${\mathcal{P}\mathrm{sh}(\mathbf{SmlSm}/S)_{(*)}}$${\mathcal{P}\mathrm{sh}(\mathbf{Sm}/S)_{(*)}}$$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{\rho^{*}\cong\lambda_{\sharp}}$$\scriptstyle{\rho_{*}\cong\lambda^{*}\cong\omega_{\sharp}}$$\scriptstyle{\lambda_{*}\cong\omega^{*}}$$\scriptstyle{\omega_{*}}$

and

(4.0.4)
${\mathcal{P}\mathrm{sh}(\mathbf{lSm}/S)_{(*)}}$${\mathcal{P}\mathrm{sh}(\mathbf{Sm}/S)_{(*)}.}$$\scriptstyle{\rho^{*}\cong\lambda_{\sharp}}$$\scriptstyle{\rho_{*}\cong\lambda^{*}\cong\omega_{\sharp}}$$\scriptstyle{\lambda_{*}\cong\omega^{*}}$$\scriptstyle{\omega_{*}}$

We now recall the construction of the motivic homotopy categories due to Morel
and Voevodsky [70] in our context.

######  Definition 4.0.2.

Suppose $\tau$ is a topology on $\mathbf{Sch}$ generated by a family of
morphisms in $\mathbf{Sm}$. Let

$\mathcal{H}_{(\ast),\tau}\colon\mathbf{Sch}^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})$

be the $\mathbf{Sm}$-premotivic $\infty$-categories associated with
$\mathcal{S}\mathrm{hv}_{\tau,\mathbb{A}^{1}}(\mathbf{Sm}/-)_{(\ast)}$.

Let

$\mathcal{SH}_{S^{1},\tau},\mathcal{SH}_{\tau}\colon\mathbf{Sch}^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})$

be the $S^{1}$ (resp. $T$) stabilization. We often omit $\tau$ in this
notation when $\tau=Nis$.

By e.g., Proposition 2.2.8 we have equivalences of $\infty$-categories

(4.0.5)
$\mathcal{SH}(S)\simeq\operatorname{colim}_{\mathcal{P}\mathrm{r}^{\mathrm{L}}}(\mathcal{H}_{\ast}(S)\xrightarrow{\Sigma_{\mathbb{P}^{1}}}\mathcal{H}_{\ast}(S)\xrightarrow{\Sigma_{\mathbb{P}^{1}}}\cdots)$

and

(4.0.6) $\mathcal{SH}(S)\simeq\mathop{\rm
lim}(\cdots\xrightarrow{\Omega_{\mathbb{P}^{1}}}\mathcal{H}_{\ast}(S)\xrightarrow{\Omega_{\mathbb{P}^{1}}}\mathcal{H}_{\ast}(S)).$

The idea of contracting the projective line $\mathbb{P}^{1}$ instead of the
affine line $\mathbb{A}^{1}$ was considered in [10]. It is not clear how to
compare this theory directly with our construction, but we offer a variant of
Ayoub’s construction, in which we further localize with respect to all
projections $X\times\mathbb{P}^{n}\to X$ for all $X\in\mathbf{Sm}/S$ and
integers $n\geq 1$. We refer to this localization as the
_$\mathbb{P}^{\bullet}$ -localization_. The definition of the
$\mathbb{P}^{\bullet}$-localized motivic homotopy category is completely
straightforward.

######  Definition 4.0.3.

Let

$\mathcal{H}^{\mathbb{P}^{\bullet}}_{(\ast)}\colon\mathbf{Sch}^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})$

be the $\mathbf{Sm}$-premotivic $\infty$-category associated with
$\mathcal{S}\mathrm{hv}_{Nis,\mathbb{P}^{\bullet}}(\mathbf{Sm}/-)_{(\ast)}$.
Let

$\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}\colon\mathbf{Sch}^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})$

be its $S^{1}$-stabilization. We can similarly define
$\mathcal{H}^{\mathbb{P}^{1}}$, $\mathcal{H}_{\ast}^{\mathbb{P}^{1}}$, and
$\mathcal{SH}_{S^{1}}^{\mathbb{P}^{1}}$.

######  Question 4.0.4.

Is there an equivalence of $\mathbf{Sm}$-premotivic $\infty$-categories

$\mathcal{SH}_{S^{1}}^{\mathbb{P}^{1}}\simeq\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}?$

######  Example 4.0.5.

Let $X$ be a scheme. If $\mathcal{F}$ be a quasi-coherent sheaf on $X$, we
regard $\mathcal{F}$ as a Nisnevich sheaf on $\mathbf{Sm}/X$ associated with
the presheaf given by

$\underline{\mathcal{F}}(Y):=\Gamma(Y,(Y\to X)^{*}\mathcal{F}).$

If $X=\operatorname{Spec}(A)$ is affine and $\mathcal{F}$ corresponds to an
$A$-module $M$, then
$R\Gamma(Y,\underline{\mathcal{F}})=R\Gamma(Y,\mathcal{O}_{Y})\otimes_{A}M$;
see [10, §3]. The sheaf $\underline{\mathcal{F}}$ is $\mathbb{P}^{n}$-local
for every $n\geq 1$ by the argument for [10, Proposition 3.3].

######  Construction 4.0.6.

The functor $\rho_{\sharp}$ maps $(\mathbb{P}^{n},\mathbb{P}^{n-1})$ to
$\mathbb{P}^{n}$ for every integer $n\geq 1$ and every strict Nisnevich
distinguished square to a Nisnevich distinguished square. By Corollary 3.5.8,
we obtain a functor of $\mathbf{Sm}$-premotivic $\infty$-categories

$\rho_{\sharp}\colon\mathrm{log}\mathcal{SH}_{S^{1}}\to\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}.$

We denote its right adjoint by $\rho^{*}$.

By using the functor $\rho_{\sharp}$, various properties of motives in
$\mathrm{log}\mathcal{SH}_{S^{1}}$ can be translated into properties of
motives in $\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}$. For example, we have
the following result.

######  Proposition 4.0.7.

Suppose $S\in\mathbf{Sch}$ and $i\colon Z\to X$ is a closed immersion in
$\mathbf{Sm}/S$. Then the projection $p\colon{\mathrm{Bl}}_{Z}X\to X$ is an
equivalence in $\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}(S)$.

###### Proof.

We may assume that $i$ has codimension $d$. Theorem 3.3.5 gives a cocartesian
square

(4.0.7)
${\mathbb{P}({\mathrm{N}}_{Z}X)}$${{\mathrm{Bl}}_{Z}X}$${Z}$${X}$$\scriptstyle{p}$$\scriptstyle{i}$

in $\mathrm{log}\mathcal{SH}_{S^{1}}(S)$. Apply $\rho_{\sharp}$ to (4.0.7) to
see that (4.0.7) is a cocartesian square in
$\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}(S)$ too. Hence it remains to
check that the projection

$\mathbb{P}({\mathrm{N}}_{Z}X)\to Z$

is an equivalence in $\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}(S)$.

By induction on the number of open subsets in a trivialization of the
projective bundle $\mathbb{P}({\mathrm{N}}_{Z}X)$, we reduce to showing that
the projection

$Z\times\mathbb{P}^{n}\to Z$

is an equivalence in $\mathcal{SH}_{S^{1}}^{\mathbb{P}^{\bullet}}(S)$ for
every integer $n\geq 1$. This follows from the definition. ∎

######  Construction 4.0.8.

The functor $\omega_{\sharp}$ maps $\square$ to $\mathbb{A}^{1}$,
$\mathbb{P}^{1}$ to $\mathbb{P}^{1}$, and every strict Nisnevich distinguished
square to a Nisnevich distinguished square. This naturally induces a functor
of $\mathbf{Sm}$-premotivic $\infty$-categories

$\omega_{\sharp}\colon\mathrm{log}\mathcal{SH}\to\mathcal{SH}.$

We similarly have functors of $\mathbf{Sm}$-premotivic $\infty$-categories

$\omega_{\sharp}\colon\mathrm{log}\mathcal{H}\to\mathcal{H},\text{ and
}\omega_{\sharp}\colon\mathrm{log}\mathcal{H}_{\ast}\to\mathcal{H}_{\ast},$

and a functor of $\mathbf{Sm}$-premotivic $\infty$-categories

$\omega_{\sharp}\colon\mathrm{log}\mathcal{SH}_{S^{1}}\to\mathcal{SH}_{S^{1}}.$

Their right adjoints, denoted by $\omega^{*}$, are both fully faithful by [84,
Proposition 2.5.7]. This means there is an embedding of motivic homotopy
theories:

$(\mathbb{A}^{1}\text{-local homotopy theory})\subset(\square\text{-local
homotopy theory})$

######  Remark 4.0.9.

For the convenience of the reader, we outline the proof for
$\omega^{*}\colon\mathcal{H}(S)\to\mathrm{log}\mathcal{H}(S)$. To begin with,
the left adjoint functor
$\omega_{\sharp}\colon\mathrm{log}\mathcal{H}(S)\to\mathcal{H}(S)$ admits a
factorization

$\mathrm{log}\mathcal{H}(S)\xrightarrow{L_{\mathbb{A}^{1}}}(\mathbb{A}^{1})^{-1}\mathrm{log}\mathcal{H}(S)\xrightarrow{\omega_{\sharp}}\mathcal{H}(S),$

where $L_{\mathbb{A}^{1}}$ denotes the localization functor. Let us explain
why the latter $\omega_{\sharp}$ is an equivalence of $\infty$-categories,
which shows the claim since a right adjoint of $L_{\mathbb{A}^{1}}$ is fully
faithful. There is a functor

$\lambda_{\sharp}\colon\mathcal{H}(S)\to(\mathbb{A}^{1})^{-1}\mathrm{log}\mathcal{H}(S)$

sending $X\in\mathbf{Sm}/S$ to $\underline{X}$. We have an isomorphism of
schemes $\omega\lambda(X)\cong X$, and this implies
$\omega_{\sharp}\lambda_{\sharp}\simeq\operatorname{id}$. On the other hand,
we have an isomorphism of schemes $\lambda\omega(Y)\cong Y-\partial Y$ for all
$Y\in\mathbf{lSm}/S$. One can show $Y-\partial Y\simeq Y$ in
$(\mathbb{A}^{1})^{-1}\mathrm{log}\mathcal{H}(S)$ as in [84, Proposition
2.4.3], and this implies
$\lambda_{\sharp}\omega_{\sharp}\simeq\operatorname{id}$. Hence
$\omega_{\sharp}$ is an equivalence of $\infty$-categories.

######  Remark 4.0.10.

If $S\in\mathbf{lSch}$ has a nontrivial log structure and $f\colon X\to S$ is
a log smooth morphism of fs log schemes, then taking the complement
$X-\partial X$ is a drastic operation since it removes everything over the
boundary $\partial S$. Instead, we can remove the “vertical boundary”
$\partial_{S}X$, which is defined to be the set of points $x\in X$ such that
$\mathcal{M}_{S,f(X)}\to\mathcal{M}_{X,x}$ is not vertical in the sense of
[78, Definition I.4.3.1].

However, there is a technical difficulty of removing vertical boundaries
because there is no functor $\omega\colon\mathbf{lSm}/S\to\mathbf{lSm}/S$
sending $X\in\mathbf{lSm}/S$ to $X-\partial_{S}X$ (unless $S$ has trivial log
structure). For example, consider the morphisms

$Z:=\mathbb{A}_{\mathbb{N}}\times\mathbb{A}_{\mathbb{N}}\xrightarrow{h}Y:=\mathbb{A}_{\mathbb{N}}\times\mathbb{A}_{\mathbb{N}}\xrightarrow{g}X:=\mathbb{A}_{\mathbb{N}}\times\mathbb{A}^{1}\xrightarrow{f}S:=\mathbb{A}_{\mathbb{N}},$

where $f$ is the first projection, $g$ removes the log structure on the second
factor, and $h$ is naturally induced by the homomorphism
$\mathbb{N}^{2}\to\mathbb{N}^{2}$ sending $(x,y)$ to $(x,x+y)$. Then we have
isomorphisms of fs log schemes $X-\partial_{S}X\cong X$ and
$Y-\partial_{S}Y\cong\mathbb{A}_{\mathbb{N}}\times{\mathbb{G}}_{m}$. Since the
diagonal homomorphism $\mathbb{N}\to\mathbb{N}^{2}$ is vertical, we also have
an isomorphism $Z-\partial_{S}Z\cong Z$. Taking the existence of $\omega$ for
granted, then $f$ agrees with $\omega(Z)\to\omega(X)$ up to isomorphism.
However, unlike $\omega(Z)\to\omega(X)$, the map $f$ does not factor through
$w(Y)$.

######  Construction 4.0.11.

For $S\in\mathbf{Sch}$, let $\omega\colon\mathbf{cSm}/S\to\mathbf{Sm}/S$ be
the functor sending $X\in\mathbf{cSm}/S$ to $X-\partial X$. As in Construction
4.0.8, we have functors of $\mathbf{Sm}$-premotivic $\infty$-categories

$\omega_{\sharp}\colon\mathrm{log}\mathcal{H}^{\mathbf{Adm}}\to\mathcal{H},\text{
and
}\omega_{\sharp}\colon\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}\to\mathcal{H}_{\ast},$

and functors of $\mathbf{Sm}$-premotivic $\infty$-categories

$\omega_{\sharp}\colon\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}\to\mathrm{log}\mathcal{SH}_{S^{1}},\text{
and
}\omega_{\sharp}\colon\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}\to\mathrm{log}\mathcal{SH}.$

Let $\omega^{*}$ denote the corresponding right adjoints.

######  Construction 4.0.12.

For $S\in\mathbf{Sch}$, let $\psi\colon\mathbf{SmlSm}/S\to\mathbf{cSm}/S$ be
the inclusion functor. This maps $\square$ to $\square$, $\mathbb{P}^{1}$ to
$\mathbb{P}^{1}$, every strict Nisnevich cover to strict Nisnevich cover, and
every admissible blow-up along a smooth center to an admissible blow-up.

Using Corollary 3.5.4 we have a functor of $\mathbf{Sm}$-premotivic
$\infty$-categories

$\psi_{\sharp}\colon\mathrm{log}\mathcal{SH}\to\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}$

We similarly have a functor of $\mathbf{Sm}$-premotivic $\infty$-categories

$\psi_{\sharp}\colon\mathrm{log}\mathcal{SH}_{S^{1}}\to\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}.$

The corresponding right adjoints are denoted by $\psi^{*}$.

Observe that there is a commutative triangle

(4.0.8)
${\mathrm{log}\mathcal{SH}}$${\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}}$${\mathcal{SH}.}$$\scriptstyle{w_{\sharp}}$$\scriptstyle{\psi_{\sharp}}$$\scriptstyle{w_{\sharp}}$

We also have similar commutative triangles for
$\mathrm{log}\mathcal{SH}_{S^{1}}$.

######  Proposition 4.0.13.

Suppose $k$ is a perfect field admitting resolution of singularities. Then the
functors

$\psi_{\sharp}\colon\mathrm{log}\mathcal{SH}_{S^{1}}(k)\to\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(k)\text{
and
}\psi_{\sharp}\colon\mathrm{log}\mathcal{SH}(k)\to\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(k)$

are equivalences. In particular, every map in $\mathbf{Adm}$ is invertible in
$\mathbf{logSH}(k)$.

###### Proof.

We define the functor

$\alpha\colon\mathrm{log}\mathcal{SH}_{S^{1}}(k)\to\mathcal{P}\mathrm{sh}(\mathbf{cSm}/k,\mathcal{S}\mathrm{pt})$

by setting

$\alpha(\mathcal{F})(X):=\mathop{\rm colim}_{Y\in(\mathbf{Adm}\downarrow
X)\cap\mathbf{lSm}/k}\mathcal{F}(Y)$

for all $\mathcal{F}\in\mathrm{log}\mathcal{SH}_{S^{1}}(k)$ and
$X\in\mathbf{cSm}/k$. Here, the colimit runs over the category
$(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$, whose objects are admissible
blow-ups $Y\to X$ with $Y\in\mathbf{lSm}/k$ and whose morphisms are admissible
blow-ups $Z\to Y$ over $X$.

Suppose $Y,Y^{\prime}\in(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$. By
resolution of singularities, there exists a blow-up
$\underline{Y^{\prime\prime}}\to\underline{Y}\times_{\underline{X}}\underline{X^{\prime}}$
such that $\underline{Y^{\prime\prime}}\in\mathbf{Sm}/k$ and the preimage
$Y^{\prime\prime}-\partial Y^{\prime\prime}$ of $X-\partial X$ in
$\underline{Y^{\prime\prime}}$ is the complement of a strict normal crossing
divisor on $\underline{Y^{\prime\prime}}$. We set
$Y^{\prime\prime}:=(\underline{Y^{\prime\prime}},Y^{\prime\prime}-\partial
Y^{\prime\prime})$, which is an object of $\mathbf{SmlSm}/k$. Then the
naturally induced morphisms $Y^{\prime\prime}\to Y,Y^{\prime}$ are admissible
blow-ups, so the category $(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$ is
connected.

Suppose $f,g\colon Y^{\prime}\rightrightarrows Y$ are two morphisms in
$(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$. Since $f$ and $g$ agree on the
dense open subset $Y^{\prime}-\partial Y^{\prime}$ on $Y^{\prime}$, we have
$f=g$. Hence the category $(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$ is
filtered. Together with Theorem 3.2.24, we have
$\alpha(\mathcal{F})(X)\simeq\mathcal{F}(Y)$ whenever
$Y\in(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$.

If $X\in\mathbf{cSm}/k$, then there exists an object $Y$ of
$(\mathbf{Adm}\downarrow X)\cap\mathbf{lSm}/k$ by resolution of singularities.
Since $Y\times\square\in(\mathbf{Adm}\downarrow
X\times\square)\cap\mathbf{lSm}/k$, we have equivalences of spectra

$\alpha(\mathcal{F})(X)\simeq\mathcal{F}(Y)\simeq\mathcal{F}(Y\times\square)\simeq\alpha(\mathcal{F})(X\times\square).$

Hence $\alpha(\mathcal{F})$ is $\square$-invariant, and we obtain the
factorization

$\alpha\colon\mathrm{log}\mathcal{SH}_{S^{1}}(k)\to\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(k).$

Conversely, we have the functor

$\psi^{*}\colon\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(k)\to\mathrm{log}\mathcal{SH}_{S^{1}}(k)$

given by

$\psi^{*}(\mathcal{G})(Y):=\mathcal{G}(Y)$

for all $\mathcal{G}\in\mathrm{log}\mathcal{SH}(k)$ and $Y\in\mathbf{lSm}/k$.
There is a natural equivalence $\alpha\psi^{*}\simeq\operatorname{id}$. Due to
Theorem 3.2.24, we also have a natural equivalence
$\psi^{*}\alpha\simeq\operatorname{id}$. Hence $\psi^{*}$ is an equivalence of
$\infty$-categories. This implies that $\psi_{\sharp}$ is an equivalence of
$\infty$-categories too.

The claim about $\mathrm{log}\mathcal{SH}$ holds true by passing to the
$\mathbb{G}_{m}^{\mathrm{log}}$-stabilization. ∎

######  Remark 4.0.14.

Suppose $k$ is a perfect field admitting resolution of singularities. As
above, one can similarly show that there are equivalences of symmetric
monoidal $\infty$-categories

$\psi_{\sharp}\colon\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(k)\xrightarrow{\simeq}\mathrm{log}\mathcal{H}^{\mathbf{Adm}}(k)\text{
and
}\psi_{\sharp}\colon\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{SmAdm}}(k)\xrightarrow{\simeq}\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(k).$

######  Construction 4.0.15.

The functor $\lambda$ maps $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$ and every
Nisnevich distinguished square to a strict Nisnevich distinguished square. We
obtain functors of $\mathbf{Sm}$-premotivic $\infty$-categories:

$\lambda_{\sharp}\colon\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/-)\to\mathrm{log}\mathcal{H}$
$\lambda_{\sharp}\colon\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/-,\mathcal{S}\mathrm{pt})\to\mathrm{log}\mathcal{SH}_{S^{1}}.$

We denote the right adjoints by $\lambda^{*}$.

######  Question 4.0.16.

Is the functor

$\lambda^{*}\colon\mathrm{log}\mathcal{SH}_{S^{1}}(S)\to\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/S,\mathcal{S}\mathrm{pt})$

fully faithful for $S\in\mathbf{Sch}$?

######  Remark 4.0.17.

We can regard the essential image of $\lambda^{*}$ as a _homotopy theory of
schemes without $\mathbb{A}^{1}$-invariance_. We find it an appealing question
to describe the said image using exclusively the language of schemes with no
mention of logarithmic structures. An analogous (but less structured) problem
was considered in [16, 3.3.5].

######  Construction 4.0.18.

Suppose $S\in\mathbf{Sch}$. Let us compare $\mathrm{log}\mathcal{SH}(S)$ with
the $\infty$-category $\mathbf{MS}_{S}$ (following the notation in [1, §4]) of
$\mathbb{P}^{1}$-spectra due to Annala-Iwasa [2]. Consider the full
subcategory of $\mathcal{P}\mathrm{sh}(\mathbf{Sm}/S,\mathcal{S}\mathrm{pt})$
spanned by the presheaves satisfying Zariski descent and elementary blow-up
excision, see [1, Definition 2.1(i)] for the latter notion. Then
$\mathbf{MS}_{S}$ is defined to be the $\infty$-category of
$\mathbb{P}^{1}$-spectra in this $\infty$-category.

The functor
$\lambda_{\sharp}\colon\mathcal{P}\mathrm{sh}(\mathbf{Sm}/S,\mathcal{S}\mathrm{pt})\to\mathcal{P}\mathrm{sh}(\mathbf{lSm}/S,\mathcal{S}\mathrm{pt})$
maps $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$ and every Zariski distinguished
square to a strict Nisnevich distinguished square and every elementary blow-up
square to an elementary blow-up square. Furthermore, an elementary blow-up
square is cocartesian in $\mathrm{log}\mathcal{SH}^{\mathbf{SmAdm}}(S)$ and
hence in $\mathrm{log}\mathcal{SH}(S)$ by Proposition 3.2.20. It follows that
we have the induced (monoidal) functor of $\mathbf{Sm}$-premotivic
$\infty$-categories

$\lambda_{\sharp}\colon\mathbf{MS}^{\otimes}\to\mathrm{log}\mathcal{SH}^{\otimes}.$

We denote its right adjoint by $\lambda^{*}$. We expect this functor to be
fully faithful for $S\in\mathbf{Sch}$.

######  Remark 4.0.19.

Let us further elaborate on Remark 4.0.17. The construction of
$\mathbf{MS}^{\otimes}_{S}$ is inherently _stable_ (since it is built as the
category of $\mathbb{P}^{1}$-spectra). Our approach allows us to consider an
_unstable_ motivic homotopy theory without $\mathbb{A}^{1}$-invariance, given
by the essential image of
$\lambda^{*}\colon\mathrm{log}\mathcal{H}(S)\to\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/S,\mathcal{S})$
for any $S\in\mathbf{Sch}$, or alternatively by the essential image of

$\lambda^{*}\colon\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)\to\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/S).$

Let us provisionally denote the latter category by
$\mathcal{H}^{\mathbf{SmAdm}}(S)$. It is an interesting question to study
explicitly $\mathcal{H}^{\mathbf{SmAdm}}(S)$, without referring to log
structures. We make a couple of immediate remarks. First, every elementary
blow-up square is cocartesian in
$\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$, thus the same is true in
$\mathcal{H}^{\mathbf{SmAdm}}(S)$; that is, $\mathcal{H}^{\mathbf{SmAdm}}(S)$
is a subcategory of the category of
$\mathcal{S}\mathrm{hv}_{Nis}(\mathbf{Sm}/S,\mathcal{S})$ spanned by those
sheaves $\mathcal{F}$ satisfying elementary blow-up excision. Second, the
Gysin sequence in Remark 3.2.22 suggests that the objects of
$\mathcal{H}^{\mathbf{SmAdm}}(S)$ are local with respect to the inclusions
$\mathbb{P}^{n-1}\to\mathbb{P}^{n}$. We leave the study of this category for
future work.

##  5\. Logarithmic Picard groups and
$\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}$

The goal of this section is to discuss the cohomology of different sheaves of
_units_ associated to a log scheme, namely $\mathcal{M}^{{\mathrm{gp}}}$ and
$\mathbb{G}_{m}^{\mathrm{log}}$. Our main result is Theorem 5.2.10, proving
that for every $S\in\mathbf{Sch}$ there is an equivalence
$\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}\simeq\mathbb{P}^{\infty}$ in
$\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$ (and therefore in
$\mathrm{log}\mathcal{SH}(S,\Lambda)$ for $S\in\mathbf{lSch}$ thanks to
Corollary 3.5.4).

###  5.1. $(\mathbb{P}^{n},\mathbb{P}^{n-1})$-invariance of logarithmic Picard
groups

Let us start with a definition.

######  Definition 5.1.1.

If $X$ is a log scheme, we define the _logarithmic Picard group_ as

(5.1.1)
$\operatorname{logPic}(X):=H_{sNis}^{1}(X,\mathcal{M}^{{\mathrm{gp}}}).$

If $X$ is a scheme, we observe that there is an isomorphism of abelian groups
$\operatorname{logPic}(X)\cong\operatorname{Pic}(X)$ with the Picard group.

######  Remark 5.1.2.

Our $\operatorname{logPic}(X)$ is different from Kato’s original definition in
[49] as

$H_{{s\acute{e}t}}^{1}(X,\mathcal{M}^{{\mathrm{gp}}}),$

which we denote by $\operatorname{logPic}_{{s\acute{e}t}}(X)$. It is natural
to ask whether the canonical homomorphism of abelian groups

(5.1.2) $\operatorname{logPic}(X)\to\operatorname{logPic}_{{s\acute{e}t}}(X)$

is an isomorphism or not. This question is a log version of Hilbert’s theorem
90.

In this subsection we study the $(\mathbb{P}^{n},\mathbb{P}^{n-1})$-invariance
of $\mathcal{M}^{{\mathrm{gp}}}$, $\operatorname{logPic}$, and
$\operatorname{logPic}_{{s\acute{e}t}}$, i.e., whether the naturally induced
homomorphisms of abelian groups

(5.1.3)
$\begin{split}\mathcal{M}^{{\mathrm{gp}}}(X)\to\mathcal{M}^{{\mathrm{gp}}}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}))&,\;\operatorname{logPic}(X)\to\operatorname{logPic}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1})),\\\
\operatorname{logPic}_{{s\acute{e}t}}(X)\to&\operatorname{logPic}_{{s\acute{e}t}}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}))\end{split}$

are isomorphisms or not for, say, fs log schemes $X$.

We will show that the homomorphisms in (5.1.3) are isomorphisms in the
following two cases:

  1. (1)

$S$ is a geometrically unibranch scheme, and $X\in\mathbf{SmlSm}/S$.

  2. (2)

$X$ is a saturated log scheme such that $\underline{X}$ is the spectrum of a
field.

######  Proposition 5.1.3.

Let $X$ be a locally noetherian connected scheme. Then there is an isomorphism
of abelian groups

$\operatorname{Pic}(X)\times\mathbb{Z}\xrightarrow{\cong}\operatorname{Pic}(X\times\mathbb{P}^{n})$

sending $(\mathcal{L},d)\in\operatorname{Pic}(X)\times\mathbb{Z}$ to
$q^{*}\mathcal{L}(d)$, where $q\colon X\times\mathbb{P}^{n}\to X$ is the
projection.

###### Proof.

We refer to [34, Remarque II.4.2.7] and [71, Proposition 3, Lecture 13]. ∎

######  Proposition 5.1.4.

Let $X$ be a unibranch (resp. geometrically unibranch) scheme. Then for every
integer $i>0$, we have the vanishing

$H_{Nis}^{i}(X,\mathbb{Z})=0\text{ (resp.\
}H_{{\acute{e}t}}^{i}(X,\mathbb{Z})=0\text{)}.$

###### Proof.

For the étale case, we refer to [4, Proposition XI.3.6(2)]. The Nisnevich case
can be proven similarly, with reference to [34, Proposition IV.18.12.6]
instead of [34, Proposition IV.18.14.15]. ∎

######  Proposition 5.1.5.

Let $X$ be a geometrically unibranch irreducible scheme. Then the
homomorphisms in (5.1.3) are isomorphisms.

###### Proof.

Let $i\colon X\times\mathbb{P}^{n-1}\to X\times\mathbb{P}^{n}$ be the closed
immersion at $\infty$. There is an exact sequence of sheaves of abelian groups

$0\to\mathcal{M}_{X\times\mathbb{P}^{n}}^{{\mathrm{gp}}}\to\mathcal{M}_{X\times(\mathbb{P}^{n},\mathbb{P}^{n-1})}^{{\mathrm{gp}}}\to
i_{*}\mathbb{Z}\to 0.$

Hence we have a long exact sequence

(5.1.4)
$\begin{split}0&\to\mathcal{M}^{{\mathrm{gp}}}(X\times\mathbb{P}^{n})\to\mathcal{M}^{{\mathrm{gp}}}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}))\to
H_{{\acute{e}t}}^{0}(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})\\\
&\xrightarrow{\delta}\operatorname{Pic}(X\times\mathbb{P}^{n})\to\operatorname{Pic}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}))\to
H_{{\acute{e}t}}^{1}(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})\to\cdots.\end{split}$

Since $X\times\mathbb{P}^{n-1}$ is irreducible, there are isomorphisms of
abelian groups

(5.1.5)
$\Gamma(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})\cong\Gamma(X\times\mathbb{P}^{n-1},\mathbb{Z})\cong\mathbb{Z}.$

Furthermore, since $X\times\mathbb{P}^{n-1}$ is geometrically unibranch,
Proposition 5.1.4 implies the vanishing

(5.1.6)
$H_{Nis}^{1}(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})=H_{Nis}^{1}(X\times\mathbb{P}^{n-1},\mathbb{Z})=H_{{\acute{e}t}}^{1}(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})=H_{{\acute{e}t}}^{1}(X\times\mathbb{P}^{n-1},\mathbb{Z})=0.$

The element $1\in\mathbb{Z}$ corresponds to a section
$s\in\Gamma(X\times\mathbb{P}^{n},i_{*}\mathbb{Z})$. There is a Zariski cover
of $(\mathbb{P}^{n},\mathbb{P}^{n-1})$ consisting of

$U_{i}:=\operatorname{Spec}(\mathbb{N}(x_{n}/x_{i})\to\mathbb{Z}[x_{0}/x_{i},\ldots,x_{n}/x_{i}])$

for $0\leq i\leq n-1$ and

$U_{n}:=\operatorname{Spec}(\mathbb{Z}[x_{0}/x_{n},\ldots,x_{n}/x_{n}]).$

The section $s$ can be locally lifted to

$x_{n}/x_{i}\in\Gamma(X\times U_{i},\mathcal{M}^{\mathrm{gp}})$

for $0\leq i\leq n$. The Zariski cover $\\{X\times U_{i}\\}$ of
$X\times(\mathbb{P}^{n},\mathbb{P}^{n-1})$ defines a sequence

$0\to\bigoplus_{i}\Gamma(X\times
U_{i},\mathcal{M}^{{\mathrm{gp}}})\to\bigoplus_{i\neq
j}\Gamma(X\times(U_{i}\cap U_{j}),\mathcal{M}^{{\mathrm{gp}}}).$

The section $(x_{n}/x_{0},\ldots,x_{n}/x_{n})$ maps to a section whose
components have the form

$x_{j}/x_{i}\in\Gamma(X\times(U_{i}\cap
U_{j}),\mathcal{O}^{*})\subset\Gamma(X\times(U_{i}\cap
U_{j}),\mathcal{M}^{{\mathrm{gp}}}).$

This is exactly the element in the Čech cohomology associated with
$\mathcal{O}(1)\in\operatorname{Pic}(X\times\mathbb{P}^{n})$. Hence the
boundary map $\delta$ in (5.1.4) sends $s$ to
$\mathcal{O}(1)\in\operatorname{Pic}(X\times\mathbb{P}^{n})$. Together with
Proposition 5.1.3, (5.1.5), and (5.1.6) we conclude. ∎

######  Proposition 5.1.6.

Suppose $S$ is a geometrically unibranch scheme and $X\in\mathbf{SmlSm}/S$.
Then the homomorphisms in (5.1.3) are isomorphisms.

###### Proof.

By [34, Proposition 17.5.7], $X$ is geometrically unibranch. Hence we only
need to consider the case when $X$ is irreducible since every connected
component of $X$ is an irreducible component.

We proceed by induction on the number of irreducible components $r$ of
$\partial X$. The case $r=0$ is done in Proposition 5.1.5. If $r>0$, let $D$
be an irreducible component of $\partial X$. Let $Y$ be the fs log scheme
whose underlying scheme is $\underline{X}$ and whose log structure is the
compactifying log structure associated with the open immersion

$X-\overline{\partial X-D}\to X.$

For simplicity of notation, we set

$X^{\prime}:=X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}),\text{
}Y^{\prime}:=Y\times(\mathbb{P}^{n},\mathbb{P}^{n-1}),\text{ and
}D^{\prime}:=D\times(\mathbb{P}^{n},\mathbb{P}^{n-1}).$

Let $i\colon D\to X$ and $i^{\prime}\colon D^{\prime}\to X^{\prime}$ denote
the closed immersions. Since $D$ and $D^{\prime}$ are geometrically unibranch
by [34, Proposition 17.5.7], as in (5.1.5) and (5.1.6) there are isomorphisms
of abelian groups

(5.1.7)
$\Gamma(X,i_{*}\mathbb{Z})\cong\Gamma(X^{\prime},i_{*}^{\prime}\mathbb{Z})\cong\mathbb{Z}.$

and the vanishing

(5.1.8)
$H_{sNis}^{1}(X,i_{*}\mathbb{Z})=H_{sNis}^{1}(X^{\prime},i_{*}^{\prime}\mathbb{Z})=H_{{s\acute{e}t}}^{1}(X,i_{*}\mathbb{Z})=H_{{s\acute{e}t}}^{1}(X^{\prime},i_{*}^{\prime}\mathbb{Z})=0.$

There are exact sequences

$0\to\mathcal{M}_{Y}^{{\mathrm{gp}}}\to\mathcal{M}_{X}^{{\mathrm{gp}}}\to
i_{*}\mathbb{Z}\to 0\text{ and
}0\to\mathcal{M}_{Y^{\prime}}^{{\mathrm{gp}}}\to\mathcal{M}_{X^{\prime}}^{{\mathrm{gp}}}\to
i_{*}^{\prime}\mathbb{Z}\to 0.$

Together with (5.1.7) and (5.1.8), these give a commutative diagram

${0}$${\mathcal{M}^{{\mathrm{gp}}}(Y)}$${\mathcal{M}^{{\mathrm{gp}}}(X)}$${\mathbb{Z}}$${\operatorname{logPic}(Y)}$${\operatorname{logPic}(X)}$${0}$${0}$${\mathcal{M}^{{\mathrm{gp}}}(Y^{\prime})}$${\mathcal{M}^{{\mathrm{gp}}}(X^{\prime})}$${\mathbb{Z}}$${\operatorname{logPic}(Y^{\prime})}$${\operatorname{logPic}(X^{\prime})}$${0}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{\cong}$$\scriptstyle{c}$$\scriptstyle{d}$

with exact rows. We have a similar commutative diagram for
$\operatorname{Pic}_{{s\acute{e}t}}$. By induction, $a$ and $c$ are
isomorphisms. An application of the five lemma shows that $b$ and $d$ are
isomorphisms too. ∎

We do not know whether there is an isomorphism of abelian groups

$\operatorname{logPic}(X)\xrightarrow{\cong}\operatorname{logPic}(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}))$

for arbitrary fs log schemes $X$ and $n\geq 1$. The assumption imposed in
Proposition 5.1.6 seems too restrictive. In the following, we investigate
another case.

######  Proposition 5.1.7.

Let $X$ be a saturated log scheme such that $\underline{X}$ is the spectrum of
a field. Then the homomorphisms in (5.1.3) are isomorphisms.

###### Proof.

We set $G:=\overline{\mathcal{M}}^{{\mathrm{gp}}}(X)$ as abbreviation. The
assumption that $X$ is saturated implies that $G$ is torsion-free. Hence, [4,
Proposition XI.3.6(2)] is applicable to $G$ due to [4, Remarques XI.3.7].
Since $\underline{X}$ and $\underline{X}\times\mathbb{P}^{n-1}$ are
irreducible and geometrically unibranch, as in (5.1.5) and (5.1.6) there are
isomorphisms of abelian groups

(5.1.9)
$\Gamma(X,G)\cong\Gamma(X\times(\mathbb{P}^{n},\mathbb{P}^{n-1}),G)\cong G$

and the vanishing

(5.1.10)
$H_{sNis}^{1}(X,G)=H_{sNis}^{1}(X\times\mathbb{P}^{n-1},G)=H_{{s\acute{e}t}}^{1}(X,G)=H_{{s\acute{e}t}}^{1}(X\times\mathbb{P}^{n-1},G)=0.$

Let
$i\colon\underline{X}\times\mathbb{P}^{n-1}\to\underline{X}\times\mathbb{P}^{n}$
be the closed immersion at $\infty$. There are exact sequences

$0\to\mathcal{M}_{\underline{X}}^{\mathrm{gp}}\to\mathcal{M}^{\mathrm{gp}}_{X}\to
G\to 0\text{ and
}0\to\mathcal{M}_{\underline{X}\times(\mathbb{P}^{n},\mathbb{P}^{n-1})}^{{\mathrm{gp}}}\to\mathcal{M}^{\mathrm{gp}}_{X\times(\mathbb{P}^{n},\mathbb{P}^{n-1})}\to
i_{*}G\to 0.$

As abbreviations, we set $Y:=\underline{X}$,
$X^{\prime}=X\times(\mathbb{P}^{n},\mathbb{P}^{n-1})$, and
$Y^{\prime}:=\underline{X}\times(\mathbb{P}^{n},\mathbb{P}^{n-1})$. Together
with (5.1.9) and (5.1.10), these give a commutative diagram with exact rows:

${0}$${\mathcal{M}^{{\mathrm{gp}}}(Y)}$${\mathcal{M}^{{\mathrm{gp}}}(X)}$${G}$${\operatorname{logPic}(Y)}$${\operatorname{logPic}(X)}$${0}$${0}$${\mathcal{M}^{{\mathrm{gp}}}(Y^{\prime})}$${\mathcal{M}^{{\mathrm{gp}}}(X^{\prime})}$${G}$${\operatorname{logPic}(Y^{\prime})}$${\operatorname{logPic}(X^{\prime})}$${0.}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{\cong}$$\scriptstyle{c}$$\scriptstyle{d}$

We have a similar commutative diagram for
$\operatorname{Pic}_{{s\acute{e}t}}$. By Proposition 5.1.5, $a$ and $c$ are
isomorphisms. The five lemma lets us conclude that also $b$ and $d$ are
isomorphisms. ∎

###  5.2. The infinite projective space

Morel-Voevodsky showed the equivalence

(5.2.1) $\mathbb{P}^{\infty}\xrightarrow{\simeq}\mathrm{B}{\mathbb{G}}_{m}$

in $\mathcal{H}(S)$, where $S\in\mathbf{Sch}$, see [70, Proposition 3.7, p.
138]. As argued in this text, $\mathbb{G}_{m}^{\mathrm{log}}$ replaces
${\mathbb{G}}_{m}$ in $\square$-motivic homotopy theory. The purpose of this
section is to show that there is an equivalence

(5.2.2) $\mathbb{P}^{\infty}\to\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}$

in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$ for every $S\in\mathbf{Sch}$.

There is an explicit description of $\mathbb{A}^{1}$ -homotopies that yields
the equivalence (5.2.1) in [74, §2]. Our method of establishing (5.2.2) is to
“compactify” their argument.

######  Definition 5.2.1.

In this subsection, we use the abbreviation

$V_{n}:=({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),H+E)$

for every integer $n\geq 1$, where $O$ is the point $[0:\cdots:0:1]$ in
$\mathbb{P}^{n}$, $H$ is the divisor $\mathbb{P}^{n}-\mathbb{A}^{n}$, and $E$
is the exceptional divisor. Observe that the log structure on $V_{n}$ is the
compactifying log structure associated with the open immersion
$\mathbb{A}^{n}-0\to{\mathrm{Bl}}_{O}(\mathbb{P}^{n})$.

There is a closed immersion

$\mathbb{P}^{n}\to\mathbb{P}^{n+1}$

sending $[a_{1}:\cdots:a_{n+1}]$ to $[a_{1}:\cdots:a_{n}:0:a_{n+1}]$. Since
the preimage of $O\in\mathbb{P}^{n+1}$ is $O\in\mathbb{P}^{n}$, there is a
naturally induced morphism

(5.2.3)
${\mathrm{Bl}}_{O}(\mathbb{P}^{n})\to{\mathrm{Bl}}_{O}(\mathbb{P}^{n+1}).$

This is an extension of the closed immersion

(5.2.4) $\mathbb{A}^{n}-0\to\mathbb{A}^{n+1}-0$

sending $(a_{1},\ldots,a_{n})$ to $(a_{1},\ldots,a_{n},0)$. Combine (5.2.3)
and (5.2.4) to produce a strict closed immersion

(5.2.5) $i\colon V_{n}\to V_{n+1}.$

Finally, we form a colimit in the category of presheaves

$V_{\infty}:=\mathop{\rm colim}_{n}V_{n}.$

######  Lemma 5.2.2.

Let $S\in\mathbf{Sch}$ and
$\operatorname{pt}=\mathcal{F}_{0}\to\mathcal{F}_{1}\to\cdots$ be an infinite
sequence in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$. Suppose for every
integer $i\geq 0$, there exists $j>i$ such that
$\mathcal{F}_{i}\to\mathcal{F}_{j}$ is equivalent to the constant morphism
given by the composite

$\mathcal{F}_{i}\to\operatorname{pt}\to\mathcal{F}_{1}\to\cdots\to\mathcal{F}_{j}.$

Then $\mathop{\rm colim}_{i}\mathcal{F}_{i}$ is equivalent to
$\operatorname{pt}$ in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$.

###### Proof.

Due to Corollary 2.7.2, it suffices to show that

$\operatorname{Hom}_{\mathbf{logH}^{\mathbf{SmAdm}}(S)}(\Sigma_{S^{1}}^{n}X,\mathop{\rm
colim}_{i}\mathcal{F}_{i})$

is trivial for all $X\in\mathbf{lSm}/S$ and integer $n\geq 0$. By a standard
reduction, we can assume that $S=\operatorname{Spec}(A)$ is affine, and
noetherian. Since $X$ is a compact object (see Remark 2.7.4), it suffices to
show that

(5.2.6) $\mathop{\rm
colim}_{i}\operatorname{Hom}_{\mathbf{logH}^{\mathbf{SmAdm}}(S)}(\Sigma_{S^{1}}^{n}X,\mathcal{F}_{i})$

is trivial. If $f\colon\Sigma_{S^{1}}^{n}X\to\mathcal{F}_{i}$ is any morphism,
the composite

$\Sigma_{S^{1}}^{n}X\to\mathcal{F}_{i}\to\mathcal{F}_{j}$

is equivalent to the constant morphism given by the composite

$\Sigma_{S^{1}}^{n}X\to\operatorname{pt}\to\mathcal{F}_{1}\to\cdots\to\mathcal{F}_{j}.$

This shows the equivalence in (5.2.6). ∎

For every integer $n\geq 1$, let $v_{1},\ldots,v_{n}$ be the points
$(1,0,\ldots,0)$, $\ldots$, $(0,\ldots,0,1)$ in $\mathbb{A}^{n}-0$.

######  Lemma 5.2.3.

Suppose $S\in\mathbf{Sch}$. For every integer $n\geq 1$, the inclusion

$i\colon V_{n}\rightarrow V_{n+1}$

in (5.2.5) is equivalent to the constant map to $v_{n+1}\in V_{n+1}$ in
$\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$.

###### Proof.

Let $u\colon S\to\operatorname{Spec}(\mathbb{Z})$ be the structure morphism.
If we show the claim for $\operatorname{Spec}(\mathbb{Z})$, then we can use
$u^{*}$ to deduce the claim for $S$. Hence we may assume
$S=\operatorname{Spec}(\mathbb{Z})$.

The closed immersion $v$ in (5.2.4) is $\mathbb{A}^{1}$-homotopic to a
constant map via the morphism

$h\colon(\mathbb{A}^{n}-0)\times\mathbb{A}^{1}\to\mathbb{A}^{n+1}-0$

sending $(a_{1},\ldots,a_{n},t)$ to $(ta_{1},\ldots,ta_{n},1-t)$. We use the
coordinates $\\{x_{1},\ldots,x_{n}\\}$ for $\mathbb{A}^{n}$, $t$ for
$\mathbb{A}^{1}$, and $\\{x_{1},\ldots,x_{n},y\\}$ for $\mathbb{A}^{n+1}$. The
morphism $h$ induces a homomorphism

$u\colon\mathbb{Z}(x_{1},\ldots,x_{n},y)\to\mathbb{Z}(x_{1},\ldots,x_{n},t)$

sending $x_{i}$ to $tx_{i}$ for $1\leq i\leq n$ and $y$ to $1-t$.

We note that $h$ extends to a morphism

$g\colon{\mathrm{Bl}}_{E\times[1:1]+H\times[0:1]}({\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1})\to{\mathrm{Bl}}_{O}(\mathbb{P}^{n+1}),$

where $E$ is the exceptional divisor on $\mathbb{P}^{n}$, and
$H:=\mathbb{P}^{n}-\mathbb{A}^{n}$. To show this, we will define $g$ Zariski
locally as follows. The scheme ${\mathrm{Bl}}_{O}(\mathbb{P}^{n+1})$ has a
Zariski cover consisting of spectra of the rings

$\displaystyle
A_{i}:=\mathbb{Z}[x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},y/x_{i}],\;B:=\mathbb{Z}[y,x_{1}/y,\ldots,x_{n}/y],$
$\displaystyle
C_{i}:=\mathbb{Z}[1/x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},y/x_{i}],\;D:=\mathbb{Z}[1/y,x_{1}/y,\ldots,x_{n}/y]$

for $1\leq i\leq n$. The scheme
${\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1}$ has a Zariski cover
consisting of spectra of the rings

$\displaystyle
E_{i}:=\mathbb{Z}[x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},1/(1-t)],\;F_{i}:=\mathbb{Z}[x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},t,1/t],$
$\displaystyle
G_{i}:=\mathbb{Z}[1/x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},t,1/(1-t)],\;H_{i}:=\mathbb{Z}[1/x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},1/t]$

for $1\leq i\leq n$. The scheme
${\mathrm{Bl}}_{E\times[1:1]+H\times[0:1]}({\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1})$
has a Zariski cover consisting of spectra of the rings $E_{i}$, $H_{i}$, and

$\displaystyle
F_{i}^{\prime}:=\mathbb{Z}[x_{i},(1-t)/x_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},1/t],$
$\displaystyle
F_{i}^{\prime\prime}:=\mathbb{Z}[1-t,x_{i}/(1-t),x_{1}/x_{i},\ldots,x_{n}/x_{i},1/t],$
$\displaystyle
G_{i}^{\prime}:=\mathbb{Z}[1/x_{i},tx_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},1/(1-t)],$
$\displaystyle
G_{i}^{\prime\prime}:=\mathbb{Z}[t,1/(tx_{i}),x_{1}/x_{i},\ldots,x_{n}/x_{i},1/(1-t)]$

for $1\leq i\leq n$. For convenience, we include the following expressions

$\displaystyle
u(A_{i})=\mathbb{Z}[tx_{i},x_{1}/x_{i},\ldots,x_{n}/x_{i},(1-t)/(tx_{i})],$
$\displaystyle u(B_{i})=\mathbb{Z}[1-t,tx_{1}/(1-t),\ldots,tx_{n}/(1-t)],$
$\displaystyle
u(C_{i})=\mathbb{Z}[1/(tx_{i}),x_{1}/x_{i},\ldots,x_{n}/x_{i},(1-t)/(tx_{i})],$
$\displaystyle u(D_{i})=\mathbb{Z}[1/(1-t),tx_{1}/(1-t),\ldots,tx_{n}/(1-t)].$

There are inclusions

$u(D_{i})\subset E_{i},\;u(A_{i})\subset F_{i}^{\prime},\;u(B_{i})\subset
F_{i}^{\prime\prime},\;u(D_{i})\subset G_{i}^{\prime},\;u(C_{i})\subset
G_{i}^{\prime\prime},\;u(C_{i})\subset H_{i}.$

From these inclusions, we can construct the morphism $g$ that extends $h$.

Let $X$ be the fs log scheme with the underlying scheme
${\mathrm{Bl}}_{E\times[1:1]+H\times[0:1]}({\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1})$,
and whose log structure is the compactifying log structure associated with the
open immersion

$(\mathbb{A}^{n}-0)\times\mathbb{A}^{1}\to{\mathrm{Bl}}_{E\times[1:1]+H\times[0:1]}({\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1}).$

Next, $g$ gives a morphism

$f\colon Y\to V_{n+1}$

extending $h$, and there is a projection $p\colon Y\to V_{n}\times\square$.
The zero section and one section $i_{0},i_{1}\colon V_{n}\to
V_{n}\times\square$ admits lifts

$i_{0}^{\prime},i_{1}^{\prime}\colon V_{n}\to Y.$

Since $p$ is a composite of admissible blow-ups along smooth centers, we can
apply Lemma 3.2.12 to deduce that $i=fi_{1}^{\prime}$ is equivalent to the
constant map $fi_{0}^{\prime}$ in
$\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$. ∎

######  Remark 5.2.4.

We note that $h$ appearing in Lemma 5.2.3 cannot be extended to a morphism

${\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1}\to{\mathrm{Bl}}_{O}(\mathbb{P}^{n+1})$

since $g$ does not factor through the blow-down
${\mathrm{Bl}}_{O}(\mathbb{P}^{n})\times\mathbb{P}^{1}$.

######  Lemma 5.2.5.

For every $S\in\mathbf{lSch}$ and $n\geq 2$, the two constant maps

$\operatorname{pt}\rightrightarrows V_{n}$

to $v_{1}$ and $v_{n}$ are equivalent in $\mathrm{log}\mathcal{H}(S)$.

###### Proof.

Let $f\colon\mathbb{P}^{1}\to\mathbb{P}^{n}$ be the morphism given by

$[a:b]\mapsto[a:0\cdots:0:b-a:b].$

Since $f(\mathbb{A}^{1})\subset\mathbb{A}^{n}$, $f$ naturally induces a
morphism $g\colon\square\to(\mathbb{P}^{n},H)$, where $H$ is the divisor
$\mathbb{P}^{n}-\mathbb{A}^{n}$. The image of $f$ does not intersect with the
point $O:=[0:\cdots:0:1]$, so $g$ can be lifted to a morphism

$h\colon\square\to V_{n}.$

To conclude, observe that $h([0:1])=v_{n}$ and $h([1:1])=v_{1}$. ∎

######  Proposition 5.2.6.

For every $S\in\mathbf{Sch}$, the object $V_{\infty}$ is equivalent to
$\operatorname{pt}$ in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$.

###### Proof.

By Lemmas 5.2.3 and 5.2.5, the morphism $V_{n}\to V_{n+1}$ is equivalent to
the the constant map to $v_{1}$ in
$\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$. Then use Lemma 5.2.2. ∎

######  Definition 5.2.7.

Suppose $S\in\mathbf{lSch}$ and
$G\in\mathcal{S}\mathrm{hv}_{dNis}(\mathbf{lSm}/S)$ is a group object acting
on $X\in\mathcal{S}\mathrm{hv}_{dNis}(\mathbf{lSm}/S)$. Let $X/^{h}G$ denote
the homotopy colimit of the simplicial object

(5.2.7) ${\cdots}$${X\times G^{2}}$${X\times G}$${X.}$

If $X\to Y$ is a $G$-equivariant morphism in
$\mathcal{S}\mathrm{hv}_{dNis}(\mathbf{lSm}/S)$ with trivial $G$-action on
$Y$, there is a naturally induced morphism $X/^{h}G\to Y$.

######  Example 5.2.8.

For every $S\in\mathbf{lSch}$ there is no morphism

$m\colon\mathbb{G}_{m}^{\mathrm{log}}\times\mathbb{G}_{m}^{\mathrm{log}}\to\mathbb{G}_{m}^{\mathrm{log}}$

in $\mathbf{lSm}/S$ that extends the multiplication
${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}\to{\mathbb{G}}_{m}$ sending $(x,y)$ to
$xy$ since it is impossible to define $m$ at $(0,\infty)$ and $(\infty,0)$.
However, $m$ exists in $\mathcal{S}\mathrm{hv}_{dNis}(\mathbf{lSm}/S)$. To
show this, consider the morphism

$p\colon
Y:=({\mathrm{Bl}}_{(0,\infty)\cup(\infty,0)}(\mathbb{P}^{1}\times\mathbb{P}^{1}),\widetilde{H_{1}}+\widetilde{H_{2}}+\widetilde{H_{3}}+\widetilde{H_{4}}+E_{1}+E_{2})\to\mathbb{G}_{m}^{\mathrm{log}}\times\mathbb{G}_{m}^{\mathrm{log}},$

where $\widetilde{H_{i}}$ is the strict transforms of $0\times\mathbb{P}^{1}$,
$\mathbb{P}^{1}\times 0$, $\infty\times\mathbb{P}^{1}$, or
$\mathbb{P}^{1}\times\infty$, and $E_{1}$ and $E_{2}$ are the exceptional
divisors. Since $p$ is a dividing cover, $p$ is an equivalence as a morphism
of sheaves. We now claim that there exists a morphism $n\colon
Y\to\mathbb{G}_{m}^{\mathrm{log}}$ extending the multiplication morphism
${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}\to{\mathbb{G}}_{m}$. To show this,
glue the morphisms of schemes associated with the six homomorphisms

$\mathbb{Z}[x]\to\mathbb{Z}[x,y],\;\mathbb{Z}[xy,1/y],\;\mathbb{Z}[x,1/(xy)],\;\mathbb{Z}[1/x,xy],\;\mathbb{Z}[1/(xy),y],\;\mathbb{Z}[1/x,1/y]$

sending $x$ to $xy$, and check that the morphisms are compatible with the log
structures. We set $m$ to be the composite $np^{-1}$.

The unit of $\mathbb{G}_{m}^{\mathrm{log}}$ is defined to be the one section
$u\colon 1\to\mathbb{G}_{m}^{\mathrm{log}}$. The morphism
$\mathbb{P}^{1}\to\mathbb{P}^{1}$ sending $[a:b]$ to $[b:a]$ sends
${\mathbb{G}}_{m}$ onto ${\mathbb{G}}_{m}$. By [78, Proposition III.1.6.2],
this naturally induces a morphism
$i\colon\mathbb{G}_{m}^{\mathrm{log}}\to\mathbb{G}_{m}^{\mathrm{log}}$.

With the construction described above, the fact that ${\mathbb{G}}_{m}$ is a
group object readily implies that $\mathbb{G}_{m}^{\mathrm{log}}$ is a group
object in $\mathcal{S}\mathrm{hv}_{dNis}(\mathbf{lSm}/S)$.

######  Lemma 5.2.9.

Suppose $S\in\mathbf{lSch}$. There is an equivalence

(5.2.8)
$V_{n+1}/^{h}\mathbb{G}_{m}^{\mathrm{log}}\xrightarrow{\simeq}\mathbb{P}^{n}$

in $\mathrm{log}\mathcal{H}(S)$.

###### Proof.

The scheme $\mathbb{P}^{n}$ has a Zariski cover consisting of

$U_{i}:=\operatorname{Spec}(\mathbb{Z}[x_{0}/x_{i},\ldots,x_{n}/x_{i}])$

for $0\leq i\leq n$. The fs log scheme $V_{n+1}$ has a Zariski cover
consisting of

$V_{i}:=\operatorname{Spec}(\mathbb{N}(x_{i})\to\mathbb{Z}[x_{i},x_{0}/x_{i},\ldots,x_{n}/x_{i}])$

and

$V_{i}^{\prime}:=\operatorname{Spec}(\mathbb{N}(1/x_{i})\to\mathbb{Z}[1/x_{i},x_{0}/x_{i},\ldots,x_{n}/x_{i}])$

for $0\leq i\leq n$, where $\mathbb{N}(x_{i})$ and $\mathbb{N}(1/x_{i})$
denotes the free monoid generated by $x$ and $1/x_{i}$. There is a canonical
isomorphism of fs log schemes

$V_{i}\cup V_{i}^{\prime}\cong U_{i}\times\mathbb{G}_{m}^{\mathrm{log}}.$

If $I$ is a subset of $\\{0,\ldots,n\\}$, we set

$U_{I}:=\cap_{i\in I}U_{i},\text{ }V_{I}:=\cap_{i\in I}V_{i},\text{ and
}V_{I}^{\prime}:=\cap_{i\in I}V_{i}^{\prime}.$

Then there is a canonical isomorphism of fs log schemes

(5.2.9) $V_{I}\cup V_{I}^{\prime}\cong
U_{I}\times\mathbb{G}_{m}^{\mathrm{log}}.$

Glue these all together to have a morphism

$f\colon V_{n+1}\to\mathbb{P}^{n}.$

Furthermore, from (5.2.9), we have a $\mathbb{G}_{m}^{\mathrm{log}}$-action on
$V_{n+1}$. The morphism $f$ is $\mathbb{G}_{m}^{\mathrm{log}}$-equivariant
with trivial $\mathbb{G}_{m}^{\mathrm{log}}$-action on $\mathbb{P}^{n}$. This
constructs the morphism in (5.2.8). We show that it is an equivalence. By
considering the Čech nerve of $\mathbb{P}^{n}$ associated with
$\\{U_{i}\\}_{1\leq i\leq n}$, it suffices to show

$(U_{I}\times\mathbb{G}_{m}^{\mathrm{log}})/^{h}\mathbb{G}_{m}^{\mathrm{log}}\simeq
U_{I}$

for all subsets $I$ of $\\{0,\ldots,n\\}$. This follows since the simplicial
diagram (5.2.7) defining
$(U_{I}\times\mathbb{G}_{m}^{\mathrm{log}})/^{h}\mathbb{G}_{m}^{\mathrm{log}}$
admits an evident retraction. ∎

######  Theorem 5.2.10.

For every $S\in\mathbf{Sch}$ there is an equivalence

$\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}\simeq\mathbb{P}^{\infty}$

in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$, which is natural in $S$.

###### Proof.

Colimits preserve equivalences, so Lemma 5.2.9 yields an equivalence

(5.2.10)
$V_{\infty}/^{h}\mathbb{G}_{m}^{\mathrm{log}}\simeq\mathbb{P}^{\infty}$

in $\mathrm{log}\mathcal{H}^{\mathbf{SmAdm}}(S)$. Proposition 5.2.6 and [70,
Proposition 2.14, p. 74] finish the proof. ∎

######  Corollary 5.2.11.

For every $S\in\mathbf{lSch}$, there is an equivalence

$\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}\simeq\mathbb{P}^{\infty}$

in $\mathrm{log}\mathcal{SH}^{\rm eff}(S,\Lambda)$ and
$\mathrm{log}\mathcal{SH}(S,\Lambda)$, which is natural in $S$.

###### Proof.

The claim holds for $\underline{S}$ by Proposition 3.5.4 and Theorem 5.2.10.
We deduce the claim for $S$ by pulling back. ∎

##  6\. $K$-theory of fs log schemes

###  6.1. Setting and summary of the results

Morel and Voevodsky [70, Propositions 3.7 and 3.9] proved that the algebraic
$K$-theory of regular schemes is represented geometrically by the infinite
Grassmannian $\mathbb{Z}\times{\mathrm{Gr}}$ in the unstable motivic homotopy
category $\mathcal{H}(S)_{\ast}$ for any noetherian scheme $S$ of finite Krull
dimension. This result was later refined by Voevodsky in [98], where the Bott
periodic $\mathbb{P}^{1}$-spectrum ${\mathbf{KGL}}$ was constructed. In [98]
Voevodsky announced that ${\mathbf{KGL}}$ represents homotopy invariant
algebraic $K$-theory for schemes in the _stable_ motivic homotopy category.
Cisinski completed the proof in [28, Théorème 2.20]. See also [81] for a
detailed discussion.

The main goal of this section is to construct a logarithmic $K$-theory
spectrum $\mathbf{logKGL}$ in $\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$ for
any regular scheme $S$ (see Definition 6.5.6), and show that there is an
equivalence

(6.1.1)
$\mathbf{logKGL}\simeq(\mathbb{L}_{sNis,\square\cup\mathbf{Adm}}\mathbb{Z}\times{\mathrm{Gr}},\mathbb{L}_{sNis,\square\cup\mathbf{Adm}}\mathbb{Z}\times{\mathrm{Gr}},\ldots)$

in $\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$. Here the pointed log motivic
space $\mathbb{L}_{sNis,\square\cup\mathbf{Adm}}\mathbb{Z}\times{\mathrm{Gr}}$
can be viewed as a logarithmic replacement for the infinite Grassmannian
employed by Morel-Voevodsky [70]. Our proof of the equivalence (6.1.1) uses
the singular functor $\operatorname{Sing}$ discussed in §6.3. This is the root
cause for assuming that $S$ is a regular scheme in (6.1.1). At the end of this
section, we will compare our logarithmic $K$-theory with Niziol’s $K$-theories
of fs log schemes defined in [77]. Elementary computations verify that the two
theories are different.

The main application of the above-mentioned result will be discuss in §8.6,
where we will use the geometric representability of logarithmic $K$-theory for
constructing a logarithmic cyclotomic trace map, as a morphism internal to our
motivic homotopy theory. Note that the corresponding construction would
dramatically fail in $\mathcal{SH}(S)$.

######  Remark 6.1.1.

Let
$\mathrm{K}\in\mathcal{P}\mathrm{sh}(\mathbf{Sch}_{noeth},\mathcal{S}\mathrm{pt})$
denote the Bass $K$-theory presheaf of spectra. If $X\in\mathbf{Sch}$, then

$\pi_{n}\mathrm{K}(X)\cong\mathrm{K}_{n}(X)$

for all integers $n$, where $\mathrm{K}_{n}(X)$ is the $n$th Bass $K$-group of
perfect complexes on $X$.

Due to [70, Propositions 3.9], there is a canonical equivalence of sheaves of
pointed spaces

(6.1.2)
$\Omega_{S^{1}}^{\infty}\mathrm{K}\simeq\mathbb{L}_{Nis}\Omega\mathrm{B}\big{(}\coprod_{n\geq
0}{\mathrm{BGL}}_{n}\big{)},$

where $\mathbb{L}_{Nis}$ is the Nisnevich localization functor, and
$\Omega_{S^{1}}^{\infty}$ is the infinite suspension functor.

If $X\in\mathbf{Sch}_{noeth}$ is regular, then the Bass $K$-theory spectrum
$\mathrm{K}(X)$ and $\Omega_{S^{1}}^{\infty}$ participates in an equivalence
of spectra

(6.1.3)
$\mathrm{K}(X)\simeq(\Omega_{S^{1}}^{\infty}\mathrm{K}(X),\Omega_{\mathbb{G}_{m}}\Omega_{S^{1}}^{\infty}\mathrm{K}(X),\Omega_{\mathbb{G}_{m}}^{2}\Omega_{S^{1}}^{\infty}\mathrm{K}(X),\ldots),$

where the bonding maps

$\Sigma_{S^{1}}\Omega_{\mathbb{G}_{m}}^{n}\Omega_{S^{1}}^{\infty}\mathrm{K}(X)\to\Omega_{\mathbb{G}_{m}}^{n+1}\Omega_{S^{1}}^{\infty}\mathrm{K}(X)$

are naturally induced by the map

$\Omega_{S^{1}}^{\infty}\mathrm{K}\to\Omega_{\mathbb{P}^{1}}\Omega_{S^{1}}^{\infty}\mathrm{K}$

appearing in the projective bundle formula [95, Theorem 7.3] for $K$-theory.
Restricting $\mathrm{K}$ to $\mathbf{Sm}/S^{\mathrm{op}}$ defines an object of
$\mathcal{SH}_{S^{1}}(S)$, since $\mathrm{K}(-)$ is $\mathbb{A}^{1}$-invariant
for regular schemes and satisfies Nisnevich descent.

###  6.2. The singular functor

In this subsection, we begin by reviewing the singular functor in [22, §6].

Throughout, $\mathbf{T}$ is a small ordinary category with finite limits,
$\operatorname{pt}$ denotes the initial object, and $\mathcal{A}$ is a set of
morphisms in $\mathbf{T}$ that admits a calculus of right fractions. Moreover,
$I$ is an interval object of $\mathbf{T}$ in the sense of [70, §2.3], i.e., an
object of $\mathbf{T}$ together with morphisms

$i_{0},i_{1}\colon\operatorname{pt}\to I,\;p\colon I\to\operatorname{pt}$

in $\mathbf{T}$ and with a morphism

$\mu\colon I\times I\to I$

in $\mathbf{T}[\mathcal{A}^{-1}]$ satisfying the identities

$\displaystyle\mu\circ(i_{0}\times\operatorname{id})=(\operatorname{id}\times
i_{0})\circ\mu=i_{0}p,$
$\displaystyle\mu\circ(i_{1}\times\operatorname{id})=(\operatorname{id}\times
i_{1})\circ\mu=\operatorname{id}.$

We also assume that $i_{0}\amalg
i_{1}\colon\operatorname{pt}\amalg\operatorname{pt}\to I$ is a monomorphism.

There is a localization functor

$v\colon\mathbf{T}\rightarrow\mathbf{T}[\mathcal{A}^{-1}]$

inducing an adjunction

${\mathcal{P}\mathrm{sh}(\mathbf{T})}$${\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}]).}$$\scriptstyle{v_{\sharp}}$$\scriptstyle{v^{*}}$$\scriptstyle{v_{*}}$

Here $v_{\sharp}$ is left adjoint to $v^{*}$, and $v^{*}$ is left adjoint to
$v_{*}$. Let $\mathcal{A}\downarrow X$ denote the over category of $X$, whose
objects are the morphisms $f\colon Y\to X$ in $\mathcal{A}$, with the obvious
convention for maps. The functors $v_{\sharp}$ and $v^{*}$ are then given
objectwise by

(6.2.1) $v_{\sharp}\mathcal{F}(X)\cong\mathop{\rm
colim}_{Y\in\mathcal{A}\downarrow X}\mathcal{F}(Y),\quad
v^{*}\mathcal{G}(X):=\mathcal{G}(X)$

for any presheaf of Kan complexes
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{T})$,
$\mathcal{G}\in\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}])$, and
$X\in\mathbf{T}$.

For abbreviation, let $I$ be the set of projections $X\times I\to X$ for all
$X\in\mathbf{T}$.

######  Proposition 6.2.1.

The unit $\operatorname{id}\to v^{*}v_{\sharp}$ is an $\mathcal{A}$-local
equivalence.

###### Proof.

We refer to [22, Proposition 6.1.11]. ∎

By assumption, $I$ is an interval object of $\mathbf{T}[\mathcal{A}^{-1}]$ in
the sense of [70, §2.3]. Recall that there is a functor

$\operatorname{Sing}^{I}\colon\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}])\to\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}]).$

that is defined as follows: let $\Delta_{I}^{\bullet}$ be the simplicial
object of $\mathbf{T}[\mathcal{A}^{-1}]$ defined objectwise by $[n]\mapsto
I^{n}$ (see [70, p.88], or [22, 6.2]). Then we have the $\infty$-functor

$\mathcal{F}\mapsto\underline{\operatorname{Map}}(y(\Delta_{I}^{\bullet}),\mathcal{F})\in\operatorname{Fun}(\Delta^{\mathrm{op}},\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}]).$

We let $\operatorname{Sing}^{I}(\mathcal{F})$ be the geometric realization of
$\underline{\operatorname{Map}}(y(\Delta_{I}^{\bullet}),\mathcal{F})$. Using
e.g., [63, A.2.9.30] one can readily verify that it agrees with [22,
Definition 6.2.1].

######  Definition 6.2.2.

The singular functor

$\operatorname{Sing}^{I\cup\mathcal{A}}\colon\mathcal{P}\mathrm{sh}(\mathbf{T})\to\mathcal{P}\mathrm{sh}(\mathbf{T})$

is given by $v^{*}\operatorname{Sing}^{I}v_{\sharp}$.

######  Definition 6.2.3.

Let $f,g\colon\mathcal{F}\to\mathcal{G}$ be two morphisms in
$\mathcal{P}\mathrm{sh}(\mathbf{T})$. An _elementary
$I\cup\mathcal{A}$-homotopy between $f$ and $g$_ is a morphism

$h\colon v_{\sharp}\mathcal{F}\times I\to v_{\sharp}\mathcal{G}$

in $\mathcal{P}\mathrm{sh}(\mathbf{T}[\mathcal{A}^{-1}])$ such that

$h\circ i_{0}=f\text{ and }h\circ i_{1}=g.$

Two morphisms are called _$I\cup\mathcal{A}$ -homotopic_ if they can be
connected by a finite number of elementary $I\cup\mathcal{A}$-homotopies.

A morphism $f\colon\mathcal{F}\to\mathcal{G}$ in
$\mathcal{P}\mathrm{sh}(\mathbf{T})$ is called a _strict
$I\cup\mathcal{A}$-homotopy equivalence_ if there exists a morphism
$g\colon\mathcal{G}\to\mathcal{F}$ in $\mathcal{P}\mathrm{sh}(\mathbf{T})$
such that the compositions $f\circ g$ and $g\circ f$ are both
$I\cup\mathcal{A}$-homotopic to the respective identities.

######  Proposition 6.2.4.

The following statements hold true.

  1. (1)

The morphism

$i_{0}\colon\operatorname{pt}\to I$

is a strict $I\cup\mathcal{A}$-homotopy equivalence.

  2. (2)

If $f$ and $g$ are $I\cup\mathcal{A}$-homotopic morphisms in
$\mathcal{P}\mathrm{sh}(\mathbf{T})$, then
$\operatorname{Sing}^{I\cup\mathcal{A}}(f)$ and
$\operatorname{Sing}^{I\cup\mathcal{A}}(g)$ are homotopic.

  3. (3)

For every $X\in\mathbf{T}$, the morphism

$\operatorname{Sing}^{I\cup\mathcal{A}}(X)\to\operatorname{Sing}^{I\cup\mathcal{A}}(X\times\square)$

naturally induced by $i_{0}$ is an equivalence equivalence of presheaves of
Kan complexes.

  4. (4)

For every $\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{T})$, the canonical
morphism

$\mathcal{F}\to\operatorname{Sing}^{I\cup\mathcal{A}}(\mathcal{F})$

is an $I\cup\mathcal{A}$-local equivalence.

###### Proof.

We refer to [22, Propositions 6.2.3, 6.2.4, 6.2.6, Corollary 6.2.5]. ∎

###  6.3. Applications of $\operatorname{Sing}$

Our theory of $\operatorname{Sing}$ is mainly applied to the case when
$\mathbf{T}=\mathbf{cSm}/S$, where $S$ is a scheme, $I=\square$, and
$\mathcal{A}$ is the set of admissible blow-ups $\mathbf{Adm}$ in
$\mathbf{cSm}/S$.

######  Proposition 6.3.1.

For every $S\in\mathbf{Sch}$, the class of admissible blow-ups in
$\mathbf{cSm}/S$ admits a calculus of right fractions.

###### Proof.

It is obvious that the class of admissible blow-ups is closed under
compositions.

The class of admissible blow-ups is closed under pullbacks, so the class of
admissible blow-ups satisfies the right Ore condition.

Suppose $f,g\colon Y\to X$ and $h\colon X\to X^{\prime}$ be morphisms in
$\mathbf{cSm}/S$ such that $hf=hg$. Then $f$ and $g$ agree on the dense open
subscheme $Y-\partial Y$ of $Y$, so we have $f=g$. This shows that the class
of admissible blow-ups satisfies the right cancellability condition. ∎

######  Proposition 6.3.2.

For every $S\in\mathbf{Sch}$ and presheaf
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{cSm}/S)$, the canonical morphism

$\mathcal{F}\to\operatorname{Sing}^{\square\cup\mathbf{Adm}}(\mathcal{F})$

is a $\square\cup\mathbf{Adm}$-local equivalence.

###### Proof.

Immediate from Proposition 6.2.4(4). ∎

######  Proposition 6.3.3.

Suppose $S\in\mathbf{Sch}$ and
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{cSm}/S)$ is of the form
$\mathop{\rm colim}\mathcal{F}_{i}$, where each $\mathcal{F}_{i}$ is
representable by a smooth and proper scheme over $S$. Then the unit map

$\mathcal{F}\rightarrow\omega^{*}\omega_{\sharp}\mathcal{F}$

is a $\square\cup\mathbf{Adm}$-local equivalence.

###### Proof.

By Proposition 6.2.1, there is a $\square\cup\mathbf{Adm}$-local equivalence

$\mathcal{F}\to v^{*}v_{\sharp}\mathcal{F}.$

Use (6.2.1) to have an equivalence of Kan complexes

$v^{*}v_{\sharp}\mathcal{F}(Y)\cong\mathop{\rm
colim}_{Y^{\prime}\in\mathbf{Adm}\downarrow Y}\mathcal{F}(Y^{\prime}).$

On the other hand, there is an equivalence

$\omega^{*}\omega_{\sharp}\mathcal{F}(Y)\cong\mathcal{F}(Y-\partial Y).$

Hence we only need to show that the map

$\mathop{\rm colim}_{Y^{\prime}\in\mathbf{Adm}\downarrow
Y}\mathcal{F}(Y^{\prime})\to\mathcal{F}(Y-\partial Y)$

is an equivalence. We only need to show this for each $\mathcal{F}_{i}$. Hence
we only need to show that the map

(6.3.1) $\mathop{\rm colim}_{Y^{\prime}\in\mathbf{Adm}\downarrow
Y}\operatorname{Hom}(Y^{\prime},X)\to\operatorname{Hom}(Y-\partial Y,X)$

is an isomorphism for every smooth and proper scheme $X$ over $S$.

Suppose $f\colon Y-\partial Y\to X$ is a morphism of schemes. Let
$\underline{Y^{\prime}}$ be the closure of the image of the graph morphism
$Y-\partial Y\to(Y-\partial Y)\times\underline{X}$ in
$\underline{Y}\times\underline{X}$. By setting
$Y^{\prime}:=(\underline{Y^{\prime}},Y^{\prime}-\partial Y^{\prime})$ and
$Y^{\prime}-\partial Y^{\prime}$ as the preimage of $Y-\partial Y$ in
$Y^{\prime}$, we obtain an admissible blow-up $p$ participating in a
commutative diagram of fs log schemes

${Y^{\prime}}$${Y}$${X.}$$\scriptstyle{p}$$\scriptstyle{g}$$\scriptstyle{f}$

This shows that (6.3.1) is surjective.

It remains to show that (6.3.1) is injective. Suppose $Y^{\prime}$ and
$Y^{\prime\prime}$ are two admissible blow-ups of $Y$ with two morphisms
$f^{\prime}\colon Y^{\prime}\to X$ and $f^{\prime\prime}\colon
Y^{\prime\prime}\to X$ such that the restrictions of $f^{\prime}$ and
$f^{\prime\prime}$ to $Y-\partial Y$ are equal. We need to show that there
exists a common admissible blow-up $Y^{\prime\prime\prime}$ of $Y^{\prime}$
and $Y^{\prime\prime}$ such that we have the same morphism from $f^{\prime}$
and $f^{\prime\prime}$ by composing $Y^{\prime\prime\prime}\to Y^{\prime}$ and
$Y^{\prime\prime\prime}\to Y^{\prime\prime}$. A common admissible blow-up
always exists since $\mathbf{Adm}\downarrow Y$ admits a calculus of right
fractions by Proposition 6.3.1, so we may assume that
$Y^{\prime}=Y^{\prime\prime}$. To conclude, observe that $Y-\partial Y$ is
dense in $Y^{\prime}$ and $X$ is separated. ∎

######  Lemma 6.3.4.

For $\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{cSm}/S)$, where
$S\in\mathbf{Sch}$, there is a natural isomorphism

$\operatorname{Sing}^{\square\cup\mathbf{Adm}}\omega^{*}\mathcal{F}\cong\omega^{*}\operatorname{Sing}^{\mathbb{A}^{1}}\mathcal{F}.$

###### Proof.

Let $\mathcal{G}\in\mathcal{P}\mathrm{sh}(\mathbf{cSm}/S)$. Recall that
$\operatorname{Sing}^{\square\cup\mathbf{Adm}}\mathcal{G}$ is defined as the
geometric realization of the simplicial presheaf

$[n]\mapsto\operatorname{Sing}_{n}^{\square\cup\mathbf{Adm}}\mathcal{G}=v^{*}\operatorname{Sing}_{n}^{\square}v_{\sharp}\mathcal{G}=v^{*}\underline{\operatorname{Map}}(\square^{n},v_{\sharp}\mathcal{G}).$

For every $X\in\mathbf{cSm}/S$, we have then

(6.3.2)
$\operatorname{Sing}_{n}^{\square\cup\mathbf{Adm}}\omega^{*}\mathcal{F}(X)=\mathop{\rm
colim}_{Y\in\mathbf{Adm}\downarrow(X\times\square^{n})}\omega^{*}\mathcal{F}(Y)=\mathop{\rm
colim}_{Y\in\mathbf{Adm}\downarrow(X\times\square^{n})}\mathcal{F}(Y-\partial
Y).$

If $Y\to X\times\square^{n}$ is an admissible blow-up, then the morphism

$Y-\partial Y\to(X-\partial X)\times\mathbb{A}^{n}$

is an isomorphism of schemes. Hence we have

(6.3.3) $\mathop{\rm colim}_{Y\in\mathbf{Adm}\downarrow
X\times\square^{n}}\mathcal{F}(Y-\partial Y)\cong\mathcal{F}((X-\partial
X)\times\mathbb{A}^{n})\cong\operatorname{Sing}_{n}^{\mathbb{A}^{1}}\mathcal{F}(X-\partial
X)=\omega^{*}\operatorname{Sing}_{n}^{\mathbb{A}^{1}}\mathcal{F}(X).$

Combine (6.3.2) and (6.3.3) to have an isomorphism of presheaves

$\operatorname{Sing}_{n}^{\square\cup\mathbf{Adm}}\omega^{*}\mathcal{F}\cong\omega^{*}\operatorname{Sing}_{n}^{\mathbb{A}^{1}}\mathcal{F}$

To conclude, observe that the degeneracy and face maps for
$\operatorname{Sing}_{n}^{\square\cup\mathbf{Adm}}\omega^{*}\mathcal{F}$ and
$\omega^{*}\operatorname{Sing}_{n}^{\mathbb{A}^{1}}\mathcal{F}$ are compatible
with this isomorphism. ∎

###  6.4. Infinite Grassmannians and $\Omega_{S^{1}}^{\infty}\mathrm{K}$

In the following, we review how to obtain the infinite dimensional
Grassmannians from the finite dimensional ones.

######  Definition 6.4.1.

Fix integers $0<r<n$. Let ${\mathrm{Gr}}(r,n)$ denote the $(r,n)$
Grassmannian, that is, the scheme representing the functor
$S\mapsto{\mathrm{Gr}}(r,n)(S)$, where ${\mathrm{Gr}}(r,n)(S)$ is the set of
isomorphism classes of surjections $\mathcal{O}^{\oplus n}_{S}\to\mathcal{Q}$,
where $\mathcal{Q}$ is a locally free $\mathcal{O}_{S}$-module of rank $n-r$,
see [93, Tag 089R].

We equip ${\mathrm{Gr}}(r,n)$ with the trivial log structure. In particular,
for every log scheme $X$, there is a bijection

$\operatorname{Hom}(X,{\mathrm{Gr}}(r,n))\cong\operatorname{Hom}(\underline{X},{\mathrm{Gr}}(r,n)).$

There are canonical morphisms

${\mathrm{Gr}}(r,n)\to{\mathrm{Gr}}(r,n+1)\text{ and
}{\mathrm{Gr}}(r,n)\to{\mathrm{Gr}}(r+1,n+1)$

taking $V\subset\mathcal{O}^{n}$ to
$V\oplus\\{0\\}\subset\mathcal{O}^{n}\oplus\mathcal{O}$ and
$V\oplus\mathcal{O}\subset\mathcal{O}^{n}\oplus\mathcal{O}$, where
$\mathcal{O}$ denotes the standard linear space of dimension $1$. From these,
we can form colimits

${\mathrm{Gr}}(r,\infty):=\mathop{\rm colim}_{n\geq r}{\mathrm{Gr}}(r,n)\text{
and }{\mathrm{Gr}}:=\mathop{\rm colim}_{r}{\mathrm{Gr}}(r,\infty).$

Observe that ${\mathrm{Gr}}(r,\infty)$ and ${\mathrm{Gr}}$ are filtered
colimits of smooth schemes. There is a natural morphism

${\mathrm{Gr}}(r,n)\times{\mathrm{Gr}}(r^{\prime},n^{\prime})\to{\mathrm{Gr}}(r+r^{\prime},n+n^{\prime})$

sending $(V,W)$ to $V\oplus W$. This gives a (non-commutative) monoid
structure on the coproduct $\coprod_{r=1}^{\infty}{\mathrm{Gr}}(r,\infty)$ in
the category of log schemes.

######  Proposition 6.4.2.

If $X$ is a regular affine noetherian scheme, then the canonical morphism

(6.4.1)
$\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}})(X)\to\Omega_{S^{1}}^{\infty}\mathrm{K}(X)$

is an equivalence of Kan complexes. In particular, if $Y$ is a regular
noetherian scheme, then the canonical morphism

(6.4.2)
$L_{Nis}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}})(Y)\to\Omega_{S^{1}}^{\infty}\mathrm{K}(Y)$

is an equivalence of Kan complexes.

###### Proof.

For (6.4.1), see the proof of [70, Proposition 3.16, p. 141]. This implies
that (6.4.2) is a weak equivalence since $\Omega_{S^{1}}^{\infty}\mathrm{K}$
satisfies Nisnevich descent. ∎

######  Proposition 6.4.3.

Suppose $S$ is a regular noetherian scheme. Then for all $X\in\mathbf{cSm}/S$,
there is a canonical equivalence of spaces

$\Omega_{S^{1}}^{\infty}\mathrm{K}(X-\partial
X)\simeq\operatorname{Map}_{\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)}(X_{+},\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}).$

###### Proof.

For every $X\in\mathbf{cSm}/S$, we have an equivalence

$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}(X)\simeq\Omega_{S^{1}}^{\infty}\mathrm{K}(X-\partial
X).$

The $\mathbb{A}^{1}$-invariance of $K$-theory implies that
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$ is $\square$-local. If $Y\to X$
is an admissible blow-up, then $Y-\partial Y\to X-\partial X$ is an
isomorphism of schemes. It follows that
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$ is $\mathbf{Adm}$-local.
Furthermore, together with Lemma 2.3.23, we deduce that
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$ is strict Nisnevich local since
$\mathrm{K}$ is Nisnevich local. In conclusion,
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$ is
$(sNis,\square\cup\mathbf{Adm})$-local. This gives the desired equivalence. ∎

The following is the most important result in this subsection, which shows
that the unstable $K$-theory spectrum
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$ admits a geometric model in our
logarithmic setting.

######  Theorem 6.4.4.

Suppose $S$ is a regular scheme. Then there is a canonical equivalence

(6.4.3)
$\mathbb{Z}\times{\mathrm{Gr}}\to\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$

in $\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$.

###### Proof.

Due to Proposition 6.3.3, there is a $\square\cup\mathbf{Adm}$-local
equivalence

(6.4.4)
$\mathbb{Z}\times{\mathrm{Gr}}\simeq\omega^{*}(\mathbb{Z}\times{\mathrm{Gr}}).$

Apply Proposition 6.3.2 to have a $\square\cup\mathbf{Adm}$-local equivalence

(6.4.5)
$\omega^{*}(\mathbb{Z}\times{\mathrm{Gr}})\simeq\operatorname{Sing}^{\square\cup\mathbf{Adm}}\omega^{*}(\mathbb{Z}\times{\mathrm{Gr}}).$

Furthermore, Lemma 6.3.4 gives an equivalence

(6.4.6)
$\operatorname{Sing}^{\square\cup\mathbf{Adm}}\omega^{*}(\mathbb{Z}\times{\mathrm{Gr}})\cong\omega^{*}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}}).$

Next we combine (6.4.4), (6.4.5), and (6.4.6) to deduce the
$(sNis,\square\cup\mathbf{Adm})$-local equivalence

(6.4.7) $\mathbb{Z}\times{\mathrm{Gr}}\simeq
L_{sNis}\omega^{*}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}}).$

There is a canonical morphism of presheaves

(6.4.8)
$\omega^{*}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}})\to\omega^{*}L_{Nis}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}}).$

If $x$ is a point of $X\in\mathbf{cSm}/S$, then there exists an open affine
neighborhood $U$ of $x$ in $X$ such that the restriction $\partial U$ of the
Cartier divisor $\partial X$ to $U$ is principal. We note that $U-\partial U$
is affine. By Proposition 6.4.2, the morphism of Kan complexes

$\omega^{*}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}})(U)\to\omega^{*}L_{Nis}\operatorname{Sing}^{\mathbb{A}^{1}}(\mathbb{Z}\times{\mathrm{Gr}})(U)$

obtained from (6.4.8) becomes $\Omega_{S^{1}}^{\infty}\mathrm{K}(U-\partial
U)\xrightarrow{\simeq}\Omega_{S^{1}}^{\infty}\mathrm{K}(U-\partial U)$. This
means that $L_{sNis}$ turns (6.4.8) into a pointwise weak equivalence.
Together with (6.4.2) and (6.4.7), we obtain the desired equivalence. ∎

###  6.5. The logarithmic $K$-theory spectrum

Next we explain how to construct a $(2,1)$-periodic log motivic spectrum from
a pointed log space satisfying the following condition.

######  Definition 6.5.1.

A pointed log motivic space $\mathcal{F}\in\mathrm{log}\mathcal{H}_{\ast}(S)$
satisfies the _loops $\mathbb{P}^{1}$-bundle formula_ if there exists an
equivalence

$\alpha_{\mathcal{F}}\colon\mathcal{F}\to\Omega_{\mathbb{P}^{1}}\mathcal{F}$

in $\mathrm{log}\mathcal{H}_{\ast}(S)$.

######  Construction 6.5.2.

If a pointed log motivic space
$\mathcal{F}\in\mathrm{log}\mathcal{H}_{\ast}(S)$ satisfies the loops
$\mathbb{P}^{1}$-bundle formula, then by adjunction we have a morphism

$\beta_{\mathcal{F}}\colon\Sigma_{\mathbb{P}^{1}}\mathcal{F}\to\mathcal{F}.$

We use $\beta_{\mathcal{F}}$ as the bonding maps in the log
$\mathbb{P}^{1}$-spectrum

$\mathrm{Bott}(\mathcal{F}):=(\mathcal{F},\mathcal{F},\ldots)\in\mathrm{log}\mathcal{SH}(S).$

This is called _the logarithmic motivic Bott spectrum of $\mathcal{F}$_. By
construction, there are canonical equivalences in
$\mathrm{log}\mathcal{SH}(S)$

(6.5.1)
$\mathrm{Bott}(\mathcal{F})\simeq\Sigma_{\mathbb{P}^{1}}\mathrm{Bott}(\mathcal{F})$

and

(6.5.2)
$\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}^{\infty}\mathrm{Bott}(\mathcal{F})\simeq(\mathcal{F},\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}\mathcal{F},\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}^{2}\mathcal{F},\ldots).$

The bonding maps in (6.5.2) are induced by
$\mathcal{F}\to\Omega_{\mathbb{P}^{1}}\mathcal{F}\simeq\Omega_{S^{1}}\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}\mathcal{F}$.

######  Construction 6.5.3.

If a pointed log motivic space
$\mathcal{F}\in\mathrm{log}\mathcal{H}_{\ast}(S)$ satisfies the loops
$\mathbb{P}^{1}$-bundle formula, then we can view $\mathrm{Bott}(\mathcal{F})$
as a $(\mathbb{P}^{1})^{\wedge 2}$-spectrum

(6.5.3) $(\mathcal{F},\mathcal{F},\ldots)$

with bonding maps given by the composite
$\Sigma_{\mathbb{P}^{1}}^{2}\mathcal{F}\to\Sigma_{\mathbb{P}^{1}}\mathcal{F}\to\mathcal{F}$.
We can also view $\mathrm{Bott}(\mathcal{F})\wedge\mathrm{Bott}(\mathcal{F})$
as a $(\mathbb{P}^{1})^{\wedge 2}$-spectrum

(6.5.4) $(\mathcal{F}\wedge\mathcal{F},\mathcal{F}\wedge\mathcal{F},\ldots),$

where the bonding maps are given by

$\mathcal{F}\wedge\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}\xrightarrow{\simeq}\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathcal{F}\wedge\mathbb{P}^{1}\xrightarrow{\beta_{\mathcal{F}}\wedge\beta_{\mathcal{F}}}\mathcal{F}\wedge\mathcal{F}.$

Suppose $\mathcal{F}$ is a homotopy commutative monoid in
$\mathrm{log}\mathcal{H}_{\ast}(S)$ with multiplication
$\mu\colon\mathcal{F}\wedge\mathcal{F}\to\mathcal{F}$ such that the diagram

(6.5.5)
${\mathcal{F}\wedge\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}}$${\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathcal{F}\wedge\mathbb{P}^{1}}$${\mathcal{F}\wedge\mathcal{F}}$${\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}}$${\mathcal{F}}$$\scriptstyle{\simeq}$$\scriptstyle{\mu}$$\scriptstyle{\beta_{\mathcal{F}}\wedge\beta_{\mathcal{F}}}$$\scriptstyle{\mu}$$\scriptstyle{\beta_{\mathcal{F}}^{2}}$

commutes. Then, since the bonding maps are compatible, $\mu$ induces a map

$\mathrm{Bott}(\mathcal{F})\wedge\mathrm{Bott}(\mathcal{F})\to\mathrm{Bott}(\mathcal{F})$

Furthermore, the assumptions on $\mathcal{F}$ ensures
$\mathrm{Bott}(\mathcal{F})$ is a homotopy commutative monoid in
$\mathrm{log}\mathcal{SH}(S)$.

The composite of the two maps in the upper horizontal row of (6.5.5) is
equivalent to the composite

(6.5.6)
$\mathcal{F}\wedge\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}\xrightarrow{\alpha_{\mathcal{F}}\wedge\alpha_{\mathcal{F}}}\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}\xrightarrow{\simeq}\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\mathbb{P}^{1}\wedge\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\mathbb{P}^{1}\xrightarrow{\gamma\wedge\gamma}\mathcal{F}\wedge\mathcal{F},$

where
$\gamma\colon\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\mathbb{P}^{1}\to\mathcal{F}$
is the canonical map. The adjoint of the composite of the two right-most
morphisms in (6.5.6) takes the form

$\nu\colon\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\Omega_{\mathbb{P}^{1}}\mathcal{F}\to\Omega_{(\mathbb{P}^{1})^{\wedge
2}}(\mathcal{F}\wedge\mathcal{F}).$

To check that (6.5.5) commutes, it suffices to check that the diagram

(6.5.7)
${\mathcal{F}\wedge\mathcal{F}}$${\Omega_{\mathbb{P}^{1}}\mathcal{F}\wedge\Omega_{\mathbb{P}^{1}}\mathcal{F}}$${\Omega_{(\mathbb{P}^{1})^{\wedge
2}}(\mathcal{F}\wedge\mathcal{F})}$${\mathcal{F}}$${\Omega_{(\mathbb{P}^{1})^{\wedge
2}}(\mathcal{F})}$$\scriptstyle{\mu}$$\scriptstyle{\alpha_{\mathcal{F}}\wedge\alpha_{\mathcal{F}}}$$\scriptstyle{\nu}$$\scriptstyle{\mu}$$\scriptstyle{\alpha_{\mathcal{F}}^{2}}$

commutes by adjunction.

######  Remark 6.5.4.

The above discussions on the loops $\mathbb{P}^{1}$-bundle formula and
logarithmic motivic Bott spectrum can be adapted in the setting of
$\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$ and
$\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$.

######  Proposition 6.5.5.

Suppose $S$ is a regular scheme. Then
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}\in\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$
satisfies the loops $\mathbb{P}^{1}$-bundle formula. A similar result holds
for
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}\in\mathrm{log}\mathcal{H}_{\ast}(S)$
too.

###### Proof.

We focus on the case of
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}\in\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$
since the proofs are similar. By the projective bundle formula for $K$-theory
of schemes, we have an equivalence of presheaves of spaces

(6.5.8)
$\Omega_{S^{1}}^{\infty}\mathrm{K}\simeq\Omega_{\mathbb{P}^{1}}\Omega_{S^{1}}^{\infty}\mathrm{K}.$

For every $X\in\mathbf{lSm}/S$, we have a canonical isomorphism of schemes

$(X\times\mathbb{P}^{1})-\partial(X\times\mathbb{P}^{1})\cong(X-\partial
X)\times\mathbb{P}^{1}.$

This implies that there is a natural equivalence

(6.5.9)
$\omega^{*}\Omega_{\mathbb{P}^{1}}\simeq\Omega_{\mathbb{P}^{1}}\omega^{*}.$

Apply $\omega^{*}$ to (6.5.8), and use (6.5.9) to obtain the desired
equivalence. ∎

######  Definition 6.5.6.

Let $S$ be a regular scheme. Proposition 6.5.5 shows that
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}\in\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$
satisfies the loops $\mathbb{P}^{1}$-bundle formula. With reference to
Construction 6.5.2, we define the _logarithmic $K$-theory spectrum of $S$_ to
be the Bott periodic $\mathbb{P}^{1}$-spectrum

(6.5.10)
$\mathbf{logKGL}:=\mathrm{Bott}(\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K})=(\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K},\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K},\ldots)\in\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S).$

By Theorem 6.4.4, we have an equivalence in
$\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$

(6.5.11)
$\mathbf{logKGL}\simeq(\mathbb{L}_{sNis,\square\cup\mathbf{Adm}}\mathbb{Z}\times{\mathrm{Gr}},\mathbb{L}_{sNis,\square\cup\mathbf{Adm}}\mathbb{Z}\times{\mathrm{Gr}},\ldots).$

We also have the logarithmic $K$-theory spectrum

(6.5.12)
$\mathbf{logKGL}:=\mathrm{Bott}(\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K})\in\mathrm{log}\mathcal{SH}(S).$

Using the functor
$\psi^{*}\colon\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)\to\mathrm{log}\mathcal{SH}(S)$
in 4.0.12, we can relate (6.5.10) and (6.5.12) by the equivalence

(6.5.13) $\psi^{*}\mathbf{logKGL}\simeq\mathbf{logKGL}$

since the triangle (4.0.8) commutes.

######  Theorem 6.5.7.

Suppose $S$ is a regular scheme. For all $X\in\mathbf{cSm}/S$ and integer
$d\in\mathbb{Z}$, there are canonical equivalences of spectra

(6.5.14) $\mathrm{K}(X-\partial
X)\simeq\operatorname{map}_{\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(S)}(\Sigma_{S^{1}}^{\infty}X_{+},\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}^{\infty}\mathbf{logKGL})$

and

(6.5.15) $\mathrm{K}(X-\partial
X)\simeq\operatorname{map}_{\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)}(\Sigma_{\mathbb{P}^{1}}^{d}\Sigma_{\mathbb{P}^{1}}^{\infty}X_{+},\mathbf{logKGL}).$

Similar results hold for $\mathrm{log}\mathcal{SH}_{S^{1}}(S)$ and
$\mathrm{log}\mathcal{SH}(S)$ too.

###### Proof.

We focus on the cases of $\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(S)$
and $\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$ since the proofs are similar.
Due to (6.5.1), we may assume $d=0$. Then, by adjunction, (6.5.15) is a
consequence of (6.5.14).

By (6.5.2), there is an equivalence in
$\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(S)$

(6.5.16)
$\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}^{\infty}\mathbf{logKGL}\simeq(\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K},\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K},\ldots).$

This gives an equivalence of spectra

(6.5.17)
$\begin{split}&\operatorname{map}_{\mathrm{log}\mathcal{SH}_{S^{1}}^{\mathbf{Adm}}(k)}(\Sigma_{S^{1}}^{\infty}X_{+},\Omega_{\mathbb{G}_{m}^{\mathrm{log}}}^{\infty}\mathbf{logKGL})\\\
\simeq&(\Omega_{S^{1}}^{\infty}\mathrm{K}(X-\partial
X),\Omega_{\mathbb{G}_{m}}\Omega_{S^{1}}^{\infty}\mathrm{K}(X-\partial
X),\ldots).\end{split}$

Compare with (6.1.3) to conclude. ∎

######  Proposition 6.5.8.

The motivic space $\mathbb{Z}\times{\mathrm{Gr}}$ is a homotopy commutative
monoid in $\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$. If $S$ is a
regular scheme, $\mathbf{logKGL}$ is a homotopy commutative monoid in
$\mathrm{log}\mathcal{SH}^{\mathbf{Adm}}(S)$.

###### Proof.

The morphism
${\mathrm{Gr}}(r,n)\times{\mathrm{Gr}}(s,l)\to{\mathrm{Gr}}(r+s,n+l)$ sending
any pair $(V\in\mathcal{O}^{n},W\in\mathcal{O}^{l})$ to $V\oplus W$ naturally
induces a morphism
$\mu\colon(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})\to\mathbb{Z}\times{\mathrm{Gr}}$.
Together with Theorem 6.4.4, we have a map
$\mu^{\prime}\colon(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})\to\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$.
The square

${(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})}$${(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})}$${(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})}$${\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}}$$\scriptstyle{\mu\wedge\operatorname{id}}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\scriptstyle{\mu}$$\scriptstyle{\mu}$

commutes since the corresponding square removing $\omega^{*}$ in the lower
right corner commutes in $\mathcal{H}_{\ast}(k)$. This shows that $\mu$
satisfies the associative axiom. One can similarly show that $\mu$ satisfies
the unital and commutative axioms, and hence $\mathbb{Z}\times{\mathrm{Gr}}$
is a homotopy commutative monoid in
$\mathrm{log}\mathcal{H}_{\ast}^{\mathbf{Adm}}(S)$.

It remains to check that (6.5.5) commutes for
$\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}$. In other words, we need to show
that the diagram

${(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}}$${(\mathbb{Z}\times{\mathrm{Gr}})\wedge\mathbb{P}^{1}\wedge(\mathbb{Z}\times{\mathrm{Gr}})\wedge\mathbb{P}^{1}}$${(\mathbb{Z}\times{\mathrm{Gr}})\wedge(\mathbb{Z}\times{\mathrm{Gr}})}$${(\mathbb{Z}\times{\mathrm{Gr}})\wedge\mathbb{P}^{1}\wedge\mathbb{P}^{1}}$${\omega^{*}\Omega_{S^{1}}^{\infty}\mathrm{K}}$

commutes. This follows from the commutativity of the corresponding square in
$\mathcal{H}_{\ast}(k)$, where $\omega^{*}$ in the lower right corner is
removed. ∎

######  Definition 6.5.9.

Suppose $S$ is a regular scheme and $f\colon X\to S$ is a morphism in
$\mathbf{lSch}$. The _logarithmic $K$-theory of $S$_ is the mapping spectrum

$\mathrm{logK}(X):=\operatorname{map}_{\mathrm{log}\mathcal{SH}(X)}(\Sigma_{\mathbb{P}^{1}}^{\infty}X_{+},f^{*}\mathbf{logKGL}).$

For every integer $n\in\mathbb{Z}$, we set

$\mathrm{logK}_{n}(X):=\pi_{n}\mathrm{logK}(X).$

If $f$ is log smooth, then we have equivalences of spectra

(6.5.18)
$\mathrm{logK}(X)\simeq\operatorname{map}_{\mathrm{log}\mathcal{SH}(S)}(\Sigma_{\mathbb{P}^{1}}^{\infty}X_{+},\mathbf{logKGL})\simeq\mathrm{K}(X-\partial
X),$

where the right equivalence is due to Theorem 6.5.7. In particular, if
$S\in\mathbf{Sm}/k$, then we have an equivalence of spectra

$\mathrm{logK}(S)\simeq\mathrm{K}(S).$

One may ask if this is true in a more general situation.

######  Conjecture 6.5.10.

For every $S\in\mathbf{Sch}$ there exists an equivalence of spectra

$\mathrm{logK}(S)\simeq\mathrm{K}(S).$

######  Remark 6.5.11.

There exist examples of non-regular schemes $S$ and $X\in\mathbf{lSm}/S$ such
that

$\mathrm{logK}(X)\not\simeq\mathrm{K}(X-\partial X).$

Indeed, the left-hand side is $\square$-invariant, but the right-hand side is
not.

######  Remark 6.5.12.

Niziol defined $K$-groups $\mathrm{K}^{\mathrm{Niz}}_{n}(X_{\tau})$ for
various topologies $\tau$ and fs log schemes $X$ in [77]. Some of these
$K$-groups are computed when $X$ is regular and $\partial X$ is a strict
normal crossing divisor in [77, Corollary 4.17]. In particular, if $k$ is a
field, then for every integer $n\geq 0$ there are isomorphisms of abelian
groups

$\mathrm{K}^{\mathrm{Niz}}_{n}(\square_{k{\acute{e}t}})\cong\mathrm{K}_{n}(k)\otimes((\mathbb{Q}/\mathbb{Z})^{\prime}-0)\text{
and
}\mathrm{K}^{\mathrm{Niz}}_{n}(\square_{kfl})\cong\mathrm{K}_{n}(k)\otimes(\mathbb{Q}/\mathbb{Z}-0),$

where, if $k$ has characteristic $p$, we set

$(\mathbb{Q}/\mathbb{Z})^{\prime}:=\oplus_{\ell\neq
p}\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}.$

Hence the groups $\mathrm{K}^{\mathrm{Niz}}_{n}(\square_{k{\acute{e}t}})$ and
$\mathrm{K}^{\mathrm{Niz}}_{n}(\square_{kfl})$ are non-isomorphic to
$\mathrm{K}_{n}(k)$.

On the other hand, suppose $X$ is an fs log scheme log smooth over a regular
scheme. Due to (6.5.18) and $\mathbb{A}^{1}$-invariance of $K$-theory for
regular schemes, there are isomorphisms of abelian groups

$\mathrm{logK}_{n}(X\times\square)\cong\mathrm{K}_{n}(X\times\mathbb{A}^{1})\cong\mathrm{K}_{n}(X).$

Hence our logarithmic $K$-group $\mathrm{K}_{n}^{log}(X)$ is non-isomorphic to
Niziol’s $K$-groups $\mathrm{K}^{\mathrm{Niz}}_{n}(X_{k{\acute{e}t}})$ and
$\mathrm{K}^{\mathrm{Niz}}_{n}(X_{kfl})$.

##  7\. Oriented cohomology theories

Orientations in $\mathbb{A}^{1}$-homotopy theory can be used to give a
systematic proof of the projective bundle formula for various cohomology
theories including homotopy algebraic $K$-theory and motivic cohomology.
Furthermore, one can associate Chern and Thom classes as in the classical
intersection theory. We refer to [97], [80], and [32] for these developments.

The purpose of this section is to prove various analogous results in our
logarithmic setting, without $\mathbb{A}^{1}$-invariance. As discussed in the
introduction, the scope of applications of these results goes beyond the
logarithmic world, since it gives automatically results for virtually all
known non-$\mathbb{A}^{1}$-homotopy invariant cohomology theories (for
example: crystalline, Hodge, prismatic and syntomic cohomology). In presence
of an orientation, the Thom space term in the Gysin sequence of Remark 3.2.22
can be trivialized in light of the Thom isomorphism theorem 7.3.8 below.

In this section, we also present a complete computation of the cohomology the
Grassmannians ${\mathrm{Gr}}(r,n)$: our proof uses in essential way the
$(\mathbb{P}^{\bullet},\mathbb{P}^{\bullet-1})$-invariance of any cohomology
theory representable in $\mathrm{log}\mathcal{SH}$, and once again does not
follow from the analogous computation in the $\mathbb{A}^{1}$-setting.

We begin by defining precisely what we mean by cohomology theory associated
with a homotopy commutative monoid $\mathbf{E}$.

######  Definition 7.0.1.

Suppose $S\in\mathbf{lSch}$ and let $\mathbf{E}$ be a homotopy commutative
monoid in $\mathbf{logSH}(S)$. For integers $p$ and $q$, we define the
_$\mathbf{E}$ -cohomology_ of $X\in\mathbf{lSm}/S$ by

(7.0.1)
$\mathbf{E}^{p,q}(X):=\operatorname{Hom}_{\mathbf{logSH}(S)}(\Sigma_{T}^{\infty}X_{+},\Sigma^{p,q}\mathbf{E}).$

For any morphism $Y\to X$ in $\mathbf{lSm}/S$, we define

$\mathbf{E}^{p,q}(X/Y):=\operatorname{Hom}_{\mathbf{logSH}(S)}(\Sigma_{T}^{\infty}X/Y,\Sigma^{p,q}\mathbf{E}).$

We set $\mathbf{E}^{p,q}:=\mathbf{E}^{p,q}(\operatorname{pt})$ and
$\mathbf{E}^{**}(X):=\bigoplus_{p,q\in\mathbb{Z}}\mathbf{E}^{p,q}(X)$ for
abbreviation.

######  Remark 7.0.2.

For $X,X^{\prime}\in\mathbf{lSm}/S$ and integers $p$, $q$, $r$, and $s$, any
two maps $u\colon\Sigma_{T}^{\infty}X_{+}\to\Sigma^{p,q}\mathbf{E}$ and
$v\colon\Sigma_{T}^{\infty}X_{+}^{\prime}\to\Sigma^{r,s}\mathbf{E}$ give

$\Sigma_{T}^{\infty}(X\times
X^{\prime})_{+}\xrightarrow{\simeq}\Sigma_{T}^{\infty}X_{+}\wedge\Sigma_{T}^{\infty}X_{+}^{\prime}\xrightarrow{u\wedge
v}\Sigma^{p,q}\mathbf{E}\wedge\Sigma^{r,s}\mathbf{E}\to\Sigma^{p+r,q+s}\mathbf{E}.$

From this, we have a homomorphism

(7.0.2)
$\mathbf{E}^{p,q}(X)\times\mathbf{E}^{r,s}(X^{\prime})\to\mathbf{E}^{p+r,q+s}(X\times
X^{\prime}).$

When a morphism $\alpha\colon X^{\prime}\to X\times X^{\prime}$ is given,
compose (7.0.2) with $\alpha^{*}$ to obtain the _cup product_

(7.0.3)
$\smile\colon\mathbf{E}^{p,q}(X)\times\mathbf{E}^{r,s}(X^{\prime})\to\mathbf{E}^{p+r,q+s}(X^{\prime}).$

We are particularly interested in the case when $X=X^{\prime}$ and $\alpha$ is
the diagonal morphism. We similarly have the cup product for
$\mathbf{E}^{**}(X/Y)$ for every morphism $Y\to X$ in $\mathbf{lSm}/S$ by
considering the diagonal map $X/Y\to X/Y\wedge X/Y$.

This cup product satisfies the axioms of rings since $\mathbf{E}$ satisfies
similar axioms, so $\mathbf{E}^{**}(X)$ and $\mathbf{E}^{**}(X/Y)$ are graded
rings. Let $\mathbf{1}$ denote the unit in these rings. Since (7.0.2) is
functorial in $X$, there is a naturally induced graded ring homomorphism

$f^{*}\colon\mathbf{E}^{**}(X)\to\mathbf{E}^{**}(Y)$

for every morphism $f\colon Y\to X$ in $\mathbf{logH}(S)$. We have a similar
homomorphism for a map $X^{\prime}/Y^{\prime}\to X/Y$ in
$\mathbf{logH}_{\ast}(S)$.

###  7.1. Orientations

Throughout this subsection, we fix $S\in\mathbf{Sch}$ and a homotopy
commutative monoid $\mathbf{E}$ in $\mathbf{logSH}(S)$.

######  Definition 7.1.1.

There are closed immersions

$\mathbb{P}^{1}\to\mathbb{P}^{2}\to\cdots\to\mathbb{P}^{n}\to\cdots,$

where $\mathbb{P}^{n}\to\mathbb{P}^{n+1}$ maps $[x_{0}:\cdots:x_{n}]$ to
$[x_{0}:\cdots:x_{n}:0]$. Hence we can regard $\mathbb{P}^{1}$ as a closed
subscheme of $\mathbb{P}^{\infty}$. Furthermore, we can let the distinguished
points of $\mathbb{P}^{n}$ and $\mathbb{P}^{\infty}$ be $1\in\mathbb{P}^{1}$.
A _Chern orientation_ (or simply _orientation_) of $\mathbf{E}$ is a class
$c_{\infty}\in\mathbf{E}^{2,1}(\mathbb{P}^{\infty}/\operatorname{pt})$ whose
restriction to $\mathbb{P}^{1}/\operatorname{pt}$ is the class of
$\mathbf{E}^{2,1}(\mathbb{P}^{1}/\operatorname{pt})$ given by

$(\mathbb{P}^{1}/\operatorname{pt})\wedge\mathbf{1}\colon\Sigma_{T}^{\infty}\mathbb{P}^{1}/\operatorname{pt}\to(\mathbb{P}^{1}/\operatorname{pt})\wedge\mathbf{E}=\Sigma^{2,1}\mathbf{E}.$

For abbreviation, we set
$\Sigma^{2,1}\mathbf{1}:=(\mathbb{P}^{1}/\operatorname{pt})\wedge\mathbf{1}$.

######  Remark 7.1.2.

One may also define a Chern orientation of $\mathbf{E}$ as a class
$c_{\infty}$ whose restriction to $\mathbb{P}^{1}/\operatorname{pt}$ is
$-\Sigma^{2,1}\mathbf{1}$, see [80, Definition 1.2] for example. There is a
one-to-one correspondence between this definition and ours, which sends
$c_{\infty}$ to $-c_{\infty}$.

######  Definition 7.1.3.

A Chern orientation $c_{\infty}$ can be expressed as a morphism
$\Sigma_{T}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt}\to\Sigma^{2,1}\mathbf{E}$
in $\mathbf{logSH}(S)$. Hence for every $X\in\mathbf{lSm}/S$, there are
canonical maps

(7.1.1)
$\begin{split}\operatorname{Hom}_{\mathbf{logSH}_{S^{1}}(S)}(\Sigma_{S^{1}}^{\infty}X_{+},\Sigma_{S^{1}}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt})&\to\operatorname{Hom}_{\mathbf{logSH}(S)}(\Sigma_{T}^{\infty}X_{+},\Sigma_{T}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt})\\\
&\to\operatorname{Hom}_{\mathbf{logSH}(S)}(\Sigma_{T}^{\infty}X_{+},\Sigma^{2,1}\mathbf{E})=\mathbf{E}^{2,1}(X).\end{split}$

We have maps

(7.1.2)
$\begin{split}\operatorname{logPic}(X)=H_{sNis}^{1}(X,\mathbb{G}_{m}^{\mathrm{log}})\to&\operatorname{Hom}_{\mathbf{logSH}_{S^{1}}(S)}(\Sigma_{S^{1}}^{\infty}X_{+},\Sigma_{S^{1}}^{\infty}\mathrm{B}\mathbb{G}_{m}^{\mathrm{log}}/\operatorname{pt})\\\
\cong&\operatorname{Hom}_{\mathbf{logSH}_{S^{1}}(S)}(\Sigma_{S^{1}}^{\infty}X_{+},\Sigma_{S^{1}}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt}),\end{split}$

where the last isomorphism is due to Theorem 5.2.10. Combine with (7.1.1) to
have a canonical map

$c_{1}\colon\operatorname{logPic}(X)\to\mathbf{E}^{2,1}(X),$

which we call the _first Chern class_. There is a canonical homomorphism
$\operatorname{Pic}(\underline{X})=\operatorname{logPic}(\underline{X})\to\operatorname{logPic}(X)$.
By composition we obtain the canonical map

$c_{1}\colon\operatorname{Pic}(\underline{X})\to\mathbf{E}^{2,1}(X).$

In particular, since every line bundle $\mathcal{L}$ determines a class of
$\operatorname{Pic}(\underline{X})$, we can define
$c_{1}(\mathcal{L})\in\mathbf{E}^{2,1}(X)$.

The map (7.1.2) sends the class
$\mathcal{O}(-1)\in\operatorname{Pic}(\mathbb{P}^{1})$ to the class

$\alpha\in\operatorname{Hom}_{\mathbf{logSH}_{S^{1}}(S)}(\Sigma_{S^{1}}^{\infty}(\mathbb{P}^{1})_{+},\Sigma_{S^{1}}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt})$

induced by the inclusion $\mathbb{P}^{1}\to\mathbb{P}^{\infty}$. The map
(7.1.1) sends the class $\beta$ to the restriction of $c_{\infty}$ to
$\mathbb{P}^{1}/\operatorname{pt}$. Hence we have the formula

(7.1.3) $c_{1}(\mathcal{O}(-1))=\Sigma^{2,1}\mathbf{1}.$

Let $\mathcal{T}_{1}$ be the tautological bundle on $\mathbb{P}^{\infty}$. The
map (7.1.2) sends the class
$\mathcal{T}_{1}\in\operatorname{Pic}(\mathbb{P}^{\infty})$ to the class

$\beta\in\operatorname{Hom}_{\mathbf{logSH}_{S^{1}}(S)}(\Sigma_{S^{1}}^{\infty}(\mathbb{P}^{\infty})_{+},\Sigma_{S^{1}}^{\infty}\mathbb{P}^{\infty}/\operatorname{pt})$

induced by the quotient map
$(\mathbb{P}^{\infty})_{+}\to\mathbb{P}^{\infty}/\operatorname{pt}$. The map
(7.1.1) sends the class $\alpha$ to $c_{\infty}$. Hence we have the formula

(7.1.4) $c_{1}(\mathcal{T}_{1})=c_{\infty}.$

If $f\colon Y\to X$ is a morphism in $\mathbf{lSm}/S$, then there is a
commutative diagram

${\operatorname{logPic}(X)}$${\mathbf{E}^{2,1}(X)}$${\operatorname{logPic}(Y)}$${\mathbf{E}^{2,1}(Y).}$$\scriptstyle{f^{*}}$$\scriptstyle{c_{1}}$$\scriptstyle{f^{*}}$$\scriptstyle{c_{1}}$

In particular, for every line bundle $\mathcal{L}$ over $X$, we have the
formula

(7.1.5) $f^{*}(c_{1}(\mathcal{L}))=c_{1}(f^{*}(\mathcal{L})).$

######  Lemma 7.1.4.

Let $\mathcal{T}_{1,n}$ be the tautological bundle on $\mathbb{P}^{n}$. Then
$\mathbb{P}(\mathcal{T}_{1,n}\oplus\mathcal{O})$ is isomorphic to the blow-up
of $\mathbb{P}^{n+1}$ along a point, and $\mathbb{P}(\mathcal{T}_{1,n})$ is
the exceptional divisor on $\mathbb{P}(\mathcal{T}_{1,n}\oplus\mathcal{O})$.

###### Proof.

The scheme $\mathbb{P}(\mathcal{T}_{1,n}\oplus\mathcal{O})$ admits a Zariski
cover consisting of the spectra of

$\mathbb{Z}[x_{0}/x_{i},\ldots,x_{n}/x_{i},x_{i}]\text{ and
}\mathbb{Z}[x_{0}/x_{i},\ldots,x_{n}/x_{i},1/x_{i}]$

for $0\leq i\leq n$. The divisor $\mathbb{P}(\mathcal{T}_{1,n})$ is given
locally by the divisor $x_{0}/x_{i}=0$ for $1\leq i\leq n$. Let $O$ be the
point $[0:\cdots:0:1]$ in $\mathbb{P}^{n+1}$. The scheme
${\mathrm{Bl}}_{O}\mathbb{P}^{n+1}$ admits a Zariski cover consisting of the
spectra of

$\mathbb{Z}[x_{1}/x_{i},\ldots,x_{n+1}/x_{i},x_{i}]\text{ and
}\mathbb{Z}[x_{1}/x_{i},\ldots,x_{n+1}/x_{i},1/x_{i}]$

for $1\leq i\leq n+1$. By identifying $x_{n+1}$ with $x_{0}$, we have the
required claim. ∎

######  Lemma 7.1.5.

Let $\mathcal{T}_{1}$ be the tautological bundle on $\mathbb{P}^{\infty}$, and
let $z\colon\mathbb{P}^{\infty}\to\mathcal{T}_{1}$ be the zero section. Then
there is an exact sequence

(7.1.6)
$0\to\mathbf{E}^{**}(\operatorname{pt})\to\mathbf{E}^{**}(\mathbb{P}^{\infty})\xrightarrow{z^{*}}\mathbf{E}^{**}(\mathrm{Th}(\mathcal{T}_{1}))\to
0.$

###### Proof.

Let $\mathcal{T}_{1,n}$ be the tautological bundle on $\mathbb{P}^{n}$. From
Theorem 3.3.5 and Lemma 7.1.4, we have an isomorphism

$\mathbb{P}(\mathcal{T}_{1,n}\oplus\mathcal{O})/\mathbb{P}(\mathcal{T}_{1,n})\xrightarrow{\cong}\mathbb{P}^{n+1}/\operatorname{pt}$

in $\mathbf{logSH}(S)$. If we compose with the zero section
$(\mathbb{P}^{n})_{+}\to\mathbb{P}(\mathcal{T}_{1,n}\oplus\mathcal{O})/\mathbb{P}(\mathcal{T}_{1,n})$,
then $(\mathbb{P}^{n})_{+}\to\mathbb{P}^{n+1}/\operatorname{pt}$ factors
through $(\mathbb{P}^{n+1})_{+}$. Take colimits to have morphisms

$(\mathbb{P}^{\infty})_{+}\to\mathbb{P}(\mathcal{T}_{1}\oplus\mathcal{O})/\mathbb{P}(\mathcal{T}_{1})\xrightarrow{\cong}\mathbb{P}^{\infty}/\operatorname{pt}$

whose composite morphism is equal to the quotient map in $\mathbf{logSH}(S)$.
Proposition 3.2.19 finishes the proof. ∎

######  Lemma 7.1.6.

Let
$\overline{i}_{1},\ldots,\overline{i}_{n}\colon\mathbb{P}^{1}\to\mathbb{P}^{n}$
be the closed immersions sending $[a_{0}:a_{1}]$ to
$[a_{0}:0:\cdots:0:a_{1}]$, $\ldots$, $[0:\cdots:0:a_{0}:a_{1}]$. Then
$\overline{i}_{1},\ldots,\overline{i}_{n}$ are all equal in
$\mathbf{logSH}_{S^{1}}(S)$.

###### Proof.

Without loss of generality it suffices to show that $\overline{i}_{1}$ and
$\overline{i}_{2}$ are equal in $\mathbf{logSH}_{S^{1}}(S)$. Both
$\overline{i}_{1}$ and $\overline{i}_{2}$ factors through the closed immersion
$\mathbb{P}^{2}\to\mathbb{P}^{n}$ sending $[a_{0}:a_{1}:a_{2}]$ to
$[a_{0}:a_{1}:0:\cdots:0:a_{2}]$, so we only need to consider the case $n=2$.
Let $\overline{i}\colon\mathbb{P}^{1}\to\mathbb{P}^{2}$ be the closed
immersion sending $[a_{0}:a_{1}]$ to $[a_{0}:a_{0}:a_{1}]$. We will show that
$\overline{i}_{1}$ and $\overline{i}$ are equal in
$\mathbf{logSH}_{S^{1}}(S)$.

There is a morphism

$h\colon\mathbb{P}^{1}\times\mathbb{A}^{1}\to\mathbb{P}^{2}$

sending $([a_{0}:a_{1}],t)$ to $[a_{0}:ta_{0}:a_{1}]$. This naturally induces
a ring homomorphism

$u\colon\mathbb{Z}(x,y)\to\mathbb{Z}(x,t)$

sending $x$ to $x$ and $y$ to $xt$. The scheme $\mathbb{P}^{2}$ has a Zariski
cover consisting of the spectra of

$A_{0}:=\mathbb{Z}[x,y],\;A_{1}:=\mathbb{Z}[1/x,y/x],\;A_{2}:=\mathbb{Z}[1/y,x/y].$

The blow-up
${\mathrm{Bl}}_{0\times\infty}(\mathbb{P}^{1}\times\mathbb{P}^{1})$ has a
Zariski cover consisting of the spectra of

$\displaystyle
B_{1}:=\mathbb{Z}[x,t],\;B_{2}:=\mathbb{Z}[1/x,t],\;B_{3}:=\mathbb{Z}[xt,1/t],$
$\displaystyle B_{4}:=\mathbb{Z}[x,1/(xt)],\;B_{5}:=\mathbb{Z}[1/x,1/t].$

We have inclusions

$u(A_{0})\subset B_{1},\;u(A_{1})\subset B_{2},\;u(A_{0})\subset
B_{3},\;u(A_{2})\subset B_{4},\;u(A_{2})\subset B_{5}.$

From the above, we deduce that $h$ extends to a morphism

$g\colon{\mathrm{Bl}}_{0\times\infty}(\mathbb{P}^{1}\times\mathbb{P}^{1})\to\mathbb{P}^{2}.$

Let $X$ be the fs log scheme whose underlying scheme is
${\mathrm{Bl}}_{0\times\infty}(\mathbb{P}^{1}\times\mathbb{P}^{1})$ and whose
log structure is the compactifying log structure associated with the open
immersion
$\mathbb{P}^{1}\times\mathbb{A}^{1}\to{\mathrm{Bl}}_{0\times\infty}(\mathbb{P}^{1}\times\mathbb{P}^{1})$.
From $g$ and $h$, we can form a morphism

$f\colon X\to\mathbb{P}^{2}.$

There is also an admissible blow-up $p\colon X\to\mathbb{P}^{1}\times\square$
along a smooth center. The zero section and one section
$\mathbb{P}^{1}\rightrightarrows\mathbb{P}^{1}\times\square$ can be lifted to
closed immersions $s_{0},s_{1}\colon\mathbb{P}^{1}\rightrightarrows X$. From
Lemma 3.2.12, we deduce $\overline{i}_{1}=\overline{i}_{2}$ in
$\mathbf{logSH}_{S^{1}}(S)$.

By permuting coordinates, we can deduce $\overline{i}_{2}=\overline{i}$ too.
Hence we have $\overline{i}_{1}=\overline{i}_{2}$. ∎

######  Lemma 7.1.7.

We set $X:={\mathrm{Bl}}_{[0:0:1]}(\mathbb{P}^{2})$, which has a Zariski cover
consisting of the spectra of

$\mathbb{Z}[x/y,y],\;\mathbb{Z}[y/x,x],\;\mathbb{Z}[x/y,1/y],\;\mathbb{Z}[y/x,1/x].$

Let $i_{1},i_{2}\colon\mathbb{P}^{1}\rightrightarrows X$ be the strict
transforms of the closed immersions
$\mathbb{P}^{1}\rightrightarrows\mathbb{P}^{2}$ sending $[x_{0}:x_{1}]$ to
$[0:x_{0}:x_{1}]$ and $[x_{0}:x_{1}:0]$. Let $i_{3}\colon\mathbb{P}^{1}\to X$
be the closed immersion from the exceptional divisor given by the
homomorphisms

$\mathbb{Z}[x/y,y]\to\mathbb{Z}[t],\;x/y\mapsto t,\;y\mapsto 0$

and

$\mathbb{Z}[y/x,x]\to\mathbb{Z}[1/t],\;y/x\mapsto 1/t,\;x\mapsto 0.$

If a class $a\in\mathbf{E}^{**}(X)$ satisfies $i_{2}^{*}a=0$, then we have the
formula

$i_{1}^{*}a=-i_{3}^{*}a.$

###### Proof.

There is a commutative diagram

(7.1.7)
${\mathbb{P}^{1}}$${\operatorname{pt}}$${\mathbb{P}^{1}}$${X}$${\mathbb{P}^{1}}$${X,}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{q}$$\scriptstyle{i_{3}}$

where $q$ is given by the homomorphisms

$\mathbb{Z}[t]\to\mathbb{Z}[x/y,y],\mathbb{Z}[x/y,1/y],\;t\mapsto x/y$

and

$\mathbb{Z}[1/t]\to\mathbb{Z}[y/x,x],\mathbb{Z}[y/x,1/x],\;1/t\mapsto y/x.$

Let $p\colon X\to\mathbb{P}^{2}$ be the projection. Since
$qi_{3}=\operatorname{id}$, we have $i_{3}^{*}(a-q^{*}i_{3}^{*}a)=0$. From
Proposition 3.3.5, we have

(7.1.8) $a-q^{*}i_{3}^{*}a=p^{*}b$

for some $b\in\mathbf{E}^{**}(\mathbb{P}^{2})$. Permute coordinates in Lemma
7.1.6 to have $i_{1}^{*}p^{*}b=i_{2}^{*}p^{*}b$. Together with (7.1.8), we
have the formula

(7.1.9)
$i_{1}^{*}a-i_{1}^{*}q^{*}i_{3}^{*}a=i_{2}^{*}a-i_{2}^{*}q^{*}i_{3}^{*}a.$

Since $i_{2}^{*}a=0$, we have $i_{1}^{*}q^{*}i_{3}^{*}a=0$ from (7.1.7).
Combined with (7.1.9), we have

$i_{1}^{*}a=-i_{2}^{*}q^{*}i_{3}^{*}a.$

This implies the claimed formula since $qi_{2}=\operatorname{id}$. ∎

######  Definition 7.1.8.

Let $\mathcal{T}_{1}$ be the tautological bundle on $\mathbb{P}^{\infty}$. A
_Thom orientation_ of $\mathbf{E}$ is a class
$t_{\infty}\in\mathbf{E}^{2,1}(\mathrm{Th}(\mathcal{T}_{1}))$ such that the
restriction of $t_{\infty}$ to the distinguished point of
$\mathbb{P}^{\infty}$ is equal to the class of $\mathbf{E}^{2,1}(T)$ given by

$-T\wedge\mathbf{1}\colon T\to T\wedge\mathbf{E}=\Sigma^{2,1}\mathbf{E}.$

From (7.1.6), we have an isomorphism of graded rings

(7.1.10)
$z^{*}\colon\mathbf{E}^{**}(\mathbb{P}^{\infty}/\operatorname{pt})\xrightarrow{\cong}\mathbf{E}^{**}(\mathrm{Th}(\mathcal{T}_{1})).$

######  Proposition 7.1.9.

Let $\mathcal{T}_{1}$ be the tautological bundle on $\mathbb{P}^{\infty}$. If
$t_{\infty}\in\mathbf{E}^{**}(\mathrm{Th}(\mathcal{T}_{1}))$ is a class, then
$t_{\infty}$ is a Thom orientation of $\mathbf{E}$ if and only if
$z^{*}t_{\infty}\in\mathbf{E}^{**}(\mathbb{P}^{\infty}/\operatorname{pt})$ is
a Chern orientation of $\mathbf{E}$.

###### Proof.

Let $\mathcal{T}_{1,1}$ be the tautological bundle on $\mathbb{P}^{1}$. There
is a commutative diagram with cartesian squares

${\operatorname{pt}}$${\mathbb{P}^{1}}$${\mathbb{P}^{\infty}}$${\mathbb{P}^{1}}$${\mathbb{P}(\mathcal{T}_{1,1}\oplus\mathcal{O})}$${\mathbb{P}(\mathcal{T}_{1}\oplus\mathcal{O})}$${\operatorname{pt}}$${\mathbb{P}^{1}}$${\mathbb{P}^{\infty},}$$\scriptstyle{i_{0}}$$\scriptstyle{v}$$\scriptstyle{z}$$\scriptstyle{u}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{i_{0}}$

where $i_{0}$ is the zero section, and $p$ and $q$ are the projections. The
restriction of $t_{\infty}$ to $\mathbb{P}^{1}$ gives a class
$t_{1}\in\mathbf{E}^{2,1}(\mathrm{Th}(\mathcal{T}_{1,1}))$. Let
$\overline{t}_{1}$ be the image of $t_{1}$ in
$\mathbf{E}^{2,1}(\mathbb{P}(\mathcal{T}_{1,1}\oplus\mathcal{O}))$, whose
restriction to $\mathbb{P}(\mathcal{T}_{1,1})$ is zero. The blow-down of
$\mathbb{P}(\mathcal{T}_{1,1}\oplus\mathcal{O})$ along
$\mathbb{P}(\mathcal{T}_{1,1})$ is isomorphic to $\mathbb{P}^{2}$ due to Lemma
7.1.4. Hence we can apply Lemma 7.1.7 to our situation, and deduce the formula

(7.1.11) $u^{*}\overline{t}_{1}=-v^{*}\overline{t}_{1}.$

The class $t_{\infty}$ is a Thom orientation if and only if the restriction of
$-v^{*}\overline{t}_{1}$ to $\mathbb{P}^{1}/\operatorname{pt}$ is equal to
$\mathbb{P}^{1}/\operatorname{pt}\wedge\mathbf{1}$. The class
$z^{*}t_{\infty}$ is a Chern orientation if and only if the restriction of
$u^{*}\overline{t}_{1}$ to $\mathbb{P}^{1}/\operatorname{pt}$ is equal to
$\mathbb{P}^{1}/\operatorname{pt}\wedge\mathbf{1}$. From (7.1.11), we deduce
that these are equivalent. ∎

######  Definition 7.1.10.

Let $t_{\infty}$ be a Thom orientation. The above $z^{*}t_{\infty}$ is called
the _Chern orientation associated with $t_{\infty}$_.

######  Proposition 7.1.11.

There is a one-to-one correspondence between Chern orientations of
$\mathbf{E}$ and Thom orientations of $\mathbf{E}$.

###### Proof.

Combine Lemma 7.1.5 and Proposition 7.1.9. ∎

###  7.2. Cohomology of projective spaces

Throughout this subsection, we fix $S\in\mathbf{Sch}$ and a homotopy
commutative monoid $\mathbf{E}$ in $\mathbf{logSH}(S)$. In this subsection, we
will present several results about projective spaces in
$\mathrm{log}\mathcal{SH}_{S^{1}}(S)$ (or for simplicity in the homotopy
category $\mathbf{logSH}_{S^{1}}(S)$). Our approach culminates with a proof of
the projective bundle formula. The following theorem is a fundamental property
of our motivic category.

######  Proposition 7.2.1.

In $\mathrm{log}\mathcal{SH}_{S^{1}}(S)$, there is an equivalence

(7.2.1)
$\mathbb{P}^{n}/\mathbb{P}^{n-1}\cong(\mathbb{P}^{1}/\operatorname{pt})^{\wedge
n}.$

###### Proof.

In $\mathbf{SH}_{S^{1}}(k)$, there are isomorphisms

$\mathbb{P}^{n}/\mathbb{P}^{n-1}\xleftarrow{\cong}\mathbb{P}^{n}/(\mathbb{P}^{n}-O)\xrightarrow{\cong}(\mathbb{P}^{1}/\mathbb{A}^{1})^{\wedge
n}\xleftarrow{\cong}(\mathbb{P}^{1}/\operatorname{pt})^{\wedge n},$

where $O:=[0:\cdots:0:1]\in\mathbb{P}^{n}$. We need to “compactify” these. We
can replace $\mathbb{A}^{1}$ by $\square$ and $\mathbb{P}^{n}-O$ by
$({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),E)$, where $E$ is the exceptional divisor.
Since $({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),E)$ is a $\square$-bundle on
$\mathbb{P}^{n-1}$ by [22, Lemma 7.1.7], the projection
$({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),E)\to\mathbb{P}^{n-1}$ is an equivalence
in $\mathrm{log}\mathcal{SH}_{S^{1}}(S)$. Hence it remains to construct the
second morphism in

$\mathbb{P}^{n}/\mathbb{P}^{n-1}\xleftarrow{\cong}\mathbb{P}^{n}/({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),E)\to(\mathbb{P}^{1}/\square)^{\wedge
n}\xleftarrow{\cong}(\mathbb{P}^{1}/\operatorname{pt})^{\wedge n}$

and to show that it is an equivalence in
$\mathrm{log}\mathcal{SH}_{S^{1}}(S)$.

Let $\overline{H}_{i}$ be the divisor $x_{i}=0$ on $\mathbb{P}^{n}$ for $1\leq
i\leq n$, and set

$\overline{\mathcal{V}}:=\mathrm{horn}(\mathbb{P}^{n},\overline{H}_{1}+\cdots+\overline{H}_{n}).$

Proposition 3.2.5 gives an equivalence

(7.2.2) $({\mathrm{Bl}}_{O}\mathbb{P}^{n},E)\xleftarrow{\simeq}\mathop{\rm
colim}(\overline{\mathcal{V}})$

in $\mathrm{log}\mathcal{SH}_{S^{1}}(S)$, which induces an equivalence

(7.2.3)
$\mathbb{P}^{n}/({\mathrm{Bl}}_{O}\mathbb{P}^{n},E)\xleftarrow{\simeq}\mathbb{P}^{n}/\mathop{\rm
colim}(\overline{\mathcal{V}})$

in $\mathbf{logSH}_{S^{1}}(S)$. From (A.0.1) and Proposition A.0.8, we have an
equivalence

(7.2.4) $\mathbb{P}^{n}/\mathop{\rm
colim}(\overline{\mathcal{V}})\xrightarrow{\simeq}(\mathbb{P}^{1}/\square)^{\wedge
n}$

in $\mathbf{logSH}_{S^{1}}(S)$. Combine (7.2.3) and (7.2.4) to finish the
construction. ∎

Morel proved the following result in [68] for $\mathbb{A}^{1}$-homotopy
theory.

######  Lemma 7.2.2.

There is a commutative diagram

${\mathbb{P}^{n}/\operatorname{pt}}$${(\mathbb{P}^{n}/\operatorname{pt})^{\wedge
n}}$${\mathbb{P}^{n}/\mathbb{P}^{n-1}}$${(\mathbb{P}^{1}/\operatorname{pt})^{\wedge
n}}$$\scriptstyle{\Delta}$$\scriptstyle{\overline{i}_{1}\wedge\cdots\wedge\overline{i}_{1}}$

in $\mathbf{logSH}_{S^{1}}(S)$, where the left vertical map is the obvious
one, the lower horizontal map is (7.2.1), $\Delta$ is the diagonal morphism,
and $\overline{i}_{1}\colon\mathbb{P}^{1}\to\mathbb{P}^{n}$ is the closed
immersion sending $[a_{0}:a_{1}]$ to $[a_{0}:0:\cdots:0:a_{1}]$.

###### Proof.

We keep using the same notation as in the proof of Proposition 7.2.1. Let
$\overline{i}_{1},\ldots,\overline{i}_{n}\colon\mathbb{P}^{1}\to\mathbb{P}^{n}$
be the closed immersions in Lemma 7.1.6, which are all equal in
$\mathbf{logSH}_{S^{1}}(S)$. It suffices to show that the diagram

${\mathbb{P}^{n}/\operatorname{pt}}$${(\mathbb{P}^{n}/\operatorname{pt})^{\wedge
n}}$${\mathbb{P}^{n}/\mathbb{P}^{n-1}}$${(\mathbb{P}^{1}/\operatorname{pt})^{\wedge
n},}$$\scriptstyle{\Delta}$$\scriptstyle{\overline{i}_{1}\wedge\cdots\wedge\overline{i}_{n}}$

in $\mathbf{logSH}_{S^{1}}(S)$ commutes. For every $i$, there is an
isomorphism of fs log schemes
$\overline{V}_{i}:=(\mathbb{P}^{n},\overline{H}_{i})\cong(\mathbb{P}^{n},\mathbb{P}^{n-1})$.
This means
$\mathbb{P}^{n}/\operatorname{pt}\cong\mathbb{P}^{n}/\overline{V_{i}}$ in
$\mathbf{logSH}_{S^{1}}(S)$. Hence the obvious map

$(\mathbb{P}^{n}/\operatorname{pt})^{\wedge
n}\to(\mathbb{P}^{n}/\overline{V}_{1})\wedge\cdots\wedge(\mathbb{P}^{n}/\overline{V}_{n})$

is an isomorphism in $\mathbf{logSH}_{S^{1}}(S)$. Moreover, by strict
Nisnevich descent, there is an isomorphism

$\mathbb{P}^{1}/\square\cong\mathbb{A}^{1}/\mathbb{A}_{\mathbb{N}}$

in $\mathbf{logSH}_{S^{1}}(S)$. From the proof of Proposition 7.2.1, we see
that the diagram

${\mathbb{P}^{n}/\operatorname{pt}}$${\mathbb{P}^{n}/\mathbb{P}^{n-1}}$${\mathbb{P}^{n}/({\mathrm{Bl}}_{O}(\mathbb{P}^{n}),E)}$${\mathbb{P}^{n}/\mathop{\rm
colim}(\overline{\mathcal{V}})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

in $\mathbf{logSH}_{S^{1}}(S)$ commutes, where the maps from
$\mathbb{P}^{n}/\operatorname{pt}$ are the obvious ones. Hence it suffices to
show that the diagram

(7.2.5)
${\mathbb{P}^{n}/\operatorname{pt}}$${(\mathbb{P}^{n}/\overline{V}_{1})\wedge\cdots\wedge(\mathbb{P}^{n}/\overline{V}_{n})}$${\mathbb{P}^{n}/\mathop{\rm
colim}(\overline{\mathcal{V}})}$${(\mathbb{P}^{1}/\square)^{\wedge
n}}$${(\mathbb{A}^{1}/\mathbb{A}_{\mathbb{N}})^{\wedge
n}}$$\scriptstyle{\Delta}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{i_{1}\wedge\cdots\wedge
i_{n}}$

in $\mathbf{logSH}_{S^{1}}(S)$ commutes, where $i_{1},\ldots,i_{n}$ are the
restrictions of $\overline{i}_{1},\ldots,\overline{i}_{n}$.

Let $q\colon\mathbb{A}^{n}\to(\mathbb{P}^{n})^{n}$ be the morphism given by

$(a_{1},\ldots,a_{n})\mapsto([a_{1}:0:\cdots:0:1],\ldots,[0:\cdots:0:a_{n}:1]).$

Form the cubical horn $\mathcal{Z}$ from
$(\mathbb{P}^{n}/\overline{V}_{1})\wedge\cdots\wedge(\mathbb{P}^{n}/\overline{V}_{n})$
using Proposition A.0.8. Then there is a commutative square

${(\mathbb{P}^{n})^{n}/\mathop{\rm
colim}(\mathcal{Z})}$${(\mathbb{P}^{n}/\overline{V}_{1})\wedge\cdots\wedge(\mathbb{P}^{n}/\overline{V}_{n})}$${\mathbb{A}^{n}/\mathop{\rm
colim}(\mathcal{V})}$${(\mathbb{A}^{1}/\mathbb{A}_{\mathbb{N}})^{\wedge
n}.}$$\scriptstyle{q}$$\scriptstyle{\cong}$$\scriptstyle{i_{1}\wedge\cdots\wedge
i_{n}}$$\scriptstyle{\cong}$

There is a morphism

$\mathbb{A}^{n}\times\mathbb{A}^{1}\to(\mathbb{P}^{n})^{n}$

sending $(a_{1},\ldots,a_{n},t)$ to

$([a_{1}:ta_{2}:\ldots:ta_{n}:1],\ldots,[ta_{1}:\ldots:ta_{n-1}:a_{n}:1]).$

Let $\Delta$ be the fan associated with the toric variety
$(\mathbb{P}^{n})^{n}$. This morphism is a morphism associated with a morphism
of fans $\mathbb{N}^{n}\times\mathbb{N}^{1}\to\Delta$ whose morphism of
lattices $h\colon\mathbb{Z}^{n}\times\mathbb{Z}\to(\mathbb{Z}^{n})^{r}$ sends
$(a_{1},\ldots,a_{n},t)$ to

$((a_{1},t+a_{2},\ldots,t+a_{n}),\ldots,(a_{1}+t,\ldots,a_{n-1}+t,a_{n})).$

Let $\Theta$ be the fan associated with the toric variety $\mathbb{P}^{1}$.
According to toric resolution of singularities [31, Theorem 11.1.9], there is
a morphism of fans $u\colon\Sigma\to\Delta$ such that $\Sigma$ is a refinement
of $\mathbb{N}^{n}\times\Theta$ and the morphism of lattices associated with
$u$ is $h$. Hence we have a commutative diagram

(7.2.6)
${\underline{\mathbb{A}}_{\Sigma}}$${(\mathbb{P}^{n})^{n}}$${\mathbb{A}^{n}\times\mathbb{A}^{1}}$${\mathbb{A}^{n}\times\mathbb{P}^{1},}$

where $\underline{\mathbb{A}}_{\Sigma}$ denotes the toric variety associated
with $\Sigma$.

We set
$V_{i}:=(\mathbb{A}^{n},H_{i})\simeq\overline{V}_{i}\times_{\mathbb{P}^{n}}\mathbb{A}^{n}$
for all $1\leq i\leq n$. Let $X^{\prime}$ (resp. $V_{i}^{\prime}$) be the fs
log scheme whose underlying scheme is $\underline{\mathbb{A}}_{\Sigma}$, and
whose log structure is associated with the open immersion
$\mathbb{A}^{n}\times\mathbb{A}^{1}\to\underline{\mathbb{A}}_{\Sigma}$ (resp.
$(V_{i}-\partial
V_{i})\times\mathbb{A}^{1}\to\underline{\mathbb{A}}_{\Sigma}$), and let
$H_{i}^{\prime}$ be the strict transform of $H_{i}$ in
$\underline{\mathbb{A}}_{\Sigma}$ for all $1\leq i\leq n$. The morphisms
$X^{\prime}\to\mathbb{A}^{n}\times\square$ and $V_{i}^{\prime}\to
V_{i}\times\square$ for all $1\leq i\leq n$ are admissible blow-ups along
smooth centers. Hence the maps

$\mathop{\rm colim}(\mathcal{V}^{\prime})\to\mathop{\rm
colim}(\mathcal{V})\times\square\text{ and
}X^{\prime}\to\mathbb{A}^{n}\times\square$

are isomorphisms in $\mathbf{logSH}_{S^{1}}(S)$, where

$\mathcal{V}:=\mathrm{horn}(\mathbb{A}^{n},H_{1}+\cdots+H_{n})\text{ and
}\mathcal{V}^{\prime}:=\mathrm{horn}(X,H_{1}^{\prime}+\cdots+H_{n}^{\prime}).$

It follows that the map

(7.2.7) $X^{\prime}/\mathop{\rm
colim}(\mathcal{V}^{\prime})\to(\mathbb{A}^{n}/\mathop{\rm
colim}(\mathcal{V}))\times\square$

is an isomorphism in $\mathbf{logSH}_{S^{1}}(S)$.

By considering the suitable compactifying log structures for the schemes in
(7.2.6), we obtain the map

$X^{\prime}/\mathop{\rm
colim}(\mathcal{V}^{\prime})\to(\mathbb{P}^{n})^{n}/\mathop{\rm
colim}(\mathcal{Z}).$

Let $q^{\prime}\colon\mathbb{A}^{n}\to(\mathbb{P}^{n})^{n}$ be the restriction
of the diagonal map. The restriction of $X^{\prime}\to(\mathbb{P}^{n})^{n}$ at
$t=0$ and $t=1$ are exactly $q$ and $q^{\prime}$. It follows that the two
naturally induced maps

$q,q^{\prime}\colon\mathbb{A}^{n}/\mathop{\rm
colim}(\mathcal{V})\rightrightarrows(\mathbb{P}^{n})^{n}/\mathop{\rm
colim}(\mathcal{Z})$

are equal in $\mathbf{logSH}_{S^{1}}(S)$ since we can apply the argument in
Lemma 3.2.12 to the diagram

${\mathbb{A}^{n}/\mathop{\rm colim}(\mathcal{V})}$${X^{\prime}/\mathop{\rm
colim}(\mathcal{V}^{\prime})}$${(\mathbb{P}^{n})^{n}/\mathop{\rm
colim}(\mathcal{Z})}$${(\mathbb{A}^{n}/\mathop{\rm
colim}(\mathcal{V}))\times\square,}$$\scriptstyle{u_{1}^{\prime}}$$\scriptstyle{u_{0}^{\prime}}$

where $u_{0}^{\prime}$ and $u_{1}^{\prime}$ are the lifts of the zero section
and one section.

Finally, by combining the commutative diagrams (7.2.6) and

${\mathbb{P}^{n}/\operatorname{pt}}$${(\mathbb{P}^{n})^{n}/\mathop{\rm
colim}(\mathcal{Z})}$${\mathbb{P}^{n}/\mathop{\rm
colim}(\overline{\mathcal{V}})}$${\mathbb{A}^{n}/\mathop{\rm
colim}(\mathcal{V})}$$\scriptstyle{\Delta}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\cong}$

in $\mathbf{logSH}_{S^{1}}(S)$, we deduce that (7.2.5) commutes. ∎

######  Lemma 7.2.3.

Suppose $\mathbf{E}$ is oriented. Let $i\colon\mathbb{P}^{1}\to\mathbb{P}^{n}$
be the closed immersion sending $[x_{0}:x_{1}]$ to $[x_{0}:x_{1},0,\ldots,0]$,
let $p\colon\mathbb{P}^{n}\to\operatorname{pt}$ and
$q\colon\mathbb{P}^{1}\to\operatorname{pt}$ be the structure morphisms, and
let $\mathcal{T}$ be the tautological bundle on $\mathbb{P}^{n}$. Then the
diagram

${\mathbf{E}^{*-2,*-1}}$${\mathbf{E}^{**}(\mathbb{P}^{n})}$${\mathbf{E}^{**}(\mathbb{P}^{1})}$$\scriptstyle{\cong}$$\scriptstyle{\psi_{n}}$$\scriptstyle{i^{*}}$

commutes, where the right vertical isomorphism sends
$x\in\mathbf{E}^{*-2,*-1}$ to $q^{*}(x)\smile\Sigma^{2,1}\mathbf{1}$, and
$\psi_{n}$ sends $x\in\mathbf{E}^{*-2,*-1}$ to $p^{*}(x)\smile
c_{1}(\mathcal{T})$.

###### Proof.

Let $\mathcal{L}$ be the tautological bundle on $\mathbb{P}^{1}$. From
(7.1.5), we have the formula

$i^{*}\psi_{n}(x)=q^{*}(x)\smile c_{1}(\mathcal{L}).$

To conclude the proof, we appeal to (7.1.3). ∎

######  Construction 7.2.4.

Suppose $\mathbf{E}$ is oriented and $\mathcal{E}\to X$ is a rank $d+1$ vector
bundle with $X\in\mathbf{lSm}/S$. Let $\mathcal{L}$ be the tautological line
bundle on $\mathcal{E}$. There is a map

(7.2.8)
$\rho_{\mathcal{E}}\colon\bigoplus_{i=0}^{d}\mathbf{E}^{*-i,*-2i}(X)\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}))$

sending $(x_{0},\ldots,x_{d})$ to $\sum_{i=0}^{d}p^{*}(x_{i})\smile
c_{1}(\mathcal{L})\smile\cdots\smile c_{1}(\mathcal{L})$ ($i$ copies of
$c_{1}(\mathcal{L})$). For abbreviation, we often omit the symbol $\smile$,
and then write the expression above as

$\sum_{i=0}^{d}p^{*}(x_{i})c_{1}(\mathcal{L})^{i}.$

If $U$ is an open subscheme of $X$ with
$\mathcal{E}_{U}:=\mathcal{E}\times_{X}U$, we consider the projection
$q\colon\mathcal{E}/\mathcal{E}_{U}\to X/U$ and the morphism

$\Delta_{i}\colon\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})\to\mathbb{P}(\mathcal{E})^{i+1}/(\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i})\cong\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i}$

naturally induced by the diagonal morphism
$\mathbb{P}(\mathcal{E})\to\mathbb{P}(\mathcal{E})^{i+1}$ for every integer
$i\geq 1$. We have homomorphisms

(7.2.9)
$\begin{split}\mathbf{E}^{*-2i,*-i}(X/U)&\xrightarrow{q^{*}}\mathbf{E}^{*-2i,*-i}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U}))\\\
&\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i})\xrightarrow{\Delta_{i}^{*}}\mathbf{E}^{**}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U}))\end{split}$

where the second homomorphism is constructed from (7.0.2) and
$c_{1}(\mathcal{L})\in\mathbf{E}^{2,1}(\mathbb{P}(\mathcal{E}))$. Collect
(7.2.9) for every $i=0,\ldots,d$ to form the homomorphism

$\rho_{\mathcal{E},U}\colon\bigoplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(X/U)\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})).$

######  Lemma 7.2.5.

Suppose $\mathbf{E}$ is oriented. With the above notation, the diagram

(7.2.10)
${\oplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(X/U)}$${\oplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(X)}$${\oplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(U)}$${\oplus_{i=0}^{d}\mathbf{E}^{*-2i+1,*-i}(X/U)}$${\mathbf{E}^{**}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U}))}$${\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}))}$${\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}_{U}))}$${\mathbf{E}^{*+1,*}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U}))}$$\scriptstyle{\rho_{\mathcal{E},U}}$$\scriptstyle{\rho_{\mathcal{E}}}$$\scriptstyle{\delta}$$\scriptstyle{\rho_{\mathcal{E}_{U}}}$$\scriptstyle{\rho_{\mathcal{E},U}}$$\scriptstyle{\delta}$

commutes.

###### Proof.

The left and middle squares of (7.2.10) commute by the functoriality of the
construction. Let $j\colon U\to X$ be the open immersion, and
$\Delta_{i,U}\colon\mathbb{P}(\mathcal{E}_{U})\to\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i}$
be the restriction of the diagonal map for every integer $i\geq 1$. There is a
commutative diagram

(7.2.11)
${\mathbf{E}^{*-2i,*-i}(U)}$${\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i})}$${\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}_{U}))}$${\mathbf{E}^{*-2i+1,*-i}(X/U)}$${\mathbf{E}^{*+1,*}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})\times\mathbb{P}(\mathcal{E})^{i})}$${\mathbf{E}^{*+1*}(\mathbb{P}(\mathcal{E})/\mathbb{P}(\mathcal{E}_{U})),}$$\scriptstyle{\delta}$$\scriptstyle{\Delta_{i,U}^{*}}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\Delta_{i}^{*}}$

where the lower row of (7.2.11) comes from (7.2.9), and the upper left
horizontal homomorphism of (7.2.11) is constructed from
$j^{*}\colon\mathbf{E}^{*-2i,*-i}(U)\to\mathbf{E}^{*-2i,*-i}(X)$, (7.0.2), and
$c_{1}(\mathcal{L})\in\mathbf{E}^{2,1}(\mathbb{P}(\mathcal{E}))$. We set
$\mathcal{L}_{U}:=j^{*}(\mathcal{L})$, which is the tautological bundle on
$\mathbb{P}(\mathcal{E}_{U})$. Since
$j^{*}c_{1}(\mathcal{L})=c_{1}(\mathcal{L}_{U})$ by (7.1.5), the collection of
the composite of the homomorphisms in the upper row of (7.2.11) for
$i=0,\ldots,d$ constructs $\rho_{\mathcal{E}_{U}}$. This shows that the right
square of (7.2.10) commutes. ∎

######  Theorem 7.2.6 (Projective bundle theorem).

Suppose that $\mathbf{E}$ is oriented. With the above notation, the map
(7.2.8) is an isomorphism.

###### Proof.

There exists a Zariski cover $\\{U_{i}\\}_{1\leq i\leq n}$ of $X$ such that
$\mathcal{E}\times_{X}U_{i}$ is a trivial bundle on $U_{i}$ for every $1\leq
i\leq n$. We proceed by induction on $n$. The case $n=1$ is done in
Proposition 7.2.7 below. Suppose that the claim holds for $n-1$. We set
$U:=U_{1}\cup\cdots\cup U_{n-1}$. By induction, the map
$\rho_{\mathcal{E}_{U}}$ in Lemma 7.2.5 is an isomorphism. Therefore, by the
five lemma, it suffices to show that the map $\rho_{\mathcal{E},U}$ in Lemma
7.2.5 is an isomorphism. By excision, it suffices to show that the map

$\oplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(U_{n}/U_{n}\cap
U)\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}_{U_{n}})/\mathbb{P}(\mathcal{E}_{U_{n}\cap
U}))$

is an isomorphism, where $\mathcal{E}_{U_{n}}:=\mathcal{E}\times_{X}U_{n}$ and
$\mathcal{E}_{U_{n}\cap U}:=\mathcal{E}\times_{X}(U_{n}\cap U)$. Again by the
five lemma and Lemma 7.2.5, it suffices to show that the maps

$\oplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(V)\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}_{V}))$

are isomorphisms for $V=U_{n}$ and $V=U_{n}\cap U$. This follows from the
induction hypothesis. ∎

######  Proposition 7.2.7.

Suppose that $\mathbf{E}$ is oriented. Then the map

$\rho_{d}\colon\bigoplus_{i=0}^{d}\mathbf{E}^{*-2i,*-i}(X)\to\mathbf{E}^{**}(X\times\mathbb{P}^{d})$

is an isomorphism for every $X\in\mathbf{lSm}/S$, where
$\rho_{d}:=\rho_{\mathcal{E}}$, and $\mathcal{E}$ is the rank $d+1$ trivial
bundle on $X$.

###### Proof.

We only deal with the case where $X=\operatorname{pt}$ since the general case
follows in a similar way. It suffices to show that the map
$\widetilde{\rho}_{d}\colon\bigoplus_{i=1}^{d}\mathbf{E}^{*-2i,*-i}\to\mathbf{E}^{**}(\mathbb{P}^{d}/\operatorname{pt})$
naturally induced by $\rho_{d}$ is an isomorphism. Let
$i\colon\mathbb{P}^{d-1}\to\mathbb{P}^{d}$ be the closed immersion sending
$[a_{0}:\cdots:a_{d-1}]$ to $[a_{0}:\cdots:a_{d-1}:0]$. There is a naturally
induced diagram

(7.2.12)
${0}$${\mathbf{E}^{*-2d,*-d}}$${\oplus_{i=1}^{d}\mathbf{E}^{*-2i,*-i}}$${\oplus_{i=1}^{d-1}\mathbf{E}^{*-2i,*-i}}$${0}$${\mathbf{E}^{**}(\mathbb{P}^{d}/\mathbb{P}^{d-1})}$${\mathbf{E}^{**}(\mathbb{P}^{d}/\operatorname{pt})}$${\mathbf{E}^{**}(\mathbb{P}^{d-1}/\operatorname{pt})}$$\scriptstyle{\cong}$$\scriptstyle{q}$${(a)}$$\scriptstyle{\widetilde{\rho}_{d}}$$\scriptstyle{r}$${(b)}$$\scriptstyle{\widetilde{\rho}_{d-1}}$$\scriptstyle{i^{*}}$

with exact rows, where $q$ and $r$ are the obvious inclusion and projection
maps.

We will show that (7.2.12) commutes. Let $\mathcal{L}_{d}$ be the tautological
line bundle on $\mathbb{P}^{d}$. To show the square (b) commutes, let
$(x_{1},\ldots,x_{d})$ be an element of
$\oplus_{i=1}^{d}\mathbf{E}^{*-2i,*-i}$. We need to show the formula

$i^{*}\big{(}\sum_{i=1}^{d}x_{i}c_{1}(\mathcal{L}_{d})^{i}\big{)}=\sum_{i=1}^{d-1}x_{i}c_{1}(\mathcal{L}_{d-1})^{i}.$

Since $i^{*}(\mathcal{L}_{d})\cong\mathcal{L}_{d-1}$, we are reduced to
showing the vanishing

(7.2.13) $c_{1}(\mathcal{L}_{d-1})^{d}=0.$

For every integer $n\geq 1$, let
$\psi_{n}\colon\mathbf{E}^{*-2,*-1}\to\mathbf{E}^{**}(\mathbb{P}^{n}/\operatorname{pt})$
be the homomorphism in Lemma 7.2.3. There is a diagram

(7.2.14)
${\mathbf{E}^{*-2d,*-d}}$${\mathbf{E}^{*-2d,*-d}}$${\mathbf{E}^{**}((\mathbb{P}^{1}/\operatorname{pt})^{\wedge
d})}$${\mathbf{E}^{**}((\mathbb{P}^{d}/\operatorname{pt})^{\wedge
d})}$${\mathbf{E}^{**}((\mathbb{P}^{d-1}/\operatorname{pt})^{\wedge
d})}$${\mathbf{E}^{**}(\mathbb{P}^{d}/\mathbb{P}^{d-1})}$${\mathbf{E}^{**}(\mathbb{P}^{d}/\operatorname{pt})}$${\mathbf{E}^{**}(\mathbb{P}^{d-1}/\operatorname{pt}),}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\psi_{d}^{\wedge
d}}$$\scriptstyle{\cong}$${(d)}$${(e)}$$\scriptstyle{\psi_{d-1}^{\wedge
d}}$${(c)}$$\scriptstyle{\Delta^{*}}$${(f)}$$\scriptstyle{\Delta^{*}}$

where the homomorphisms denoted by $\Delta^{*}$ are naturally induced by the
diagonal morphisms. By Lemmas 7.2.2 and 7.2.3, $(c)$ and $(d)$ commutes. Apply
the relation (7.1.5) to the inclusion
$i\colon\mathbb{P}^{d-1}\to\mathbb{P}^{d}$ to deduce that $(e)$ commutes too.
It is trivial that the square $(f)$ commutes. The composite
$\mathbf{E}^{**}(\mathbb{P}^{d}/\mathbb{P}^{d-1})\to\mathbf{E}^{**}(\mathbb{P}^{d-1}/\operatorname{pt})$
is zero, so from the commutativity of (7.2.14) we deduce
$c_{1}(\mathcal{L}_{d-1})^{d}=0$. It follows that $(b)$ commutes. The
commutativity of $(c)$ and $(d)$ implies the commutativity of $(a)$.

We proceed by induction on $d$ to show that $\widetilde{\rho}_{d}$ is an
isomorphism for every integer $d\geq 0$. The claim is trivial if $d=0$. In the
following we assume $d>0$. If $\widetilde{\rho}_{d-1}$ is an isomorphism, then
$i^{*}$ is surjective. From the long exact sequence

$\displaystyle\mathbf{E}^{*-1,*}(\mathbb{P}^{d}/\operatorname{pt})$
$\displaystyle\xrightarrow{i^{*}}\mathbf{E}^{*-1,*}(\mathbb{P}^{d-1}/\operatorname{pt})\to\mathbf{E}^{**}(\mathbb{P}^{d}/\mathbb{P}^{d-1})$
$\displaystyle\to\mathbf{E}^{**}(\mathbb{P}^{d}/\operatorname{pt})\xrightarrow{i^{*}}\mathbf{E}^{**}(\mathbb{P}^{d-1}/\operatorname{pt}),$

we see that the sequence

$0\to\mathbf{E}^{**}(\mathbb{P}^{d}/\mathbb{P}^{d-1})\to\mathbf{E}^{**}(\mathbb{P}^{d}/\operatorname{pt})\xrightarrow{i^{*}}\mathbf{E}^{**}(\mathbb{P}^{d-1}/\operatorname{pt})\to
0$

is exact. Combine this with (7.2.12) and apply the five lemma to deduce that
$\widetilde{\rho}_{d}$ is an isomorphism too. ∎

######  Proposition 7.2.8.

Suppose $\mathbf{E}$ is oriented and $\mathcal{E}\to X$ is a vector bundle in
$\mathbf{SmlSm}/S$. Then there is an exact sequence

(7.2.15)
$0\to\mathbf{E}^{**}(\mathrm{Th}(\mathcal{E}))\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}\oplus\mathcal{O}))\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}))\to
0.$

###### Proof.

By Proposition 3.2.19, it suffices to show that
$\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}\oplus\mathcal{O}))\to\mathbf{E}^{**}(\mathbb{P}(\mathcal{E}))$
is surjective. This follows from Theorem 7.2.6. ∎

As in $\mathbb{A}^{1}$-homotopy theory, we have the following consequence of
the projective bundle theorem.

######  Proposition 7.2.9.

Suppose $\mathbf{E}$ is oriented. For all $x\in\mathbf{E}^{m,p}(X)$ and
$y\in\mathbf{E}^{n,q}(X)$ with $X\in\mathbf{lSm}/S$ and
$m,n,p,q\in\mathbb{Z}$, there is the formula

$x\smile y=(-1)^{mn}y\smile x$

in $\mathbf{E}^{m+n,p+q}(X)$.

###### Proof.

Let $\eta\colon T\wedge T\to T\wedge T$ be the permutation map, which
naturally induces a map

$\eta\colon\mathbf{E}^{**}(X)\to\mathbf{E}^{**}(X)$

for all $X\in\mathbf{lSm}/S$. In general, we have the formula

(7.2.16) $x\smile y=(-1)^{mn}\eta^{pq}y\smile x.$

Hence, it remains to show $\eta=1$.

Let $\mathcal{T}$ be the tautological bundle on $\mathbb{P}^{2}$, and we set
$\lambda:=c_{1}(\mathcal{T})$. When $x=y=\lambda$ in (7.2.16), we have
$\lambda^{2}=\eta\lambda^{2}$. Together with Theorem 7.2.6, we have $1=\eta$.
∎

###  7.3. Chern classes

Throughout this subsection, we fix $S\in\mathbf{Sch}$ and an oriented homotopy
commutative monoid $\mathbf{E}$ in $\mathbf{logSH}(S)$.

######  Definition 7.3.1.

Suppose $\mathcal{E}\to X$ is a rank $d$ vector bundle with
$X\in\mathbf{lSm}/S$. To define higher Chern classes of $\mathcal{E}$, we
imitate Grothendieck’s approach as follows. Let
$p\colon\mathbb{P}(\mathcal{E})\to X$ be the canonical projection, and let
$\mathcal{T}$ be the tautological line bundle on $\mathbb{P}(\mathcal{E})$.
For simplicity of notation, we set $\lambda:=c_{1}(\mathcal{T})$. Owing to
Theorem 7.2.6, there exist unique classes
$c_{i}(\mathcal{E})\in\mathbf{E}^{2i,i}(X)$ for all $0\leq i\leq d$ such that
$c_{0}(\mathcal{E})=1$ and

(7.3.1) $\sum_{i=0}^{d}p^{*}(c_{i}(\mathcal{E}))(-\lambda)^{d-i}=0.$

The class $c_{i}(\mathcal{E})$ is called the _$i$ -th Chern class_ of
$\mathcal{E}$. We set $c_{i}(\mathcal{E}):=0$ if $i\geq d+1$.

######  Remark 7.3.2.

Suppose $\mathcal{E}\to X$ is a line bundle in $\mathbf{lSm}/S$. Then
$\mathbf{P}(\mathcal{E})\cong X$, and (7.3.1) becomes

$c_{1}(\mathcal{E})-c_{1}(\mathcal{E})=0.$

Here, the left $c_{1}(\mathcal{E})$ is defined in Definition 7.3.1, and the
right $c_{1}(\mathcal{E})$ is defined in Definition 7.1.1. Hence, these two
definitions of the first Chern class of $\mathcal{E}$ agree.

######  Proposition 7.3.3.

Suppose $\mathcal{E}\to X$ is a trivial line bundle in $\mathbf{lSm}/S$. Then
$c_{i}(\mathcal{E})=0$ for all integers $i\geq 1$.

###### Proof.

Combine (7.2.13) and (7.3.1). ∎

######  Proposition 7.3.4.

Suppose $f\colon X^{\prime}\to X$ is a morphism in $\mathbf{lSm}/S$, and
$\mathcal{E}\to X$ is a rank $d$ vector bundle. Then, for every integer $i\geq
0$, we have the formula

$f^{*}(c_{i}(\mathcal{E}))=c_{i}(f^{*}\mathcal{E}).$

###### Proof.

Let $\mathcal{T}^{\prime}$ be the tautological line bundle of
$\mathbb{P}(f^{*}\mathcal{E})$, and we set
$\lambda^{\prime}:=c_{1}(\mathcal{T}^{\prime*})$. Apply $f^{*}$ to (7.3.1) and
use (7.1.5) to have the formula

$\sum_{i=0}^{d}p^{\prime*}(f^{*}(c_{i}(\mathcal{E})))(-\lambda^{\prime})^{d-i}=0,$

where $p^{\prime}\colon\mathbb{P}(f^{*}\mathcal{E})\to X^{\prime}$ is the
projection. To conclude, compare this with (7.3.1) for the vector bundle
$f^{*}\mathcal{E}\to X^{\prime}$. ∎

######  Proposition 7.3.5 (Splitting principle).

If $\mathcal{E}\to X$ is a rank $d$ vector bundle with $X\in\mathbf{lSm}/S$,
then there exists a strict proper smooth morphism $f\colon Y\to X$ such that
$f^{*}\colon\mathbf{E}^{**}(X)\to\mathbf{E}^{**}(Y)$ is a monomorphism and
$f^{*}\mathcal{E}$ admits a filtration of subbundles

$0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\cdots\subset\mathcal{E}_{d}=\mathcal{E}$

such that the quotient bundle $\mathcal{E}_{i}/\mathcal{E}_{i-1}$ is a line
bundle for every $1\leq i\leq d$.

###### Proof.

As in intersection theory or $\mathbb{A}^{1}$-homotopy theory, let $Y$ be the
complete flag variety associated with $\mathcal{E}$. If $f$ is the projection
$Y\to X$, then it is well known that $f^{*}\mathcal{E}$ satisfies the required
condition. Since $f$ can be factored as a tower of projective bundles, Theorem
7.2.6 shows that $f^{*}$ is a monomorphism. ∎

######  Definition 7.3.6.

Suppose $\mathcal{E}\to X$ is a rank $d$ vector bundle with
$X\in\mathbf{lSm}/S$. Let $p\colon\mathbb{P}(\mathcal{E})\to X$ and
$p^{\prime}\colon\mathbb{P}(\mathcal{E}\oplus\mathcal{O})\to X$ be the
canonical projections, and let $\mathcal{T}$ and $\mathcal{T}^{\prime}$ be the
tautological line bundles on $\mathbb{P}(\mathcal{E})$ and
$\mathbb{P}(\mathcal{E}\oplus\mathcal{O})$ for simplicity of notation. We set
$\lambda:=c_{1}(\mathcal{T})$ and
$\lambda^{\prime}:=c_{1}(\mathcal{T}^{\prime})$. Consider the class

(7.3.2)
$\overline{t}(\mathcal{E}):=\sum_{i=0}^{d}p^{\prime*}(c_{i}(\mathcal{E}))(-\lambda^{\prime})^{d-i}\in\mathbf{E}^{2d,d}(\mathbb{P}(\mathcal{E}\oplus\mathcal{O})).$

Due to (7.1.5) and (7.3.1), the restriction of $\overline{t}(\mathcal{E})$ to
$\mathbb{P}(\mathcal{E})$ is $0$. From (7.2.15), we have a unique class

$t(\mathcal{E})\in\mathbf{E}^{2d,d}(\mathrm{Th}(\mathcal{E}))$

mapping to $\overline{t}(\mathcal{E})$, which we call the _Thom class_ of
$\mathcal{E}$.

######  Proposition 7.3.7.

Let $t_{\infty}\in\mathbf{E}^{2,1}(\mathrm{Th}(\mathcal{T}_{1}))$ be a Thom
orientation, where $\mathcal{T}_{1}$ is the tautological line bundle on
$\mathbb{P}^{\infty}$. If $c_{\infty}$ is the Chern orientation associated
with $t_{\infty}$, then the Thom class $t(\mathcal{T}_{1})$ obtained from
$c_{\infty}$ coincides with $t_{\infty}$.

###### Proof.

Consider the isomorphism $z^{*}$ given in (7.1.10). We need to show
$z^{*}t_{\infty}=z^{*}t(\mathcal{T}_{1})$. The left-hand side equals
$c_{\infty}$. On the other hand, from (7.3.2), we have the formula
$z^{*}t(\mathcal{T}_{1})=c_{1}(\mathcal{T}_{1})$. An appeal to (7.1.4)
finishes the proof. ∎

Suppose $X\in\mathbf{lSm}/S$ and $\mathcal{E}\to X$ is a rank $d$ vector
bundle, and let $E$ be the exceptional divisor on
${\mathrm{Bl}}_{X}\mathcal{E}$. Using the graph morphisms of the two
projections $\mathcal{E}\to X$ and $({\mathrm{Bl}}_{X}\mathcal{E},E)\to X$, we
obtain a morphism

(7.3.3) $\mathrm{Th}(\mathcal{E})\to\mathrm{Th}(\mathcal{E})\wedge X_{+}.$

We use the latter map to define the cup product

$\smile\colon\mathbf{E}^{**}(X)\times\mathbf{E}^{**}(\mathrm{Th}(\mathcal{E}))\to\mathbf{E}^{**}(\mathrm{Th}(\mathcal{E}))$

as in (7.0.3).

######  Theorem 7.3.8 (Thom isomorphism).

For every rank $d$ vector bundle $p\colon\mathcal{E}\to X$ with
$X\in\mathbf{lSm}/S$, the cup product homomorphism

(7.3.4) $-\smile
t(\mathcal{E})\colon\mathbf{E}^{**}(X)\to\mathbf{E}^{*+2d,*+d}(\mathrm{Th}(\mathcal{E}))$

is an isomorphism.

###### Proof.

Combine Theorem 7.2.6 and (7.2.15) to have a commutative diagram with exact
rows