$ is an isomorphism on all homotopy
sheaves $\pi_{i}$ for $i\neq m$ by the 5-lemma, while the induced map on
$\pi_{m}$ is injective. But if the sequence splits Zariski locally then the
map $N\to P$ is surjective on homotopy sheaves and hence the map $\eta_{m}$ is
an equivalence. ∎

###### Proof of Proposition 6.2.1.

We first note that the exact functor $p^{*}$ underlying the Poincaré-Verdier
inclusion in (29) is t-exact, and in particular right t-exact, and hence the
Poincaré-Verdier inclusion

$(p^{*},\eta^{\mathrm{s}})\colon(\mathcal{D}^{\mathrm{qc}}(X),\Qoppa^{\mathrm{s}}_{L})\to(\mathcal{D}^{\mathrm{qc}}(\mathbb{P}^{1}\times
X),\Qoppa^{\mathrm{s}}_{p^{*}L})$

induces a Poincaré functor

$(p^{*},\eta^{\geq m})\colon(\mathcal{D}^{\mathrm{qc}}(X),\Qoppa^{\geq
m}_{L})\to(\mathcal{D}^{\mathrm{qc}}(\mathbb{P}^{1}\times X),\Qoppa^{\geq
m}_{p^{*}L})\ ,$

see §3.3. To check that this Poincaré functor remains a Poincaré-Verdier
inclusion one needs to check that the Poincaré structure map

$\eta^{\geq m}\colon\Qoppa^{\geq m}_{L}\Rightarrow\Qoppa^{\geq
m}_{p^{*}L}(p^{*}(-))$

is an equivalence. This map is already known to be an equivalence on bilinear
parts (since these are the same as those of the corresponding symmetric
Poincaré structures), and hence it will suffice to verify that $\eta^{\geq m}$
is an equivalence on linear parts. Unwinding the definitions and using that
$p^{*}$ is t-exact this amounts to the composed map

$\tau_{\geq
m}(\mathcal{E}_{L})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}p_{*}p^{*}\tau_{\geq
m}(\mathcal{E}_{L})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}p_{*}\tau_{\geq
m}(p^{*}\mathcal{E}_{L})\to p_{*}\tau_{\geq m}(\mathcal{E}_{p^{*}L})$

being an equivalence in $\mathcal{D}^{\mathrm{qc}}(X)$, and hence by Lemma
6.2.2 to the vanishing of $p_{*}\tau_{\geq
m}(p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-1))$. To see that this last term
indeed vanishes, note that the natural map $\tau_{\geq
m}(p^{*}\mathcal{E}_{L})\otimes\mathcal{O}(-1)\to\tau_{\geq
m}(p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-1))$ is an equivalence since it is
an equivalence Zariski locally, and hence the push-forward of the latter
quasi-coherent sheaf to $X$ vanishes.

Now on the projection side, since $p_{*}((-)\otimes\mathcal{O}(-1))$ is a
split Verdier projection it automatically refines to a split Poincaré-Verdier
projection

$(\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}\times X),\Qoppa^{\geq
m}_{p^{*}L})\to(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq m}_{p^{*}L},\Phi)$

where $\Phi$ is obtained by precomposing $\Qoppa^{\geq m}_{p^{*}L}$ with the
left adjoint of $p_{*}((-)\otimes\mathcal{O}(-1))$, which is given by
$p^{*}(M)\otimes\mathcal{O}(1)$. The natural transformation $\Qoppa^{\geq
m}_{p^{*}L}\Rightarrow\Qoppa^{\mathrm{s}}_{p^{*}L}$ then induces a natural
transformation

$\tau\colon\Phi\Rightarrow\Qoppa^{\mathrm{s}}_{L[-1]}\ ,$

which is an equivalence on bilinear parts. It will hence suffice to check that
$\tau$ is an $(m-1)$-connective cover on linear parts. Now, by Lemma 6.2.2, we
get that the map

$\tau_{\geq m}(\mathcal{E}_{p^{*}L})\to\tau_{\geq
m}(p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-1))=\tau_{\geq
m}(p^{*}\mathcal{E}_{L})\otimes\mathcal{O}(-1)$

induces an equivalence on mapping spectra with domain of the form
$p^{*}M\otimes\mathcal{O}(1)$ for $M\in\mathcal{D}^{\mathrm{p}}(X)$ (indeed,
the fibre of this map is the constant sheaf $\tau_{\geq
m}p^{*}\mathcal{E}_{L}=p^{*}\tau_{\geq m}\mathcal{E}_{L}$). We hence get that

$\displaystyle\Lambda_{\Phi}(M)$
$\displaystyle=\mathrm{hom}(p^{*}M\otimes\mathcal{O}(1),\tau_{\geq
m}\mathcal{E}_{p^{*}L})$
$\displaystyle=\mathrm{hom}(p^{*}M\otimes\mathcal{O}(1),\tau_{\geq
m}(p^{*}\mathcal{E}_{L})\otimes\mathcal{O}(-1))$
$\displaystyle=\mathrm{hom}(M,p_{*}(\tau_{\geq
m}p^{*}\mathcal{E}_{L}\otimes\mathcal{O}(-2))),$
$\displaystyle=\mathrm{hom}(M,p_{*}(p^{*}\tau_{\geq
m}\mathcal{E}_{L}\otimes\mathcal{O}(-2))),$
$\displaystyle=\mathrm{hom}(M,\Sigma^{-1}\tau_{\geq m}\mathcal{E}_{L}),$

and the map $\Sigma^{-1}\tau_{\geq
m}\mathcal{E}_{L}\to\Sigma^{-1}\mathcal{E}_{L}$ is indeed an
$(m-1)$-connective cover, as desired. ∎

###  6.3. Homotopy invariance of symmetric Grothendieck-Witt

As an application of the projective bundle formula (Theorem 6.1.6) and
dévissage (Theorem 5.2.1), we show here that the symmetric Grothendieck-Witt
spectrum is $\mathbb{A}^{1}$-invariant over a regular Noetherian base of
finite Krull dimension. Here, it does not matter if we consider Grothendieck-
Witt or Karoubi-Grothendieck-Witt theory: when $X$ is regular the canonical
map $\operatorname{GW}^{\mathrm{s}}(X)\to\mathbb{GW}^{\mathrm{s}}(X)$ is an
equivalence. Indeed, by Karoubi cofinality (3), this follows from the map
$\operatorname{K}(X)\to\mathbb{K}(X)$ being an equivalence, which is a well-
known consequence of $X$ being regular, see [thomason-trobaugh]*Proposition
6.8, or in more modern language of [theorem-of-the-heart]*Theorem 1.2.

######  Theorem 6.3.1 ($\mathbb{A}^{1}$-invariance).

Let $X$ be a regular Noetherian scheme of finite Krull dimension. Then the
pullback map

${\operatorname{GW}^{\mathrm{s}}(\mathcal{D}^{\mathrm{p}}(X))}$${\operatorname{GW}^{\mathrm{s}}(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{A}^{1}))}$${\mathbb{GW}^{\mathrm{s}}(\mathcal{D}^{\mathrm{p}}(X))}$${\mathbb{GW}^{\mathrm{s}}(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{A}^{1}))}$

is an equivalence.

###### Proof.

Write $p\colon X\times\mathbb{P}^{1}\to X$ and $q\colon
X\times\mathbb{A}^{1}\to X$ for the corresponding projections. Consider the
commutative diagram of Poincaré $\infty$-categories

${(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{A}^{1}),\Qoppa^{\mathrm{s}}_{q^{*}L})}$${(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})}$${(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L})}$${(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L[-1]})}$${(\mathcal{D}^{\mathrm{p}}_{X\times\\{\infty\\}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L})}$$\scriptstyle{p^{*}}$$\scriptstyle{q^{*}}$$\scriptstyle{p_{*}(-\otimes\mathcal{O}(-1))}$$\scriptstyle{\varphi}$

where the horizontal sequence is the fiber sequence of Theorem 6.1.6(4) for
$V=\mathcal{O}^{\oplus 2}$, and the vertical sequence is the localisation
sequence of Corollary 4.2.12 associated to the closed subscheme inclusion
$X\times\\{\infty\\}\subseteq X\times\mathbb{P}^{1}$. Since both the vertical
and horizontal sequences are Karoubi-Poincaré sequence we see that $q^{*}$
induces an equivalence on $\mathbb{GW}$ if and only if $\varphi$ induces an
equivalence on $\mathbb{GW}$; indeed, both statements are equivalent to the
map

$\mathbb{GW}(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L})\to\mathbb{GW}(\mathcal{D}^{\mathrm{p}}(X\times\mathbb{A}^{1}),\Qoppa^{\mathrm{s}}_{q^{*}L})\oplus\mathbb{GW}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L[-1]})$

induced by the right horizontal and top vertical arrows being an equivalence.
Now to show that $\varphi$ induces an equivalence on $\mathbb{GW}$ it will
suffice to show that the composite Poincaré functor

$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{i^{!}p^{*}L})\xrightarrow{i_{*}}(\mathcal{D}^{\mathrm{p}}_{X\times\\{\infty\\}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L})\xrightarrow{\varphi}(\mathcal{D}^{\mathrm{p}}(X),\Sigma^{-1}\Qoppa^{\mathrm{s}}_{L})$

is an equivalence, where the functor on the left is the dévissage Poincaré
functor constructed in §5.4, and which is an equivalence by Theorem 5.2.1.
Indeed, since $\varphi$ is a composite

$\big{(}\mathcal{D}^{\mathrm{p}}_{X\times\\{\infty\\}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L}\big{)}\subseteq\big{(}\mathcal{D}^{\mathrm{p}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L}\big{)}\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$

and hence it will suffice to show that the composite

$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{i^{!}p^{*}L})\xrightarrow{i_{*}}\big{(}\mathcal{D}^{\mathrm{p}}(X\times\mathbb{P}^{1}),\Qoppa^{\mathrm{s}}_{p^{*}L}\big{)}\xrightarrow{p_{*}((-)\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$

is an equivalence. Indeed, this now follows from Lemma 5.1.4. ∎

Let us finally build on a classical argument to show that on regular
Noetherian schemes, outside of the symmetric case, neither
$\operatorname{L}$-theory nor $\operatorname{GW}$-theory is homotopy
invariant.

######  Example 6.3.2.

For any $m\in\mathbb{Z}\cup\\{+\infty\\}$, the pull-back map
$\operatorname{L}(\mathbb{F}_{2},\Qoppa^{\geq
m})\to\operatorname{L}(\mathbb{F}_{2}[t],\Qoppa^{\geq m})$ is not an
equivalence.

###### Proof.

Let us start by the case $m=+\infty$, i.e. the quadratic case. The quadratic
$\operatorname{L}$-groups of rings coincide in non-negative degrees with
Ranicki’s periodic $\operatorname{L}$-groups, by [9-authors-I]*Example 2.3.17.
The $\operatorname{L}_{0}$-group is therefore the classical Witt group of non-
degenerate quadratic forms over $R$.

For any scheme $X$, consider the sections
$s_{0},s_{1}:X\to\mathbb{A}^{1}\times X$ at $0$ and $1$ of the projection
$p:\mathbb{A}^{1}\times X\to X$. We have
$s_{0}^{*}p^{*}=\mathrm{id}=s_{1}^{*}p^{*}$ on $\operatorname{L}$-groups, so
if $p^{*}$ is an isomorphism, then $s_{0}^{*}=s_{1}^{*}$, both being inverse
to $p^{*}$. We now show this cannot happen over
$X=\mathrm{spec}(\mathbb{F}_{2})$. The class of the non-singular quadratic
form $q$ of rank $2$ given by $q(x,y)=x^{2}+xy+ty^{2}$ is an element of
$\operatorname{L}_{0}(\mathbb{F}_{2}[t],\Qoppa^{\mathrm{q}})$, and it pulls
back via $s_{i}$ to the class of $q_{i}(x,y)=x^{2}+xy+ixy$ in
$\operatorname{L}_{0}(\mathbb{F}_{2},\Qoppa^{\mathrm{q}})$, whose Arf
invariant is $i\in\mathbb{F}_{2}$. Thus $s_{0}^{*}\neq s_{1}^{*}$ on
$\operatorname{L}_{0}(-,\Qoppa^{\mathrm{q}})$. Actually, by [9-authors-
II]*4.5.6, the quadratic $\operatorname{L}$-groups are 4-periodic by a natural
isomorphism commuting to these pull-backs, so $s_{0}^{*}\neq s_{1}^{*}$ on
$\operatorname{L}_{4n}(-,\Qoppa^{\mathrm{q}})$ for any $n\in\mathbb{Z}$.

When $m\in\mathbb{Z}$, for $n<<0$, we have
$\operatorname{L}_{n}(R,\Qoppa^{\geq
m})\simeq\operatorname{L}_{n}(R,\Qoppa^{\mathrm{q}})$ via the natural map, by
[9-authors-I]*Corollary 1.2.12. The quadratic case thus implies that
$s_{0}^{*}\neq s_{1}^{*}$ on $\operatorname{L}_{n}(-,\Qoppa^{\geq m})$ for
some $n<<0$. ∎

In fact, we suspect that forcing homotopy invariance on the quadratic
$\operatorname{GW}$ or any of the truncated versions might turn it into the
symmetric $\operatorname{GW}$.

##  7\. Motivic realization of localising invariants

Our goal in this section is to construct, for a qcqs scheme $S$, a lax
symmetric monoidal functor

$\mathcal{R}^{\mathrm{s}}_{S}\colon\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},{\mathcal{S}\mathrm{p}})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})),$

which takes as input a Karoubi-localising functor
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ on
Poincaré $\infty$-categories and returns a pre-motivic spectrum over $S$, that
is, a $\mathbb{P}^{1}$-spectrum object in Nisnevich sheaves of spectra. Taking
$\mathcal{F}$ to be the Karoubi-Grothendieck-Witt functor then yields a
hermitian K-theory pre-motivic spectrum, which is actually a motivic spectrum
if $S$ is regular Noetherian of finite Krull dimension. The study of this and
related motivic spectra, such as the one associated to Karoubi L-theory, will
be taken on in §8 below. The present section is, in turn, essentially
dedicated to the construction of $\mathcal{R}^{\mathrm{s}}_{S}$, which is
somewhat involved. In fact, it will be convenient to construct a slightly more
flexible version of $\mathcal{R}^{\mathrm{s}}_{S}$, which takes as input a
Karoubi-localising functor
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ and an
$S$-linear stable $\infty$-category with duality $(\mathcal{C},\mathrm{D})$.
We call the resulting pre-motivic spectrum
$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))$ the _pre-
motivic realization of $\mathcal{F}$ with coefficients in
$(\mathcal{C},\mathrm{D})$_. A motivic realization functor is then obtained by
post-composing with the $\mathbb{A}^{1}$-localisation functor
$\mathrm{Loc}_{\mathbb{A}^{1}}$, which converts any pre-motivic spectrum to a
motivic one. This is eventually achieved in the end of §7.3, see Construction
7.3.10 below. Then, in §7.4 we prove a technical lemma which says that, in a
certain precise sense, the pre-motivic spectrum
$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$ only depends on the underlying
$\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$-valued localising functor
$\Omega^{\infty}\mathcal{F}$.

###  7.1. Recollections on motivic spectra

Let $S$ be a quasi-compact and quasi-separated scheme. Recall that we denote
by $\mathrm{Sm}_{S}$ the category of smooth $S$-schemes $p\colon X\to S$. In
the present section, we always consider $\mathrm{Sm}_{S}$ as endowed with the
Nisnevich topology.

There are several equivalent ways to define the $\infty$-category of motivic
spectra over $S$. To facilitate the manoeuvre between these, let us fix
coefficient $\infty$-category $\mathcal{E}$ that is presentable. In practice,
$\mathcal{E}$ will either be the $\infty$-category of spaces, pointed spaces,
${\mathrm{E}_{\infty}}$-groups, or spectra. Recall that a Nisnevich sheaf
$\mathcal{F}\colon\mathrm{Sm}_{/S}^{\mathrm{op}}\to\mathcal{E}$ is said to be
_$\mathbb{A}^{1}$ -invariant_ if the canonical map
$\mathcal{F}(S)\to\mathcal{F}(\mathbb{A}^{1}_{S})$ is an equivalence in
$\mathcal{E}$. We then call $\mathrm{H}(S,\mathcal{E})$ the full subcategory
$\mathrm{Sh}^{\mathrm{nis}}_{\mathbb{A}^{1}}(S,\mathcal{E})\subseteq\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$
spanned by the $\mathbb{A}^{1}$-invariant Nisnevich sheaves. When
$\mathcal{E}=\mathcal{S}$, these are known as _motivic spaces_ , and one
simply writes $\mathrm{H}(S)$ for $\mathrm{H}(S,\mathcal{S})$.

When $\mathcal{E}$ is pointed, it becomes canonically tensored and cotensored
over the presentable $\infty$-category $\mathcal{S}_{*}$ of pointed spaces, in
which case $\operatorname{Psh}(S,\mathcal{E})$ becomes tensored and cotensored
over $\operatorname{Psh}(S,\mathcal{S}_{*})$ by means of Day convolution with
respect to the fibre product monoidal structure on $\mathrm{Sm}_{S}$, which
coincides in this case with the pointwise product (since the fibre product is
the cartesian product in $\mathrm{Sm}_{S}$). In particular, if $X\to S$ is a
smooth $S$-scheme equipped with an $S$-point $s\colon S\to X$, then the
pointed $S$-scheme $(X,s)$ determines via representability a presheaf of
pointed spaces, and for $\mathcal{F}\in\operatorname{Psh}(S,\mathcal{S}_{*})$,
the cotensor operation is given by

$\mathcal{F}^{(X,s)}(T)=\operatorname{fib}[\mathcal{F}(X\times_{S}T)\xrightarrow{s^{*}}\mathcal{F}(T)].$

When $X=\mathbb{P}^{1}_{S}$ and $s$ is the section at $\infty$, we also write

$\Omega_{\mathbb{P}^{1}}(\mathcal{F})=\mathcal{F}^{(\mathbb{P}^{1},\infty)}.$

By [9-authors-IV], the full subcategories
$\mathrm{H}(S,\mathcal{E})\subseteq\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})\subseteq\operatorname{Psh}(S,\mathcal{E})$
are both $\operatorname{Psh}(S,\mathcal{S}_{*})$-linear as accessible
localisations of $\operatorname{Psh}(S,\mathcal{E})$. In particular, they both
inherit the structure of being tensored and cotensored over
$\operatorname{Psh}(S,\mathcal{S}_{*})$, with the cotensor operation being
given by the same formula.

######  Definition 7.1.1.

Let $\mathcal{E}$ be a pointed presentable $\infty$-category. We define
${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})$ and
$\operatorname{SH}(S,\mathcal{E})$ to be the stabilizations of
$\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$ and $\mathrm{H}(S,\mathcal{E})$
respectively, with respect to the action of
$(\mathbb{P}^{1},\infty)\in\operatorname{Psh}(S,\mathcal{S}_{*})$. In
particular, ${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})$ and
$\operatorname{SH}(S,\mathcal{E})$ are given by the homotopy limits

${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})=\lim\big{(}\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})\xleftarrow{\Omega_{\mathbb{P}^{1}}}\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})\xleftarrow{\Omega_{\mathbb{P}^{1}}}\quad\cdots\quad\big{)}$

and

$\operatorname{SH}(S,\mathcal{E})=\lim\big{(}\mathrm{H}(S,\mathcal{E})\xleftarrow{\Omega_{\mathbb{P}^{1}}}\mathrm{H}(S,\mathcal{E})\xleftarrow{\Omega_{\mathbb{P}^{1}}}\quad\cdots\quad\big{)}\
.$

The fully-faithful inclusion
$\mathrm{H}(S,\mathcal{E})\subseteq\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$
then induces a fully-faithful inclusion
$\operatorname{SH}(S,\mathcal{E})\subseteq{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})$.
We refer to the objects of $\operatorname{SH}(S,\mathcal{E})$ as
_$\mathcal{E}$ -valued motivic spectra_, and to those of
${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})$ as _$\mathcal{E}$
-valued pre-motivic spectra_. If $\mathcal{E}={\mathcal{S}\mathrm{p}}$ then we
drop the prefix “$\mathcal{E}$-valued”.

Concretely, we may describe an $\mathcal{E}$-valued pre-motivic spectrum as a
sequence $\mathcal{X}_{0},\mathcal{X}_{1},\ldots,$ of $\mathcal{E}$-valued
Nisnevich sheaves on $S$, together with, for every $n\geq 0$, and equivalence
$f_{n}\colon\mathcal{X}_{n}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\Omega_{\mathbb{P}^{1}}\mathcal{X}_{n+1}$.
Such an $\mathcal{E}$-valued pre-motivic spectrum is an $\mathcal{E}$-valued
motivic spectrum exactly when all the Nisnevich sheaves $\mathcal{X}_{n}$ are
$\mathbb{A}^{1}$-invariant.

######  Proposition 7.1.2.

Let $\mathcal{E}$ be a pointed presentable $\infty$-category. The
$\infty$-category $\operatorname{SH}(S,\mathcal{E})$ is stable, and the
functor

$\operatorname{SH}(S,{\mathcal{S}\mathrm{p}}(\mathcal{E}))\to\operatorname{SH}(S,\mathcal{E})$

induced by
$\Omega^{\infty}\colon{\mathcal{S}\mathrm{p}}(\mathcal{E})\to\mathcal{E}$ is
an equivalence. In particular, the functors

$\operatorname{SH}(S,{\mathcal{S}\mathrm{p}})\to\operatorname{SH}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})\to\operatorname{SH}(S,\mathcal{S}_{*})$

are both equivalences.

We will consider any of the equivalent $\infty$-categories

$\operatorname{SH}(S,{\mathcal{S}\mathrm{p}})\simeq\operatorname{SH}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})\simeq\operatorname{SH}(S,\mathcal{S}_{*})$

as a model for the $\infty$-category of _motivic spectra_ over $S$.

###### Proof of Proposition 7.1.2.

Now note that since $\mathrm{H}(S,\mathcal{E})$ is closed in
$\operatorname{Psh}(S,\mathcal{E})$ under limits we have that limits in
$\mathrm{H}(S,\mathcal{E})$ are computed levelwise. This applies in particular
to taking loop objects, and so
$\operatorname{SH}(S,{\mathcal{S}\mathrm{p}}(\mathcal{E}))={\mathcal{S}\mathrm{p}}(\operatorname{SH}(S,\mathcal{E}))$
over $\operatorname{SH}(S,\mathcal{E})$. All claims hence follow once we show
that $\operatorname{SH}(S,\mathcal{E})$ is stable. This, in turn, follows from
the fact that in $\operatorname{SH}(S,\mathcal{S}_{*})$ one has the
equivalence $\mathrm{S}^{1}\wedge\mathbb{G}_{m}\simeq\mathbb{P}^{1}$. As a
consequence, universally inverting the action of $\mathbb{P}^{1}$ inverts both
the action of $\mathrm{S}^{1}$ – thus stabilizing the category in the standard
sense – and $\mathbb{G}_{m}$. ∎

The advantage of working with coefficients in ${\mathcal{S}\mathrm{p}}$ (or
more generally in any stable $\mathcal{E}$) is that the localisation functor

$\mathrm{Loc}_{\mathbb{A}^{1}}\colon\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})\to\mathrm{H}(S,{\mathcal{S}\mathrm{p}})$

is given by the explicit formula

$(\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{X})_{n}(T)=|\mathcal{X}_{n}(T\times_{S}\Delta_{S}^{\bullet})|,$

where $\Delta_{S}^{n}$ is the algebraic $n$-simplex over $S$, that is, the
closed subscheme of $\mathbb{A}^{n}\times S$ determined by the equation
$\sum_{i}x_{i}=1$. In fact, this formula always gives the reflection to the
full subcategory spanned by $\mathbb{A}^{1}$-invariant presheaves, but for a
general $\mathcal{E}$, it does not preserve the Nisnevich sheaf property.
However, when $\mathcal{E}$ is stable the sheaf property is preserved under
colimits, and hence $\mathrm{Loc}_{\mathbb{A}^{1}}$ sends Nisnevich sheaves to
Nisnevich sheaves, which implies that its universal property persists to the
sheaf context. In fact, when $\mathcal{E}$ is stable this formula also applies
for $\mathbb{P}^{1}$-spectrum objects:

######  Lemma 7.1.3.

Suppose that $\mathcal{E}$ is stable. If
$F\colon\mathrm{Sm}_{S}^{\mathrm{op}}\to\mathcal{E}$ is a Nisnevich sheaf,
then the interchange map
$\mathrm{Loc}_{\mathbb{A}^{1}}\Omega_{\mathbb{P}^{1}}F\to\Omega_{\mathbb{P}^{1}}\mathrm{Loc}_{\mathbb{A}^{1}}F$
is an equivalence. In particular, the localisation functor

$\mathrm{Loc}_{\mathbb{A}^{1}}\colon{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\mathcal{E})\to\operatorname{SH}(S,\mathcal{E})$

(which we denote by the same name) is computed $\mathbb{P}^{1}$-levelwise.

###### Proof.

The cartesian products with $\mathbb{A}^{n}$ and $\mathbb{P}^{1}$ commute, and
taking fibers commutes with colimits because in a stable $\infty$-category
finite limits commute with arbitrary colimits. ∎

######  Remark 7.1.4.

The above considerations also imply that the left adjoint to the inclusion

$\operatorname{SH}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})\subseteq{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})$

is given by the composite

${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})\xrightarrow{\iota_{*}}{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(S,{\mathcal{S}\mathrm{p}})\xrightarrow{\mathrm{Loc}_{\mathbb{A}^{1}}}\operatorname{SH}(S,{\mathcal{S}\mathrm{p}})=\operatorname{SH}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}),$

where as in §7.2, the functor $i_{*}$ is obtained by object-wise composing
with the fully-faithful inclusion
$\iota\colon\operatorname{Grp}_{{\mathrm{E}_{\infty}}}\hookrightarrow{\mathcal{S}\mathrm{p}}$
followed by the reflection functor

${\mathcal{PS}\mathrm{p}}^{\mathbb{P}^{1}}(\operatorname{Psh}(S,{\mathcal{S}\mathrm{p}}))\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})).$

###  7.2. Bounded localising invariants

Let $S$ be a quasi-compact and quasi-separated scheme.

######  Construction 7.2.1.

We denote by

$\operatorname{Mod}_{S}^{\flat}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\subseteq\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$

the full subcategory spanned by those $\mathcal{D}^{\mathrm{p}}(S)$-linear
$\infty$-categories with duality which are flat over
$\mathcal{D}^{\mathrm{p}}(S)$ in the sense of Definition 4.1.4, see Notation
3.2.3. Our previous results establish the following:

  * •

By Example 4.1.5, this full subcategory contains $\mathcal{D}^{\mathrm{p}}(X)$
for every flat qcqs $p\colon X\to S$, and so in particular for any smooth qcqs
$p$.

  * •

By Corollary 4.1.7, this full subcategory is closed under tensor products in
$\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$,
and hence inherits from
$\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$ its
symmetric monoidal structure.

Combining these facts we find that the symmetric monoidal functor

$(\mathrm{Sm}_{/S}^{\mathrm{op}})^{\otimes}\to(\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}))^{\otimes}\quad\quad[X\to
S]\mapsto(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{X})$

of (13) then factors uniquely through a symmetric monoidal functor

$(\mathrm{Sm}_{/S}^{\mathrm{op}})^{\otimes}\to\operatorname{Mod}_{S}^{\flat}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}.$

Now since the $\infty$-category $\mathrm{Sm}_{/S}$ of smooth $S$-schemes is
essentially small, we may fix a large enough regular cardinal $\kappa$ such
that the image of this symmetric monoidal functor lies in the full subcategory

$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})=\operatorname{Mod}_{S}^{\flat}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\cap\operatorname{Mod}_{S}^{\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$

spanned by the flat $\kappa$-compact $\mathcal{D}^{\mathrm{p}}(S)$-linear
$\infty$-categories with duality. Since
$\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$ is
presentably symmetric monoidal, we can ensure that
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
is closed under tensor products in
$\operatorname{Mod}_{S}^{\flat}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
by taking $\kappa$ sufficiently large. In particular,
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
inherits a symmetric monoidal structure and the above symmetric monoidal
functor further factors through a symmetric monoidal functor

(34)
$(\mathrm{Sm}_{/S}^{\mathrm{op}})^{\otimes}\to\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}\quad\quad[X\to
S]\mapsto(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{X}).$

Let us now fix a presentable $\infty$-category $\mathcal{E}$.

######  Definition 7.2.2.

We say that a functor
$\mathcal{F}\colon\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(S)}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathcal{E}$
is _bounded localising_ if it is reduced and sends bounded Karoubi squares
(see Definition 4.2.6) to fibre squares. We denote by

$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\subseteq\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$

the full subcategory spanned by the bounded localising functors.

######  Remark 7.2.3.

If $\mathcal{E}$ is a stable then a reduced functor
$\mathcal{F}\colon\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(S)}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathcal{E}$
is bounded localising if and only if it sends bounded Karoubi sequences to
exact sequences.

By [9-authors-IV], the full subcategory
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
is an accessible localisation of
$\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
with associated localisation functor

$\mathrm{Loc}_{\mathrm{b}}\colon\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E}).$

Now, by Nisnevich descent (see Corollary 4.4.3), in the left-Kan-
extension/restriction adjunction

$\operatorname{Psh}(\mathrm{Sm}_{/S},\mathcal{E})\mathbin{\leavevmode\hbox
to15.88pt{\vbox to13.5pt{\pgfpicture\makeatletter\hbox{\hskip
2.56pt\lower-3.52257pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}} {}{}{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.56pt}{3.12257pt}\pgfsys@curveto{-2.0923pt}{1.24901pt}{-1.05006pt}{0.3643pt}{0.0pt}{0.0pt}\pgfsys@curveto{-1.05006pt}{-0.3643pt}{-2.0923pt}{-1.24901pt}{-2.56pt}{-3.12257pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{12.11664pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.51663pt}{6.45831pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{12.91663pt}{0.0pt}\pgfsys@lineto{0.79999pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.4pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.19446pt}{2.19443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\perp$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
}\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E}),$

the right adjoint sends bounded localising functors to Nisnevich sheaves and
so the left adjoint sends Nisnevich local equivalences to
$\mathrm{Loc}_{\mathrm{b}}$-equivalences. By the universal property of
localisations, we now obtain a commutative square of left adjoint functors

(35)
${\operatorname{Psh}(\mathrm{Sm}_{/S},\mathcal{E})}$${\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})}$${\mathrm{Sh}^{\mathrm{nis}}(\mathrm{Sm}_{/S},\mathcal{E})}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\
.}$$\scriptstyle{(-)^{\mathrm{nis}}}$$\scriptstyle{\mathrm{Loc}_{\mathrm{b}}}$$\scriptstyle{\mathcal{T}_{S}}$

where $\mathcal{T}_{S}$ is given by left Kan extension followed by
$\mathrm{Loc}_{\mathrm{b}}$.

###  7.3. Motivic realization

In this subsection we construct our motivic realization functor, see
Constructions 7.3.4 and 7.3.10 below.

We begin by promoting the square (35) to a symmetric monoidal one, that which
will enable us to eventually pass from Nisnevich sheaves to
$\mathbb{P}^{1}$-spectrum objects therein, that is, to pre-motivic spectra
(see Definition 7.1.1). For this note that if $\mathcal{E}$ is a presentably
symmetric monoidal $\infty$-category, then the functor categories

$\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\quad\text{and}\quad\operatorname{Psh}(S,\mathcal{E})=\operatorname{Fun}(\mathrm{Sm}_{/S}^{\mathrm{op}},\mathcal{E})$

inherit symmetric monoidal structures in the form of Day convolution, and the
symmetric monoidal functor (34) induces by means of left Kan extension a
symmetric monoidal functor

(36)
$\operatorname{Psh}(S,\mathcal{E})\to\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E}).$

Note that the symmetric monoidal structure on $\mathrm{Sm}_{/S}^{\mathrm{op}}$
is the cocartesian one, and so Day convolution on
$\operatorname{Psh}(S,\mathcal{E})$ amounts to pointwise products of
presheaves.

Now, on the sheaf side, we have that $\operatorname{Psh}(S,\mathcal{E})$ is
presentable and its full subcategory
$\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})\subseteq\operatorname{Psh}(S,\mathcal{E})$
spanned by the Nisnevich sheaves is an accessible localisation. Furthermore,
this localisation is multiplicative with respect to Day convolution: indeed,
the monoidal product on $\mathrm{Sm}_{S}$ is fibre product over $S$, and
coverings families are closed under base change, see, e.g., the criterion in
[9-authors-I]*Lemma 5.3.4. In particular,
$\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$ inherits a symmetric monoidal
structure such that the sheafification functor

$(-)^{\mathrm{nis}}\colon\operatorname{Psh}(S,\mathcal{E})\to\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$

refines to a symmetric monoidal functor. Explicitly, the symmetric monoidal
product on $\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$ is given by taking
pointwise products followed by sheafification.

As explained above, the top horizontal in this square refines to symmetric
monoidal left adjoints. And since the localisation $(-)^{\mathrm{nis}}$ is
multiplicative the same holds for the left vertical functor. The same holds
for the right vertical functor:

######  Proposition 7.3.1.

The accessible localisation

$\mathrm{Loc}_{\mathrm{b}}\colon\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$

is multiplicative with respect to Day convolution. In particular,
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(S)}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
inherits a symmetric monoidal structure such that $\mathrm{Loc}_{\mathrm{b}}$
refines to a symmetric monoidal functor.

###### Proof.

Proposition 4.2.15 and [9-authors-I]*Proposition 5.3.4. ∎

By the universal property of multiplicative localisations (see [9-authors-
IV]), we now obtain a commutative square of symmetric monoidal left adjoint
functors

(37)
${\operatorname{Psh}(\mathrm{Sm}_{/S},\mathcal{E})^{\otimes}}$${\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})^{\otimes}}$${\mathrm{Sh}^{\mathrm{nis}}(\mathrm{Sm}_{/S},\mathcal{E})^{\otimes}}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})^{\otimes}}$$\scriptstyle{(-)^{\mathrm{nis}}}$$\scriptstyle{\mathrm{Loc}_{\mathrm{b}}}$$\scriptstyle{\mathcal{T}_{S}}$

whose horizontal functors are left adjoints to the corresponding restriction
functors. This yields the desired symmetric monoidal refinement of the square
(35).

######  Definition 7.3.2.

Given an object $X\to S$ of $\mathrm{Sm}_{/S}$, let us denote by
$\mathcal{E}[X]$ the $\mathcal{E}$-valued sheaf obtained by sheafifying the
presheaf

$Y\mapsto u\operatorname{Map}_{\mathrm{Sm}_{/S}}(Y,X),$

where $u\colon\mathcal{S}\to\mathcal{E}$ is the unit of $\mathcal{E}$. If
$x\colon S\to X$ is an $S$-point of $X$ and $\mathcal{E}$ is pointed then we
denote by $\mathcal{E}[X,x]$ the cofibre of the map
$\mathcal{E}[S]\xrightarrow{x}\mathcal{E}[X]$ in
$\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})$.

We can consider the functor

(38)
$\mathrm{j}_{\mathrm{bloc}}:\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$

sending $(\mathcal{C},\mathrm{D})$ to
$\mathrm{Loc}_{\mathrm{b}}\Sigma^{\infty}\operatorname{Map}((\mathcal{C},\mathrm{D}),-)_{+}$
as a co-Yoneda embedding into bounded Karoubi-localising functors to
$\mathcal{E}$.

######  Proposition 7.3.3.

Suppose that $\mathcal{E}$ is additive, so in particular pointed. Then the
symmetric monoidal functor $\mathcal{T}_{S}$ sends the object
$\mathcal{E}[\mathbb{P}^{1}_{S},\infty]$ to
$\mathrm{j}_{\mathrm{bloc}}(\mathcal{D}^{\mathrm{p}}(S),\Sigma^{-1}\mathrm{D}_{S})$.
In particular, it sends $\mathcal{E}[\mathbb{P}^{1}_{S},\infty]$ to an
_invertible_ object in
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$,
for the monoidal structure described above.

###### Proof.

Since left Kan extension sends representable functors to representable
functors, we see that $\mathcal{T}_{S}\mathcal{E}[\mathbb{P}^{1}_{S},\infty]$
is given by the cofibre of

$\mathrm{Loc}_{\mathrm{b}}u\operatorname{Map}((\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S}),-)\xrightarrow{(-)\circ\infty^{*}}\mathrm{Loc}_{\mathrm{b}}u\operatorname{Map}((\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}_{S}),\mathrm{D}_{\mathbb{P}^{1}_{S}}),-)$

where
$\infty^{*}\colon(\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}_{S}),\mathrm{D}_{\mathbb{P}^{1}_{S}})\to(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})$
is the $\mathcal{E}$-linear duality preserving functor induced by
$\infty\colon S\to\mathbb{P}^{1}_{S}$. This functor admits a retraction which
is induced by the terminal map $q\colon\mathbb{P}^{1}_{S}\to S$ in
$\mathrm{Sm}_{/S}$, and since
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
inherits from $\mathcal{E}$ the property of being additive, it follows from
the splitting lemma (see, e.g, [9-authors-II]*Lemma 1.5.12) that the cofibre
of $(-)\circ\infty^{*}$ is canonically equivalent to the fibre of the map

$\mathrm{Loc}_{\mathrm{b}}u\operatorname{Map}((\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}_{S}),\mathrm{D}_{\mathbb{P}^{1}_{S}}),-)\xrightarrow{(-)\circ
q^{*}}\mathrm{Loc}_{\mathrm{b}}u\operatorname{Map}((\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S}),-)$

induced by the duality preserving $\mathcal{E}$-linear functor
$q^{*}\colon(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})\to(\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}_{S}),\mathrm{D}_{\mathbb{P}^{1}_{S}})$.
By the bounded Karoubi sequence

$(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})\xrightarrow{q^{*}}(\mathcal{D}^{\mathrm{p}}(\mathbb{P}^{1}_{S}),\mathrm{D}_{\mathbb{P}^{1}_{S}})\to(\mathcal{D}^{\mathrm{p}}(S),\Sigma^{-1}\mathrm{D}_{S})$

underlying the split Poincaré-Verdier sequence (29) (see Example 4.2.7), we
now conclude that the fibre of $(-)\circ q^{*}$ in the $\infty$-category
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
is equivalent to
$\mathrm{Loc}_{\mathrm{b}}u\operatorname{Map}((\mathcal{D}^{\mathrm{p}}(S),\Omega\mathrm{D}_{S}),-)$,
which is therefore invertible with respect to the localised Day convolution
structure since $(\mathcal{D}^{\mathrm{p}}(S),\Sigma^{-1}\mathrm{D}_{S})$ is
invertible in
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
and $\mathrm{Loc}_{\mathrm{b}}$, $u$ and the Yoneda functor are all symmetric
monoidal. ∎

By Proposition 7.3.3 and the universal property of
$\mathbb{P}^{1}$-stabilization, if $\mathcal{E}$ is an additive presentably
symmetric monoidal $\infty$-category, the symmetric monoidal functor
$\mathcal{T}_{S}$ appearing in the lower horizontal row of (37) factors
essentially uniquely via a symmetric monoidal left adjoint functor

$\widetilde{\mathcal{T}}_{S}\colon{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))^{\otimes}\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})^{\otimes},$

whose right adjoint

$\mathcal{R}_{S}\colon\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))$

then inherits an induced lax symmetric monoidal structure.

######  Construction 7.3.4 (Motivic realization for bounded localising
functors).

Given a bounded localising functor
$\mathcal{F}\colon\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathcal{E}$
and a flat $S$-linear $\infty$-category with duality
$(\mathcal{C},\mathrm{D})\in\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$,
we will denote by

$\mathcal{R}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))\in{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))$

the image under the right adjoint $\mathcal{R}_{S}$ above of the bounded
localising functor $\mathcal{F}((-)\otimes(\mathcal{C},\mathrm{D}))$ obtained
by pre-composing $\mathcal{F}$ with $(-)\otimes(\mathcal{C},\mathrm{D})$ (and
operation which preserves bounded localising functors by Proposition 4.2.15).
We refer to $\mathcal{R}_{S}(\mathcal{F},\mathcal{C})$ as the _pre-motivic
realization_ of $\mathcal{F}$ with coefficients in $(\mathcal{C},\mathrm{D})$,
and to

$\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}_{S}(\mathcal{F};(\mathcal{C},\mathcal{D}))\in\operatorname{SH}(S,\mathcal{E})$

as the _motivic realization_ of $\mathcal{F}$ with coefficients in
$(\mathcal{C},\mathrm{D})$.

The association
$(\mathcal{F};(\mathcal{C},\mathrm{D}))\mapsto\mathcal{R}_{S}(\mathcal{F},(\mathcal{C},\mathrm{D}))$
then assembles to form a functor

$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\times\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E})).$

This functor carries a canonical lax monoidal functor: indeed, it is the
composite of the lax symmetric monoidal functor $\mathcal{R}_{S}(-)$ and the
functor
$(\mathcal{F},(\mathcal{C},\mathrm{D}))\mapsto\mathcal{F}((-)\otimes(\mathcal{C},\mathrm{D}))$,
which is nothing but the Day tensoring of $\mathcal{F}$ by
$(\mathcal{C},\mathrm{D})$, and is hence lax symmetric monoidal in its entries
as a pair. In particular, if
$\mathcal{F}\colon\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathcal{E}$
is a commutative algebra object with respect to Day convolution, that is,
$\mathcal{F}$ is a lax symmetric monoidal functor, then
$\mathcal{R}_{S}(\mathcal{F})$ inherits the structure of a commutative algebra
object in
${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))$,
and the same holds for $\mathcal{R}_{S}(\mathcal{F},(\mathcal{C},\mathrm{D}))$
for every commutative algebra object in
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$.
Similarly, if $\mathcal{F}$ is lax symmetric monoidal and
$(\mathcal{C},\mathrm{D})$ is any object of
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
then $\mathcal{R}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))$ inherits the
structure of a module over $\mathcal{R}_{S}(\mathcal{F})$.

######  Example 7.3.5.

For every $L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$ the $S$-linear
$\infty$-category with duality $(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L})$
is a module over $(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})$, and hence for
every lax symmetric monoidal bounded localising functor $\mathcal{F}$ the pre-
motivic realization
$\mathcal{R}_{S}(\mathcal{F};(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L}))$ is
a module over $\mathcal{R}_{S}(\mathcal{F})$. This example is particularly
important, and we will use the shorthand notation

$\mathcal{R}_{L}(\mathcal{F}):=\mathcal{R}_{S}(\mathcal{F};(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L}))$

to denote it. We note that for every $S$-smooth scheme $p\colon X\to S$ we
have
$(\mathcal{D}^{\mathrm{p}}(X),\mathrm{D}_{X})\otimes(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L})=(\mathcal{D}^{\mathrm{p}}(X),\mathrm{D}_{p^{*}L})$,
and so the underlying Nisnevich sheaf
$\Omega^{\infty}_{\mathbb{P}^{1}}\mathcal{R}_{L}(\mathcal{F})$ is given by
$[p\colon X\to
S]\mapsto\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{p^{*}L})$.

######  Construction 7.3.6.

The functor $\widetilde{\mathcal{T}}_{S}$ being symmetric monoidal, we have
for every $\mathcal{E}$-valued pre-motivic spectrum $E$ a commutative square

${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))\
.}$$\scriptstyle{(-)\otimes\widetilde{\mathcal{T}}_{S}(E)}$$\scriptstyle{\widetilde{\mathcal{T}}_{S}}$$\scriptstyle{(-)\otimes
E}$$\scriptstyle{\widetilde{\mathcal{T}}_{S}}$

Passing to right adjoints, we obtain a commutative square

${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))\
,}$$\scriptstyle{(-)^{\widetilde{\mathcal{T}}_{S}(E)}}$$\scriptstyle{\mathcal{R}_{S}}$$\scriptstyle{(-)^{E}}$$\scriptstyle{\mathcal{R}_{S}}$

where the vertical arrows denote the operation of cotensoring with $E$ and
$\widetilde{\mathcal{T}}_{S}(E)$, respectively. In particular, for
$\mathcal{F}\in\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
we obtain a natural equivalence

$\mathcal{R}_{S}(\mathcal{F})^{E}\simeq\mathcal{R}_{S}(\mathcal{F}^{\widetilde{\mathcal{T}}_{S}(E)}).$

In addition, the symmetry of the monoidal structure on
$\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})$
means that cotensors commute with cotensors, and hence for every
$(\mathcal{C},\mathrm{D})\in\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
we obtain a similar natural equivalence

$\mathcal{R}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))^{E}\simeq\mathcal{R}_{S}(\mathcal{F}^{\widetilde{T}_{S}(E)};(\mathcal{C},\mathrm{D})).$

Using our notation for Poincaré $\infty$-categories, given an $S$-linear
$\infty$-category with duality
$(\mathcal{C},\mathrm{D})=(\mathcal{C},\Qoppa^{\mathrm{s}}_{\mathrm{D}})$, we
denote by $(\mathcal{C},\mathrm{D})^{[n]}:=(\mathcal{C},\Sigma^{n}\mathrm{D})$
the $S$-linear $\infty$-category obtained by shifting the duality/Poincaré
structure by $n$. Given a functor
$\mathcal{F}\colon\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathcal{E}$
we denote by $\mathcal{F}^{[n]}(-):=\mathcal{F}((-)^{[n]})$ the functor
obtained by pre-composing with the $n$-shift operator. We note that if
$\mathcal{F}$ is bounded localising then $\mathcal{F}^{[n]}$ is again bounded
localising.

######  Example 7.3.7 (Bott periodicity).

Applying Construction 7.3.6 in the case of
$E=\mathcal{E}[\mathbb{P}^{1}_{S},\infty]$ and using Proposition 7.3.3 we
obtain a natural equivalence

$\Omega_{\mathbb{P}^{1}}\mathcal{R}_{S}(\mathcal{F})=\mathcal{R}_{S}(\mathcal{F}((-)\otimes(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{\mathcal{O}_{S}[-1]})=\mathcal{R}_{S}(\mathcal{F}((-)^{[-1]})=\mathcal{R}_{S}(\mathcal{F}^{[-1]}).$

Since the operations $\Omega_{\mathbb{P}^{1}}(-)$ and $(-)^{[-1]}$ are
invertible this equivalence determines an equivalence

$\Sigma_{\mathbb{P}^{1}}^{n}\mathcal{R}_{S}(\mathcal{F})=\mathcal{R}_{S}(\mathcal{F}^{[n]})$

for every integer $n\in\mathbb{Z}$. Similarly, for
$(\mathcal{C},\mathrm{D})\in\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})$
we have a natural equivalence

$\Sigma_{\mathbb{P}^{1}}^{n}\mathcal{R}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))\simeq\mathcal{R}_{S}(\mathcal{F}^{[n]};(\mathcal{C},\mathrm{D})).$

In particular, in the case of
$(\mathcal{C},\mathrm{D})=(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L})$ for
$L\in{\mathcal{P}ic}^{\mathrm{B}\mathrm{C_{2}}}(S)$ this equivalence can be
rewritten using the notation of Example 7.3.5 as

$\Sigma_{\mathbb{P}^{1}}^{n}\mathcal{R}_{L}(\mathcal{F})=\mathcal{R}_{L}(\mathcal{F}^{[n]})=\mathcal{R}_{L[n]}(\mathcal{F}).$

######  Corollary 7.3.8.

For $p\colon X\to S$ in $\mathrm{Sm}_{S}$, there is a natural equivalence

$\mathcal{R}_{S}(\mathcal{F};(\mathcal{D}^{\mathrm{p}}(X),\mathrm{D}_{X}))=p_{*}p^{*}\mathcal{R}_{S}(\mathcal{F}).$

###### Proof.

Apply Construction 7.3.6 to
$E=\Sigma^{\infty}_{\mathbb{P}^{1}}\mathcal{E}[X]$. ∎

######  Corollary 7.3.9.

The functor $\mathcal{R}_{S}$ preserves all limits. If in addition
$\mathcal{E}$ is stable, then $\mathcal{R}_{S}$ preserves all colimits.

###### Proof.

The first claim is because $\mathcal{R}_{S}$ is a right adjoint. To see the
second claim, note that if $\mathcal{E}$ is stable, then the full subcategory
of Nisnevich sheaves is closed under colimits inside all presheaves and the
functor $\Omega_{\mathbb{P}^{1}}$ commutes with colimits. It then follows that
colimits in $\mathbb{P}^{1}$-spectrum objects in Nisnevich sheaves are
computed on the level of $\mathbb{P}^{1}$-prespectrum objects in presheaves.
At the same time, when $\mathcal{E}$ is stable the full subcategory of bounded
localising functors is closed under colimits inside all functors, and so
colimits of bounded localising functors are computed levelwise. The desired
result now follows from the fact that on the level of functors and
$\mathbb{P}^{1}$-prespectra in presheaves, the functor $\mathcal{R}_{S}$ is
given levelwise by suitable evaluations, see Example 7.3.7. ∎

Finally, we now obtain motivic realization functors for Karoubi-localising
invariants of Poincaré $\infty$-categories:

######  Construction 7.3.10 (Motivic realization for Karoubi-localising
functors).

By Proposition 4.2.8, the restriction along the lax symmetric monoidal functor

(39)
$\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}\to\mathrm{Cat}^{\mathrm{p}}\quad\quad(\mathcal{C},\mathrm{D})\mapsto(\mathcal{C},\Qoppa^{\mathrm{s}}_{\mathrm{D}})$

sends Karoubi-localising functors to bounded localising functors. By means of
pre-composition we hence we obtain a lax symmetric monoidal functor

(40)
$\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},\mathcal{E})\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))\xrightarrow{\mathcal{R}_{S}}{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\mathcal{E}))$

which we denote by $\mathcal{R}^{\mathrm{s}}_{S}$. To avoid confusion, we note
that since (39) is only lax symmetric monoidal, the above right adjoint is not
part of a symmetric monoidal adjunction; more precisely, its left adjoint is
only oplax symmetric monoidal, but not symmetric monoidal. More generally, for
a Karoubi-localising functor
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{E}$ and an $S$-linear
stable $\infty$-category with duality $(\mathcal{C},\mathrm{D})$, we write

$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F};(\mathcal{C},\mathrm{D}))=\mathcal{R}_{S}(\mathcal{F}|_{\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})};(\mathcal{C},\mathrm{D}))$

for the corresponding motivic realization with coefficients. As in Example
7.3.5, for an invertible perfect complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$, we write

$\mathcal{R}^{\mathrm{s}}_{L}(\mathcal{F})=\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F};(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{L})).$

so that $\mathcal{R}^{\mathrm{s}}_{L}$ is a module over
$\mathcal{R}^{\mathrm{s}}_{S}$ with respect to Day convolution. In particular,
$\mathcal{R}_{L}(\mathcal{F})$ is a module over $\mathcal{R}_{S}(\mathcal{F})$
for any Karoubi-localising functor $\mathcal{F}$. When $L=\mathcal{O}_{S}$ we
have $\mathcal{R}^{\mathrm{s}}_{L}=\mathcal{R}^{\mathrm{s}}_{S}$.

###  7.4. The free delooping lemma

Let us now consider the case where the target $\mathcal{E}$ is either the
$\infty$-category ${\mathcal{S}\mathrm{p}}$ of spectra or the
$\infty$-category $\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$ of
${\mathrm{E}_{\infty}}$-groups. These two $\infty$-categories are related via
an adjunction

$\iota\colon\operatorname{Grp}_{{\mathrm{E}_{\infty}}}\mathbin{\leavevmode\hbox
to13.72pt{\vbox to13.5pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-3.52257pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{12.11664pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.51663pt}{6.45831pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{12.91663pt}{0.0pt}\pgfsys@lineto{0.79999pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.4pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.19446pt}{2.19443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\perp$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
}{\mathcal{S}\mathrm{p}}\nobreak\mskip 6.0mu plus
1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0mu\Omega^{\infty},$

where $\Omega^{\infty}$ is the lift of the usual loop infinity functor from
spaces to ${\mathrm{E}_{\infty}}$-groups and its left adjoint $\iota$ is the
fully-faithful embedding of $\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$ as
the full subcategory of connective spectra. In fact, this adjunction also
exhibits ${\mathcal{S}\mathrm{p}}$ as the stabilization of
$\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$, and so we could have called
$\iota$ also $\Sigma^{\infty}$, but we will stick with $\iota$ in order to
avoid confusion with $\Sigma^{\infty}$ of spaces (which is not fully-
faithful). We then have an induced adjunction

$\iota_{*}\colon{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}))\mathbin{\leavevmode\hbox
to13.72pt{\vbox to13.5pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-3.52257pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{12.11664pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.51663pt}{6.45831pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}{{{}}{{}}}{}{{}}{}{{}}{{{}}{{}}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{12.91663pt}{0.0pt}\pgfsys@lineto{0.79999pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.4pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.19446pt}{2.19443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\perp$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
}{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mu\Omega^{\infty}_{*},$

where $\Omega^{\infty}_{*}$ is given by object-wise post-composing with
$\Omega^{\infty}$ and $\iota_{*}$ is given by first object-wise post-composing
with $\iota$ and then applying the reflection functor

${\mathcal{PS}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})).$

######  Lemma 7.4.1 (Free delooping).

Let
$\mathcal{F}\colon\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\to{\mathcal{S}\mathrm{p}}$
be a bounded localising functor. Then the counit map

$\iota_{*}\Omega^{\infty}_{*}\mathcal{R}_{S}(\mathcal{F})\to\mathcal{R}_{S}(\mathcal{F})$

is an equivalence.

######  Remark 7.4.2.

Since
$\Omega^{\infty}_{*}\mathcal{R}_{S}(\mathcal{F})\simeq\mathcal{R}_{S}(\Omega^{\infty}\mathcal{F})$,
Lemma 7.4.1 says in particular that $\mathcal{R}_{S}(\mathcal{F})$ only
depends on the underlying $\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$-valued
functor $\Omega^{\infty}\mathcal{F}$. Similarly, if $\mathcal{F}$ is a
Karoubi-localising functor on $\mathrm{Cat}^{\mathrm{p}}$ then
$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$ only depends on
$\Omega^{\infty}\mathcal{F}$. This is a-priori not surprising, given that
Karoubi-localising functors are uniquely determined by their connective
covers, see [9-authors-V]. Lemma 7.4.1 is however a bit more precise, giving
the exact manner in which the eventual dependence on
$\Omega^{\infty}\mathcal{F}$ is expressed.

###### Proof.

Since the composite $\iota\circ\Omega^{\infty}$ is naturally equivalent to the
connective cover functor this amounts to checking that the map of pre-spectrum
objects in presheaves

$\tau_{\geq 0}\mathcal{R}_{S}(\mathcal{F})\to\mathcal{R}_{S}(\mathcal{F})$

is a stable pre-motivic equivalence, that is, induces an equivalences on
mapping spaces to any $\mathbb{P}^{1}$-spectrum object in sheaves. Here, the
connective cover functor $\tau_{\geq 0}$ is applied to the spectrum object
$\mathcal{R}_{S}(\mathcal{F})$ object-wise.

In what follows, let us consider $\mathbb{A}^{1},\mathbb{G}_{m}$ and
$\mathbb{P}^{1}\smallsetminus\\{0\\}$ as pointed schemes over
$\mathrm{spec}(\mathbb{Z})$ with a compatible base point given by
$1\in\mathbb{G}_{m}$ (and its images in $\mathbb{A}^{1}$ and
$\mathbb{P}^{1}\smallsetminus\\{0\\}$). We can then view them as representable
sheaves on $\mathrm{Sm}_{\mathbb{Z}}$ taking values in pointed spaces, and we
write

$B=\mathbb{A}^{1}\amalg_{\mathbb{G}_{m}}(\mathbb{P}^{1}\smallsetminus\\{0\\})\in\operatorname{Psh}(\mathrm{spec}(\mathbb{Z}),\mathcal{S}_{*})$

for the associated pushout calculated in the $\infty$-category of presheaves
of pointed spaces. Since $\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$ is
tensored and cotensored over pointed spaces we can form the associated loop
functor

$\Omega_{B}\colon\operatorname{Psh}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})\to\mathrm{Sh}^{\mathrm{nis}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}),$

and the obvious map $B\to\mathbb{P}^{1}$ induces a natural transformation

$\Omega_{\mathbb{P}^{1}}\Rightarrow\Omega_{B}.$

We now claim that for every pre-spectrum object
$\mathcal{X}_{\bullet}\in{\mathcal{PS}\mathrm{p}}^{\mathbb{P}^{1}}(\operatorname{Psh}(S,{\mathcal{S}\mathrm{p}}))$
the maps

$\mathcal{X}_{\bullet}\to\Omega_{\mathbb{P}^{1}}\mathcal{X}_{\bullet+1}\to\Omega_{B}\mathcal{X}_{\bullet+1}$

are both stable pre-motivic equivalences. To see this, we need to show that
for every
$\mathcal{Y}\in{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))$
the induced map

$\lim_{\stackrel{{\scriptstyle\longleftarrow}}{{n}}}\operatorname{Map}(\Omega_{B}\mathcal{X}_{n+1},\mathcal{Y}_{n})\to\lim_{\stackrel{{\scriptstyle\longleftarrow}}{{n}}}\operatorname{Map}(\Omega_{\mathbb{P}^{1}}\mathcal{X}_{n+1},\mathcal{Y}_{n})\to\lim_{\stackrel{{\scriptstyle\longleftarrow}}{{n}}}\operatorname{Map}(\mathcal{X}_{n},\mathcal{Y}_{n})$

are equivalences. Indeed, since the map $B\to\mathbb{P}^{1}$ is a Nisnevich
local equivalence the maps
$\mathcal{Y}_{n}\to\Omega_{\mathbb{P}^{1}}\mathcal{Y}_{n+1}\to\Omega_{B}\mathcal{Y}_{n+1}$
are both equivalences and so the map
$\operatorname{Map}(\mathcal{X}_{n+1},\mathcal{Y}_{n+1})\to\operatorname{Map}(\mathcal{X}_{n},\mathcal{Y}_{n})$
factors as a composite

$\operatorname{Map}(\mathcal{X}_{n+1},\mathcal{Y}_{n+1})\to\operatorname{Map}(\Omega_{B}\mathcal{X}_{n+1},\mathcal{Y}_{n})\to\operatorname{Map}(\Omega_{\mathbb{P}^{1}}\mathcal{X}_{n+1},\mathcal{Y}_{n})\to\operatorname{Map}(\Omega_{B}\mathcal{X}_{n+1},\mathcal{Y}_{n}).$

The three tower can hence be weaved into one big tower and the desired result
follows from cofinality.

Having shown that $\mathcal{X}_{\bullet}\to\Omega_{B}\mathcal{X}_{\bullet+1}$
is a stable pre-motivic equivalence we may also conclude that the map

$\mathcal{X}_{\bullet}\to
L_{B}\mathcal{X}_{\bullet}=\mathop{\mathrm{colim}}_{n}\Omega^{n}_{B}\mathcal{X}_{\bullet+n}$

is again a stable pre-motivic equivalence. In particular, for
$\mathcal{X}\in\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})$ the left
vertical map in the square

${\tau_{\geq 0}\mathcal{X}_{\bullet}}$${\mathcal{X}}$${L_{B}(\tau_{\geq
0}\mathcal{X}_{\bullet})}$${L_{B}(X_{\bullet})}$$\scriptstyle{\simeq}$$\scriptstyle{\eta}$

is a stable pre-motivic equivalence. To finish the proof we now show that in
the case of $\mathcal{X}_{\bullet}=\mathcal{R}_{S}(\mathcal{F})$, the map

$\eta\colon L_{B}\tau_{\geq
0}\mathcal{R}_{S}(\mathcal{F})\to\mathcal{R}_{S}(\mathcal{F})$

is a stable pre-motivic equivalence. More precisely, we claim that for every
$m,n$ and $T\in\mathrm{Sm}_{/S}$ the map

$\eta\colon\Omega^{n}_{B}\tau_{\geq
0}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\to\Omega^{n}_{B}R(\mathcal{F})_{m}(T)$

is an equivalence on homotopy groups in degrees $\geq-n$. We note that the
claim holds by construction for $n=0$. Arguing by induction, it remains to
show that if $\mathcal{Y}^{\prime}\to\mathcal{R}_{S}(\mathcal{F})_{m}$ is a
map from a presheaf $\mathcal{Y}^{\prime}$ which exhibits
$\mathcal{Y}^{\prime}(T)$ as the $(-n)$-connective cover of
$\mathcal{R}_{S}(\mathcal{F})_{m}(T)$ for every $T\in\mathrm{Sm}_{/S}$, then
the induced map
$\Omega_{B}\mathcal{Y}^{\prime}\to\Omega_{B}R(\mathcal{F})_{m}$ exhibits
$\Omega_{B}\mathcal{Y}^{\prime}(T)$ as the $(-n-1)$-connective cover of
$\Omega_{B}R(\mathcal{F})_{m}(T)$ for every $T\in\mathrm{Sm}_{/S}$. Indeed,
for $T\in\mathrm{Sm}_{/S}$, consider the commutative diagram of spectra

${\Omega_{B}\mathcal{Y}^{\prime}(T)}$${\Omega_{\mathbb{A}^{1}}\mathcal{Y}^{\prime}(T)\oplus\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{Y}^{\prime}(T)}$${\Omega_{\mathbb{G}_{m}}\mathcal{Y}^{\prime}(T)}$${\Omega_{B}R(\mathcal{F})_{m}(T)}$${\Omega_{\mathbb{A}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\oplus\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)}$${\Omega_{\mathbb{G}_{m}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)}$

whose rows are exact by construction. For every pointed scheme
$U\in\mathrm{Sm}_{/\mathrm{spec}(\mathbb{Z})}$ and presheaf $\mathcal{Z}$, the
spectrum $\Omega_{U}\mathcal{Z}(T)$ is a summand of $\mathcal{Z}(T\times U)$,
and hence the middle and right most vertical maps in the above diagrams are
isomorphisms on homotopy groups in degrees $\geq-n$ by the induction
hypothesis. By the long exact sequence in homotopy groups and the five lemma,
it will suffice to show that the square

${\pi_{-n-1}\Omega_{\mathbb{A}^{1}}\mathcal{Y}^{\prime}(T)\oplus\pi_{-n-1}\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{Y}^{\prime}(T)}$${\pi_{-n-1}\Omega_{\mathbb{G}_{m}}\mathcal{Y}^{\prime}(T)}$${\pi_{-n-1}\Omega_{\mathbb{A}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\oplus\pi_{-n-1}\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)}$${\pi_{-n-1}\Omega_{\mathbb{G}_{m}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)}$

induces an isomorphism on horizontal kernels. In the upper row, both terms are
zero by the induction hypothesis (and the fact that $\Omega_{U}\mathcal{Z}(T)$
is a summand of $\mathcal{Z}(T\times U)$ as above), so this amounts to showing
that the bottom horizontal arrow is injective. Since
$\mathcal{R}_{S}(\mathcal{F})_{m}$ is a Nisnevich sheaf, this is equivalent by
the long exact sequence in homotopy groups to the claim that the map

$\pi_{-n-1}\Omega_{\mathbb{P}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}\to\pi_{-n-1}\Omega_{\mathbb{A}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\oplus\pi_{-n-1}\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)$

is the zero map. In fact, we claim that the map of spectra

$\Omega_{\mathbb{P}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\to\Omega_{\mathbb{A}^{1}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)\oplus\Omega_{\mathbb{P}^{1}\smallsetminus\\{0\\}}\mathcal{R}_{S}(\mathcal{F})_{m}(T)$

is a null-homotopic. For this, note that the inclusions $i_{0}\colon
T\times\\{0\\}\hookrightarrow T\times\mathbb{P}^{1}$ and $i_{\infty}\colon
T\times\\{\infty\\}\hookrightarrow T\times\mathbb{P}^{1}$ are regular
codimension 1 embeddings (even if $T$ and $T\times\mathbb{P}^{1}$ are not
necessarily regular), and hence by Lemma 5.1.5 we have associated duality
preserving push-forward functors

$(i_{0})_{*}\colon(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\to(\mathcal{D}^{\mathrm{p}}(T\times\mathbb{P}^{1}),\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\quad\text{and}\quad(i_{\infty})_{*}\colon(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\to(\mathcal{D}^{\mathrm{p}}(T\times\mathbb{P}^{1}),\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}}).$

We now note that composing $(i_{\infty})_{*}$ with pullback along
$\mathbb{A}^{1}\subseteq\mathbb{P}^{1}$ yields the zero functor, and similarly
for composing $(i_{0})_{*}$ with pullback along
$\mathbb{P}^{1}\smallsetminus\\{0\\}\hookrightarrow\mathbb{P}^{1}$. In light
of Example 7.3.7 to finish the proof we now show that the sequences

$\mathcal{F}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\xrightarrow{(i_{0})_{*}}\mathcal{F}(T\times\mathbb{P}^{1},\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\xrightarrow{i_{1}^{*}}\mathcal{F}(T,\Sigma^{m}\mathrm{D}_{T})$

and

$\mathcal{F}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\xrightarrow{(i_{\infty})_{*}}\mathcal{F}(T\times\mathbb{P}^{1},\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\xrightarrow{i_{1}^{*}}\mathcal{F}(T,\Sigma^{m}\mathrm{D}_{T})$

are fibre sequences, where $i_{1}^{*}$ is pullback along the base point
inclusion $T\times\\{1\\}\hookrightarrow T\times\mathbb{P}^{1}$ we have fixed
above (and through which $\Omega_{\mathbb{P}^{1}}$ is defined). For this final
part we use the fact that $\mathcal{F}$ is in particular additive by Example
(4.2.7), and hence the split Poincaré-Verdier sequence (29) yields an exact
sequence

$\mathcal{F}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m}\mathrm{D}_{T})\xrightarrow{p^{*}}\mathcal{F}(\mathcal{D}^{\mathrm{p}}(T\times\mathbb{P}^{1}),\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-1))}\mathcal{F}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T}),$

where $p\colon T\times\mathbb{P}^{1}\to T$ is the projection. Since
$i_{1}\colon T\to T\times\mathbb{P}^{1}$ is a section of $p$ it will now
suffice to show that the composites

$(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\xrightarrow{(i_{0})_{*}}(T\times\mathbb{P}^{1},\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})$

and

$(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})\xrightarrow{(i_{\infty})_{*}}(T\times\mathbb{P}^{1},\Sigma^{m}\mathrm{D}_{T\times\mathbb{P}^{1}})\xrightarrow{p_{*}(-\otimes\mathcal{O}(-1))}(\mathcal{D}^{\mathrm{p}}(T),\Sigma^{m-1}\mathrm{D}_{T})$

are both equivalences of stable $\infty$-categories with duality. In fact,
since forgetting the duality is a conservative functor we may as well just
verify these equivalences on the level of underlying stable
$\infty$-categories. Indeed, by the projection formula, we have

$p_{*}((i_{0})_{*}(-)\otimes\mathcal{O}(-1))\simeq
p_{*}((i_{0})_{*}(-\otimes(i_{0})^{*}\mathcal{O}(-1))\simeq(-)\otimes(i_{0})^{*}\mathcal{O}(-1)\simeq\mathrm{id}$

and

$p_{*}((i_{\infty})_{*}(-)\otimes\mathcal{O}(-1))\simeq
p_{*}((i_{\infty})_{*}(-\otimes(i_{\infty})^{*}\mathcal{O}(-1))\simeq(-)\otimes(i_{\infty})^{*}\mathcal{O}(-1)\simeq\mathrm{id}.\qed$

##  8\. Hermitian K-theory as a motivic spectrum

In this section we build on the results proven so far to generalize the
construction of the hermitian K-theory motivic spectrum
$\operatorname{KQ}_{S}$ over a qcqs scheme $S$, without assuming that 2 is
invertible in $S$ (as is done in previous constructions, such as [hornbostel-
motivic] or [hermitian-bank-robbery]). Given a perfect invertible complex with
$\mathrm{C_{2}}$-action $L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$ we
also construct a motivic spectrum $\operatorname{KQ}_{L}$ of hermitian
$\operatorname{K}$-theory with coefficients in $L$, as well as corresponding
versions $\operatorname{KW}_{S}$ and $\operatorname{KW}_{L}$ of motivic Witt
spectra. We spend the majority of the section in order to establish the key
fundamental properties of these motivic spectra. In particular, we prove:

  1. (1)

$\operatorname{KQ}_{S}$ and $\operatorname{KW}_{S}$ are motivic
${\mathrm{E}_{\infty}}$-ring spectra, and the natural map
$\operatorname{KQ}_{S}\to\operatorname{KW}_{S}$ is an
${\mathrm{E}_{\infty}}$-map whose kernel is
$\operatorname{KGL}_{{\mathrm{h}\mathrm{C_{2}}}}$, where $\operatorname{KGL}$
is the motivic $\operatorname{K}$-theory spectrum. The motivic spectra
$\operatorname{KQ}_{L}$ and $\operatorname{KW}_{L}$ are modules over
$\operatorname{KQ}_{S}$ and $\operatorname{KW}_{S}$.

  2. (2)

The forgetful map $\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}$ is also one
of ${\mathrm{E}_{\infty}}$-ring spectra. It fits in the Wood fibre sequence

$\Omega_{\mathbb{P}^{1}}\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}\to\operatorname{KQ}_{S}.$

  3. (3)

The hermitian $\operatorname{K}$-theory spectrum is absolute: for any map
$T\to S$ of qcqs schemes the natural map
$f^{*}\operatorname{KQ}_{S}\to\operatorname{KQ}_{T}$ is an equivalence, see
Proposition 8.2.1.

  4. (4)

The Thom isomorphism:
$\Omega^{V}\operatorname{KQ}_{L}\simeq\operatorname{KQ}_{L\otimes\det V[-r]}$
for any rank $r$ vector bundle $V\to S$, see Proposition 8.3.1.

  5. (5)

If $S$ is Noetherian of finite Krull dimension then the motivic ring spectrum
$\operatorname{KQ}_{S}$ is _absolutely pure_ , see Theorem 8.4.2.

###  8.1. Motivic hermitian K-theory spectra

Recall the lax symmetric monoidal functor

$\mathcal{R}^{\mathrm{s}}_{S}\colon\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},{\mathcal{S}\mathrm{p}})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))$

of (40), its line bundle generalisation $\mathcal{R}^{\mathrm{s}}_{L}$ from
Construction 7.3.10, and the fact that $\mathbb{K}$, $\mathbb{GW}$ and
$\mathbb{L}$ are ${\mathrm{E}_{\infty}}$-algebra objects in
$\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},{\mathcal{S}\mathrm{p}})$
with respect to Day convolution, that is, each carries a canonical lax
symmetric monoidal structure, see [9-authors-IV], and both the universal
natural transformations $\mathbb{GW}\to\mathbb{K}$ and
$\mathbb{GW}\to\mathbb{L}$ are lax symmetric monoidal.

######  Definition 8.1.1.

We define the motivic ${\mathrm{E}_{\infty}}$-ring spectra
$\operatorname{KGL}_{S}$, $\operatorname{KQ}_{S}$ and
$\operatorname{KW}_{S},\in\operatorname{SH}(S)$ by the formulas

$\operatorname{KGL}_{S}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{K}),\qquad\operatorname{KQ}_{S}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})\quad\text{and}\quad\operatorname{KW}_{S}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{L}).$

Given an invertible perfect complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$, we also define the
motivic spectra
$\operatorname{KQ}_{L},\operatorname{KW}_{L}\in\operatorname{SH}(S)$ by the
formulas

$\operatorname{KQ}_{L}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{L}(\mathbb{GW})\quad\text{and}\quad\operatorname{KW}_{L}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{L}(\mathbb{L}).$

By construction of $\mathcal{R}^{\mathrm{s}}_{L}$, these are naturally modules
over the ${\mathrm{E}_{\infty}}$-ring spectra $\operatorname{KQ}_{S}$ and
$\operatorname{KW}_{S}$, respectively.

######  Remark 8.1.2.

When $L=\mathcal{O}_{S}$, we have
$\operatorname{KQ}_{L}=\operatorname{KQ}_{S}$ and
$\operatorname{KW}_{L}=\operatorname{KW}_{S}$.

Since $\mathbb{K}$ only depends on the underlying stable $\infty$-category, it
is invariant under shifting the Poincaré structure in the input. By Example
7.3.7, this translates to a distinguished equivalence

$\beta\colon\operatorname{KGL}_{S}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\Omega_{\mathbb{P}^{1}}\operatorname{KGL}_{S},$

which we call the Bott map. Similarly, by Ranicki periodicity the motivic
spectra $\operatorname{KQ}_{S}$ and $\operatorname{KW}_{S}$ are equipped with
equivalences of the form

$\widetilde{\beta}\colon\operatorname{KQ}_{S}\to\Omega_{\mathbb{P}^{1}}^{4}\operatorname{KQ}_{S}\quad\text{and}\quad\overline{\beta}\colon\operatorname{KW}_{S}\to\Omega_{\mathbb{P}^{1}}^{4}\operatorname{KW}_{S}.$

######  Remark 8.1.3.

Let
$\operatorname{\mathcal{GW}}^{\natural}\colon\mathrm{Cat}^{\mathrm{p}}\to\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$
be the functor given by

$\operatorname{\mathcal{GW}}^{\natural}(\mathcal{C},\Qoppa)=\operatorname{\mathcal{GW}}((\mathcal{C},\Qoppa)^{\natural})=\Omega^{\infty}\mathbb{GW}(\mathcal{C},\Qoppa),$

see [9-authors-V]. Then $\operatorname{\mathcal{GW}}^{\natural}$ is a Karoubi-
localising functor, and by Lemma 7.4.1, the canonical map

$\iota_{*}\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})\to\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})$

is an equivalence in
${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))$.
In particular, we may equally well have defined $\operatorname{KQ}_{S}$ by the
formula
$\operatorname{KQ}_{S}=\mathrm{Loc}_{\mathbb{A}^{1}}\iota_{*}\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})$,
that is, as the image of
$\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})$ under
the reflection functor

${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}))\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}_{\mathbb{A}^{1}}(S,{\mathcal{S}\mathrm{p}})).$

A similar statement holds for $\operatorname{KGL}$ and $\operatorname{KW}$.

Given a scheme $X$ and an invertible object $L\in\mathcal{D}^{\mathrm{p}}(X)$,
let

$\mathbb{HGW}^{\mathrm{s}}(X,L)=|\mathbb{GW}^{\mathrm{s}}(X\times\Delta^{\bullet},p_{\bullet}^{*}L)|,\quad\mathbb{HL}^{\mathrm{s}}(X,L)=|\mathbb{HL}^{\mathrm{s}}(X\times\Delta^{\bullet},p_{\bullet}^{*}L)|,$

and

$\mathbb{HK}(X)=|\mathbb{K}(X\times\Delta^{\bullet})|,$

where $\Delta^{n}$ is the algebraic $n$-simplex and $p_{n}\colon
X\times\Delta^{n}\to X$ is the projection. When $L$ is the structure sheaf, we
also abbreviate $\mathbb{HGW}^{\mathrm{s}}(X,\mathcal{O})$ as
$\mathbb{HGW}^{\mathrm{s}}(X)$ and $\mathbb{HL}^{\mathrm{s}}(X,\mathcal{O})$
as $\mathbb{HL}^{\mathrm{s}}(X)$. In particular, $\mathbb{HK}$ is homotopy
$\operatorname{K}$-theory, and similarly
$\mathbb{HGW}^{\mathrm{s}}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathbb{GW}^{\mathrm{s}}$
and
$\mathbb{HL}^{\mathrm{s}}=\mathrm{Loc}_{\mathbb{A}^{1}}\mathbb{L}^{\mathrm{s}}$
are the $\mathbb{A}^{1}$-invariant replacements of the Nisnevich sheaves
$\mathbb{GW}^{\mathrm{s}}$ and $\mathbb{L}^{\mathrm{s}}$, respectively. We
refer to them as _homotopy $\mathbb{GW}$-theory_ and _homotopy
$\mathbb{L}$-theory_, respectively.

######  Theorem 8.1.4.

Let $X$ be a smooth $S$-scheme and
$L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$. Then there are natural
equivalences

$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KGL}_{S})$
$\displaystyle\simeq\mathbb{HK}(X),$
$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KQ}_{L})$
$\displaystyle\simeq\mathbb{HGW}^{\mathrm{s}}(X,L[n]|_{X}),\quad\text{and}$
$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KW}_{L})$
$\displaystyle\simeq\mathbb{HL}^{\mathrm{s}}(X,L[n]|_{X}).$

In particular, $\operatorname{KGL}_{S}$ coincides (as a
$\mathbb{P}^{1}$-spectrum object in Nisnevich sheaves, and hence as a motivic
spectrum) with the usual algebraic $\operatorname{K}$-theory motivic spectrum
in a manner that identifies $\beta$ with the usual motivic Bott map.

Combining Theorem 8.1.4 with Theorem 6.3.1, we furthermore obtain:

######  Corollary 8.1.5.

In the situation of Theorem 8.1.4, if $X$ is regular Noetherian of finite
Krull dimension then the equivalences of that corollary become

$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KGL}_{S})$
$\displaystyle\simeq\operatorname{K}(X),$
$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KQ}_{L})$
$\displaystyle\simeq\operatorname{GW}^{\mathrm{s}}(X,L[n]|_{X}),\quad\text{and}$
$\displaystyle\mathrm{hom}_{\operatorname{SH}(S)}(\Sigma^{\infty}_{\mathbb{P}^{1}}X_{+},\Sigma^{n}_{\mathbb{P}^{1}}\operatorname{KW}_{L})$
$\displaystyle\simeq\operatorname{L}^{\mathrm{s}}(X,L[n]|_{X}).$

Moving on, let us exploit a bit more the properties of the motivic realization
functor $\mathcal{R}^{\mathrm{s}}_{S}$. In particular, applying it to the lax
symmetric monoidal transformation $\mathbb{GW}\to\mathbb{L}$ we obtain a map
of ${\mathrm{E}_{\infty}}$-motivic spectra

$w\colon\operatorname{KQ}_{S}\to\operatorname{KW}_{S}.$

Similarly, we write

$\operatorname{KGL}_{S}\xrightarrow{h}\operatorname{KQ}_{S}\xrightarrow{f}\operatorname{KGL}_{S}$

for the maps of motivic spectra obtained by applying
$\mathcal{R}^{\mathrm{s}}_{S}$ to the natural transformations

$\mathbb{K}\stackrel{{\scriptstyle\mathrm{hyp}}}{{\to}}\mathbb{GW}\stackrel{{\scriptstyle\mathrm{fgt}}}{{\to}}\mathbb{K}.$

In particular, since $\mathrm{fgt}$ is a lax symmetric monoidal transformation
we have that $f$ is a map of motivic ${\mathrm{E}_{\infty}}$-ring spectra, and
similarly $h$ is a map of $\mathbb{GW}$-modules. In addition, since the
functor $\mathbb{K}$ carries a $\mathrm{C_{2}}$-action in
$\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},{\mathcal{S}\mathrm{p}})$
such that the natural transformations $\mathrm{hyp}$ and $\mathrm{fgt}$ are
$\mathrm{C_{2}}$-equivariant (with respect to the trivial action on
$\mathbb{GW}$), we have that the same structure is inherited at the level of
motivic spectra, that is, $\operatorname{KGL}_{S}$ carries a
$\mathrm{C_{2}}$-structure such that $f$ and $h$ are
$\mathrm{C_{2}}$-equivariant. In particular, $h$ and $f$ induce maps

$\left(\operatorname{KGL}_{S}\right)_{{\mathrm{h}\mathrm{C_{2}}}}\xrightarrow{h_{{\mathrm{h}\mathrm{C_{2}}}}}\operatorname{KQ}_{S}\xrightarrow{f^{{\mathrm{h}\mathrm{C_{2}}}}}\left(\operatorname{KGL}_{S}\right)_{{\mathrm{h}\mathrm{C_{2}}}}.$

######  Corollary 8.1.6 (Tate sequence).

The maps $w$ and $h_{{\mathrm{h}\mathrm{C_{2}}}}$ fit into a fibre sequence

$\left(\operatorname{KGL}_{S}\right)_{{\mathrm{h}\mathrm{C_{2}}}}\xrightarrow{h_{{\mathrm{h}\mathrm{C_{2}}}}}\operatorname{KQ}_{S}\xrightarrow{w}\operatorname{KW}_{S}.$

###### Proof.

Apply $\mathcal{R}^{\mathrm{s}}_{S}$ to the Karoubi-localising fundamental
fibre sequence
$\mathbb{K}_{{\mathrm{h}\mathrm{C_{2}}}}\Rightarrow\mathbb{GW}\Rightarrow\mathbb{L}$
(see (2)) and use the fact that $\mathcal{R}^{\mathrm{s}}_{S}$ preserves
colimits when the target is stable by Lemma 7.3.9. ∎

######  Corollary 8.1.7 (Wood sequence).

The maps $f\colon\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}$ and
$\operatorname{KGL}_{S}\simeq\Sigma^{\infty}_{\mathbb{P}^{1}}\operatorname{KGL}_{S}\xrightarrow{\Sigma^{\infty}_{\mathbb{P}^{1}}h}\Sigma^{\infty}_{\mathbb{P}^{1}}\operatorname{KQ}_{S}$
fit into a fibre sequence

$\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}\to\Sigma^{\infty}_{\mathbb{P}^{1}}\operatorname{KQ}_{S}.$

###### Proof.

Apply $\mathcal{R}^{\mathrm{s}}_{S}$ to the Karoubi-localising Bott-Genauer
sequence $\mathbb{GW}\Rightarrow\mathbb{K}\Rightarrow\mathbb{GW}((-)^{[1]})$
(see (1)) and use Bott periodicity (Example 7.3.7) to identify the fibre. ∎

######  Remark 8.1.8.

It follows from the projective bundle formula of Theorem 6.1.6 and the Bott
periodicity of Example 7.3.7 that the motivic spectra
$\Omega_{\mathbb{P}^{2}}\operatorname{KQ}_{S}=\operatorname{fib}[\operatorname{KQ}_{S}^{\mathbb{P}^{2}}\to\operatorname{KQ}_{S}]$
and $\operatorname{KGL}_{S}$ coincide as $\mathbb{P}^{1}$-spectra in Nisnevich
sheaves, and hence also as motivic spectra. Unwinding the definitions in
Theorem 6.1.6, we see that under this equivalence, the map

$\Omega_{\mathbb{P}^{2}}\operatorname{KQ}_{S}\to\Omega_{\mathbb{P}^{1}}\operatorname{KQ}_{S}$

induced by a linear hyperplane inclusion
$\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{2}$ coincides with the composite

$\operatorname{KGL}_{S}\stackrel{{\scriptstyle[}}{{\simeq}}]{\beta}{\to}\Omega_{\mathbb{P}^{1}}\operatorname{KGL}_{S}\xrightarrow{\Omega_{\mathbb{P}^{1}}h}\Omega_{\mathbb{P}^{1}}\operatorname{KQ}_{S}.$

It is known that the cofibre of
$\Sigma^{\infty}_{\mathbb{P}^{1}}\mathbb{P}^{1}\hookrightarrow\Sigma^{\infty}_{\mathbb{P}^{1}}\mathbb{P}^{2}$
in $\operatorname{SH}(\mathbb{Z})$ is equivalent to
$\Sigma^{\infty}_{\mathbb{P}^{1}}(\mathbb{P}^{1}\wedge\mathbb{P}^{1})$, and
the boundary map
$\Omega\Sigma^{\infty}_{\mathbb{P}^{1}}(\mathbb{P}^{1}\wedge\mathbb{P}^{1})\to\Sigma^{\infty}_{\mathbb{P}^{1}}\mathbb{P}^{1}$
of this cofibre sequence determines, after $\mathbb{P}^{1}$ delooping, a map

$\eta\colon\Sigma^{\infty}_{\mathbb{P}^{1}}\mathbb{G}_{m}\to\SS,$

often denoted by $\eta$. We hence conclude that the boundary map

$\operatorname{KQ}_{S}\to\Sigma\Omega_{\mathbb{P}^{1}}\operatorname{KQ}_{S}=\Omega_{\mathbb{G}_{m}}\operatorname{KQ}_{S}$

of the Wood fibre sequence is induced by $\eta$.

###  8.2. Pullback invariance of hermitian K-theory

Given a map $f\colon T\to S$ of qcqs schemes, the symmetric monoidal functor
of (13) determines a commutative square of symmetric monoidal functors

(41)
${(\mathrm{Sch}_{S}^{\mathrm{op}})^{\otimes}}$${(\mathrm{Sch}_{T}^{\mathrm{op}})^{\mathrm{op}}}$${\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}}$${\operatorname{Mod}_{T}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}}$$\scriptstyle{T\times_{S}(-)}$$\scriptstyle{(\mathcal{D}^{\mathrm{p}}(T),\mathrm{D}_{T})\otimes_{(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})}(-)}$

where the vertical functors are given by the composites in (13) for $S$ and
$T$, respectively. Running through the constructions of §7.2 and §7.3, we
obtain a commutative square of symmetric monoidal functors

${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))^{\otimes}}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(T,{\mathcal{S}\mathrm{p}}))^{\otimes}}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),{\mathcal{S}\mathrm{p}})^{\otimes}}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{T}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),{\mathcal{S}\mathrm{p}})^{\otimes}}$$\scriptstyle{\widetilde{\mathcal{T}}_{S}}$$\scriptstyle{f^{*}}$$\scriptstyle{\widetilde{\mathcal{T}}_{T}}$

where the bottom horizontal map is given by left Kan extension along the
bottom horizontal arrow of (41) followed by the localisation functor

$\mathrm{Loc}_{\mathrm{b}}\colon\operatorname{Fun}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E})\to\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}^{\flat,\kappa}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),\mathcal{E}).$

Passing to right adjoints we obtain a commutative square of lax symmetric
monoidal functors

${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))^{\otimes}}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(T,{\mathcal{S}\mathrm{p}}))^{\otimes}}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),{\mathcal{S}\mathrm{p}})^{\otimes}}$${\operatorname{Fun}^{\mathrm{bloc}}(\operatorname{Mod}_{T}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}),{\mathcal{S}\mathrm{p}})^{\otimes}}$$\scriptstyle{f_{*}}$$\scriptstyle{\mathcal{R}_{S}}$$\scriptstyle{\mathcal{R}_{T}}$

where the bottom horizontal arrow is now given by pre-composition with
$(\mathcal{D}^{\mathrm{p}}(T),\mathrm{D}_{T})\otimes_{(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})}(-)$.
Given a Karoubi-localising functor
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ the
commutativity of this square and the duality preserving
$\mathcal{D}^{\mathrm{p}}(S)$-linear functor
$f^{*}\colon(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})\to(\mathcal{D}^{\mathrm{p}}(T),\mathrm{D}_{T})$
then determine a map

$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})\to\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F};(\mathcal{D}^{\mathrm{p}}(T),\mathrm{D}_{T}))=\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F}((\mathcal{D}^{\mathrm{p}}(T),\mathrm{D}_{T})\otimes_{(\mathcal{D}^{\mathrm{p}}(S),\mathrm{D}_{S})}(-))=f_{*}\mathcal{R}^{\mathrm{s}}_{T}(\mathcal{F}),$

and consequently an adjoint map

$f^{*}\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})\to\mathcal{R}^{\mathrm{s}}_{T}(\mathcal{F}).$

Applying the $\mathbb{A}^{1}$-localisation functor
$\mathrm{Loc}_{\mathbb{A}^{1}}$ and its commutativity with pullbacks we
finally get a natural map

$f^{*}\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})\to\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{T}(\mathcal{F}).$

relating the motivic realization of $\mathcal{F}$ over $T$ with the pullback
of the motivic realization of $\mathcal{F}$ over $S$. Taking
$\mathcal{F}=\mathbb{GW}$ we write

$\eta_{f}\colon f^{*}\operatorname{KQ}_{S}\to\operatorname{KQ}_{T}$

for the resulting map. Our goal in this section is the prove the following:

######  Proposition 8.2.1.

Let $f\colon T\to S$ be a map of quasi-compact quasi-separated schemes. Then
the map

$f^{*}\operatorname{KQ}_{S}\to\operatorname{KQ}_{T}$

is an equivalence of motivic spectra.

Though the motivic spectrum $\operatorname{KQ}$ is built out of symmetric
$\operatorname{GW}$-theory, the proof will require passing through genuine
$\operatorname{GW}$-theory. For this, recall from Notation 3.3.4 that for a
functor $\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{E}$, an
invertible perfect complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$ and integers
$n,m\in\mathbb{Z}$, we define $\mathcal{F}^{\geq
m,[n]}_{L}\colon\mathrm{Sm}_{S}^{\mathrm{op}}\to\mathcal{E}$ by the formula

$\mathcal{F}^{\geq
m,[n]}_{L}(X)=\mathcal{F}\left(\mathcal{D}^{\mathrm{p}}(X),(\Qoppa^{\geq
m}_{L})^{[n]}\right).$

For $m\in\\{-\infty,+\infty\\}$ we replace the superscripts $\geq-\infty$ and
$\geq+\infty$ with $\mathrm{s}$ and $\mathrm{q}$, and for $L=\mathcal{O}_{S}$
we use the subscript $S$ instead of $\mathcal{O}_{S}$. In addition, when $n=0$
we often drop it from the notation.

Using the genuine refinement of the projective line formula (Proposition
6.2.1), we now observe that Bott periodicity (Example 7.3.7) can be extended
to the case of $\mathcal{F}^{\geq m,[n]}_{L}$, if one accepts to shift $m$:

######  Proposition 8.2.2 (Genuine Bott periodicity).

If $\mathcal{F}$ is additive, for any $k\in\mathbb{N}$ and $m\in\mathbb{Z}$,
there is a canonical equivalence

$\Omega_{\mathbb{P}^{1}}^{2k}\mathcal{F}^{\geq m}_{L}\simeq\mathcal{F}^{\geq
m-k}_{(-1)^{k}L}\ .$

###### Proof.

By Proposition 6.2.1 and the additivity of $\mathcal{F}$, we have

$\Omega_{\mathbb{P}^{1}}\mathcal{F}^{\geq m,[n]}_{L}(X)\simeq\mathcal{F}^{\geq
m,[n-1]}_{L}$

and hence

$\Omega^{k}_{\mathbb{P}^{1}}\mathcal{F}^{\geq
m,[n]}_{L}(X)\simeq\mathcal{F}^{\geq m,[n-k]}_{L}.$

Combining this with Ranicki periodicity we get

$\Omega^{2k}_{\mathbb{P}^{1}}\mathcal{F}^{\geq
m}_{L}(X)\simeq\mathcal{F}^{\geq m,[-2k]}_{L}=\mathcal{F}^{\geq
m-k}_{(-1)^{k}L},$

as desired. ∎

###### Proof of Proposition 8.2.1.

Since
$f_{*}\colon\mathrm{Sh}^{\mathrm{nis}}(T,{\mathcal{S}\mathrm{p}})\to\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})$
sends $\mathbb{A}^{1}$-invariant sheaves to $\mathbb{A}^{1}$-invariant sheaves
we have that
$f^{*}\colon\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}})\to\mathrm{Sh}^{\mathrm{nis}}(T,{\mathcal{S}\mathrm{p}})$
commutes with $\mathrm{Loc}_{\mathbb{A}^{1}}$. Similarly, since $f_{*}$
commutes with post-composition with
${\mathcal{S}\mathrm{p}}\to\operatorname{Grp}_{{\mathrm{E}_{\infty}}}$ we have
after passing to left adjoint a commutative square

${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}))}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(T,\operatorname{Grp}_{{\mathrm{E}_{\infty}}}))}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(S,{\mathcal{S}\mathrm{p}}))}$${{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(T,{\mathcal{S}\mathrm{p}}))\
.}$$\scriptstyle{f^{*}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{f^{*}}$

Combining these two facts with Remark 8.1.3 we conclude that to prove the
desired claim it is enough to show that the map

$f^{*}\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})\to\mathcal{R}^{\mathrm{s}}_{T}(\operatorname{\mathcal{GW}}^{\natural})$

is an equivalence.

By Example 7.3.7 we then have that the $i$’th object in the
$\mathbb{P}^{1}$-spectrum
$\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})$ is
given by $\operatorname{\mathcal{GW}}^{\mathrm{s},[i]}_{S}$. The pullback
functor $f^{*}$ in this context is implemented by applying $f^{*}$ to
$\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{\mathcal{GW}}^{\natural})$ object-
wise and then applying to the resulting $\mathbb{P}^{1}$-prespectrum object
the $\mathbb{P}^{1}$-spectrification operation (which itself preserves
Nisnevich sheaves, since the Nisnevich site is finitary). What we need to show
is consequently that for every $m\geq 0$ the map

$\mathop{\mathrm{colim}}_{n}\Omega_{\mathbb{P}^{1}}^{n}f^{*}\operatorname{\mathcal{GW}}_{S}^{\mathrm{s},[i+n]}\to\operatorname{\mathcal{GW}}_{T}^{\mathrm{s},[i]}$

is an equivalence of Nisnevich sheaves on $T$. Now by cofinality the colimit
on $n$ can be taken in jumps of $4$, that is, we can write this map as

$\mathop{\mathrm{colim}}_{n}\Omega_{\mathbb{P}^{1}}^{4n}f^{*}\operatorname{\mathcal{GW}}_{S}^{\mathrm{s},[i+4n]}\to\operatorname{\mathcal{GW}}_{T}^{\mathrm{s},[i]}.$

At the same time, by Ranicki periodicity we have that
$\operatorname{\mathcal{GW}}_{S}^{\mathrm{s},[i+4n]}=\operatorname{\mathcal{GW}}_{S}^{\mathrm{s},[i]}$
and so we can rewrite this as

$\mathop{\mathrm{colim}}_{n}\Omega_{\mathbb{P}^{1}}^{4n}f^{*}\operatorname{\mathcal{GW}}_{S}^{\mathrm{s},[i]}\to\operatorname{\mathcal{GW}}_{T}^{\mathrm{s},[i]}.$

Finally, to check an equivalence on spectrum object it is enough to check any
infinite family of indices. Taking only the $i$’s that are divisible by 4 and
using again Ranicki periodicity we can simply forget about the $i$ and show
that the map

$\mathop{\mathrm{colim}}_{n}\Omega_{\mathbb{P}^{1}}^{4n}f^{*}\operatorname{\mathcal{GW}}_{S}^{\mathrm{s}}\to\operatorname{\mathcal{GW}}_{T}^{\mathrm{s}}$

is an equivalence of Nisnevich sheaves on $T$.

Let $Q$ be the poset whose objects are pairs $(n,m)$ with $n,m$ non-negative
integers equipped with the order relation such that
$(n,m)\leq(n^{\prime},m^{\prime})$ if and only if $n\leq n^{\prime}$ and
$n+m\leq n^{\prime}+m^{\prime}$. In particular, the projection
$Q\to\mathbb{N}$ sending $(n,m)$ to $n$ is a cartesian fibration such that for
each $n^{\prime}$ the transition functor $\tau\colon
Q_{n+1}=\mathbb{N}\to\mathbb{N}=Q_{n}$ is given by $\tau(x)=x+1$. Specifying a
functor out of $Q$ then amounts to specifying for each $n$ a functor $f_{n}$
out of $Q_{n}=\mathbb{N}$ together with natural transformations
$\eta_{n}\colon\tau^{*}f_{n-1}\Rightarrow f_{n+1}$ for $n\geq 1$. In
particular, we may consider the functor $\theta\colon
Q\to\mathrm{Sh}^{\mathrm{nis}}(S,\operatorname{Grp}_{{\mathrm{E}_{\infty}}})$
given by

$\theta(n,m)=\Omega^{4n}_{\mathbb{P}^{1}_{T}}f^{*}\operatorname{\mathcal{GW}}^{\geq-2m}_{S},$

where the maps

$\eta_{n}\colon\Omega^{4(n-1)}_{\mathbb{P}^{1}_{T}}f^{*}\operatorname{\mathcal{GW}}^{\geq-2m-2}_{S}=\Omega^{4(n-1)}_{\mathbb{P}^{1}_{T}}f^{*}\Omega^{4n}_{\mathbb{P}^{1}_{T}}\operatorname{\mathcal{GW}}^{\geq-2m}_{S}\to\Omega^{4n}_{\mathbb{P}^{1}_{T}}f^{*}\operatorname{\mathcal{GW}}^{\geq-2m}_{S}$

are given by the Beck-Chevalley map