}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Since $i$ is strict, $Q$ is cartesian in the category of log schemes. It
follows that the underlying square of schemes $\underline{Q}$ is cartesian too
due to [37, Proposition III.2.1.2].

Let us omit $R_{Zar}$ in the following for simplicity of notation. We want to
show the unit
${\mathrm{id}}\xrightarrow{ad}\underline{f}_{*}\underline{f}^{*}$ is an
equivalence. Since $\underline{i}_{*}$ is conservative, it suffices to show
that the natural transformation
$\underline{i}_{*}\xrightarrow{ad}\underline{i}_{*}\underline{f}_{*}\underline{f}^{*}$
is an equivalence. There is a commutative diagram

${\underline{i}_{*}}$${\underline{i}_{*}}$${\underline{g}_{*}\underline{g}^{*}\underline{i}_{*}}$${\underline{g}_{*}\underline{i}_{*}^{\prime}\underline{f}^{*}}$${\underline{i}_{*}\underline{f}_{*}\underline{f}^{*},}$$\scriptstyle{ad}$$\scriptstyle{{\mathrm{id}}}$$\scriptstyle{ad}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where
$\underline{g}_{*}\underline{g}^{*}\underline{i}_{*}\xrightarrow{\simeq}\underline{g}_{*}\underline{i}_{*}^{\prime}\underline{f}^{*}$
by proper base change. Hence it suffices to show the left-hand vertical map is
an equivalence. The projection formula yields an equivalence

$\underline{g}_{*}\underline{g}^{*}\mathcal{O}_{\underline{X}\times\underline{\mathbb{A}_{P}}}\otimes\mathcal{E}\xrightarrow{\simeq}\underline{g}_{*}\underline{g}^{*}\mathcal{E}$

for all
$\mathcal{E}\in\mathrm{D}^{b}(\mathrm{Coh}(\underline{X}\times\underline{\mathbb{A}_{P}}))$.
This concludes the proof since the unit

$\mathcal{O}_{\underline{X}\times\underline{\mathbb{A}_{P}}}\xrightarrow{ad}\underline{g}_{*}\underline{g}^{*}\mathcal{O}_{\underline{X}\times\underline{\mathbb{A}_{P}}}$

is an equivalence; see Proposition 6.20. ∎

##  7\. Sheaves on arrow categories

###  7.1. Topologies on arrow categories

One aspect of the dividing Nisnevich topology is that it involves non-affine
schemes. Hence to “sheafify” $\operatorname{logHH}$ with respect to the
dividing Nisnevich topology, we need to define $\operatorname{logHH}(X/S)$ for
arbitrary morphism of fs log schemes $X\to S$. For this purpose, we first view
$\operatorname{logHH}$ as a presheaf on an arrow category. In this setup, we
can sheafify with respect to the dividing Nisnevich topology.

######  Definition 7.2.

Let $\mathcal{C}$ be a category. We define the arrow category
$\mathrm{Ar}_{\mathcal{C}}$ of $\mathcal{C}$ as follows. The objects are the
morphisms $(X\to S)$ in $\mathcal{C}$. The morphisms $(X^{\prime}\to
S^{\prime})\to(X\to S)$ are the commutative diagrams

${X^{\prime}}$${S^{\prime}}$${X}$${S.}$

If fiber products are representable in $\mathcal{C}$, so are in
$\mathrm{Ar}_{\mathcal{C}}$. Indeed, if $(X^{\prime}\to S^{\prime})\to(X\to
S)$ and $(X^{\prime\prime}\to S^{\prime\prime})\to(X\to S)$ are two morphisms
in $\mathrm{Ar}_{\mathcal{C}}$, then the fiber product is
$(X^{\prime}\times_{X}X^{\prime\prime}\to
S^{\prime}\times_{S}S^{\prime\prime})$.

######  Definition 7.3.

Suppose $\mathcal{C}$ is a category with a pretopology $t$. A family of
morphisms $\\{(X_{i}\to S_{i})\to(X\to S)\\}_{i\in I}$ is called a _$t$
-cover_ if both families

$\\{X_{i}\to X\\}_{i\in I}\text{ and }\\{S_{i}\to S\\}_{i\in I}$

are $t$-covers. One can readily check that the set of $t$-covers on
$\mathrm{Ar}_{\mathcal{C}}$ is a pretopology too.

###  7.4. $\operatorname{logHH}$ as a presheaf on an arrow category

######  Definition 7.5.

An _fs semi-log ring_ is a pre-log ring $(A,M)$ such that
$\overline{M}:=M/M^{*}$ is fine and saturated and the naturally induced map

$M\to\Gamma(\operatorname{Spec}(A,M),\mathcal{M}_{\operatorname{Spec}(A,M)})$

is an isomorphism, where $\operatorname{Spec}(A,M)$ is the spectrum of $(A,M)$
[37, Definition III.1.2.3]. Let $\mathrm{SemiLog}$ denote the full category of
pre-log rings consisting of fs semi-log rings $(A,M)$ such that $A$ is
noetherian and has a finite Krull dimension.

######  Remark 7.6.

There are at least three possible variants of affine log schemes. First, one
may define an affine log scheme to be a log scheme
$X=(\underline{X},\mathcal{M}_{X})$ such that
$\underline{X}=\operatorname{Spec}(R)$ is an affine scheme. Second, following
[5, 1.1], one can add the condition that the sheaf
$\overline{\mathcal{M}}_{X}$ is generated by global sections, i.e.,
$\Gamma({X},\mathcal{M}_{X})\to\overline{\mathcal{M}}_{X,x}$ is surjective for
every point $x$. Lastly, following [26], one can ask that the identity map on
$\Gamma(X,\mathcal{M}_{X})$ is a chart for $\mathcal{M}_{X}$. The third one is
taken as our definition of an affine log scheme.

######  Proposition 7.7.

The functor $\mathrm{Spec}\colon\mathrm{SemiLog}^{op}\to\mathrm{lSch}$ is
fully faithful.

###### Proof.

Let $(A,M)$ and $(R,P)$ be fs semi-log rings. We have an isomorphism

$\mathrm{Hom}_{\mathrm{SemiLog}}((R,P),(A,M))\cong\mathrm{Hom}_{\mathrm{CRing}}(R,A)\times_{\mathrm{Hom}_{\mathrm{CMon}}(P,A)}\mathrm{Hom}_{\mathrm{CMon}}(P,M),$

where $\mathrm{CMon}$ (resp. $\mathrm{CRing}$) denotes the category of
commutative monoids (resp. commutative rings).

We set $X:=\operatorname{Spec}(A,M)$ and $S:=\operatorname{Spec}(R,P)$ for
simplicity of notation. We have isomorphisms

$\displaystyle\mathrm{Hom}_{\mathrm{lSch}}(X,S)\cong$
$\displaystyle\mathrm{Hom}_{\mathrm{lSch}}(X,\underline{S}\times_{\underline{\mathbb{A}_{P}}}\mathbb{A}_{P})$
$\displaystyle\cong$
$\displaystyle\mathrm{Hom}_{\mathrm{lSch}}(X,\underline{S})\times_{\mathrm{Hom}_{\mathrm{lSch}}(X,\underline{\mathbb{A}_{P}})}\mathrm{Hom}_{\mathrm{lSch}}(X,\underline{\mathbb{A}_{P}})$
$\displaystyle\cong$
$\displaystyle\mathrm{Hom}_{\mathrm{Sch}}(\underline{X},\underline{S})\times_{\mathrm{Hom}_{\mathrm{CMon}}(P,\Gamma(X,\mathcal{O}_{X}))}\mathrm{Hom}_{\mathrm{CMon}}(P,\Gamma(X,\mathcal{M}_{X})),$

where we use [37, Proposition III.1.2.4] for the last isomorphism. Together
with the isomorphisms $\Gamma(X,\mathcal{O}_{X})\cong A$ and
$\Gamma(X,\mathcal{M}_{X})\cong M$, we finish the proof. ∎

######  Definition 7.8.

Let $\mathrm{lAff}$ be the full subcategory of $\mathrm{lSch}$ consisting of
the spectra of fs semi-log rings. An important fact about $\mathrm{lAff}$ that
we will frequently use is that every $X\in\mathrm{lSch}$ admits a Zariski
covering $\\{U_{i}\to X\\}_{i\in I}$ such that $U_{i}\in\mathrm{lAff}$ for all
$i\in I$, see [53, Lemmas 1.3.2, 1.3.3].

The assignment

$(\operatorname{Spec}(A,M)\to\operatorname{Spec}(R,P))\mapsto\operatorname{logHH}((A,M)/(R,P))$

gives a presheaf
$\operatorname{logHH}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{S}\mathrm{CRing})$.
By composing with the canonical functor
$\mathcal{S}\mathrm{CRing}\to\mathcal{D}(\mathrm{Ab})$, we also view
$\operatorname{logHH}$ as an object of
$\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{D}(\mathrm{Ab}))$.

######  Lemma 7.9.

Let $(R,P)\xrightarrow{f}(A,M)\xrightarrow{g}(B,N)$ be homomorphisms of fs
semi-log rings. If $f$ is integral log étale, then there is an equivalence

$\operatorname{logHH}((B,N)/(R,P))\xrightarrow{\simeq}\operatorname{logHH}((B,N)/(A,M)).$

###### Proof.

According to Proposition 5.4, we have equivalences

$\operatorname{logHH}((B,N)/(A,M))\simeq
A\otimes_{\operatorname{logHH}(A,M)}\operatorname{logHH}(B,N),$
$\operatorname{logHH}((B,N)/(R,P))\simeq
R\otimes_{\operatorname{logHH}(R,P)}\operatorname{logHH}(B,N).$

Together with Corollary 5.27, we finish the proof. ∎

######  Lemma 7.10.

Let $(R,P)\xrightarrow{f}(A,M)\xrightarrow{g}(B,N)$ be homomorphisms of fs
semi-log rings. If $g$ is integral log étale, then there is an equivalence

$B\otimes_{A}\operatorname{logHH}((A,M)/(R,P))\xrightarrow{\simeq}\operatorname{logHH}((B,N)/(R,P)).$

###### Proof.

This follows from Theorem 5.24. ∎

A morphism $f\colon X\to S$ is called an _integral log étale cover_ if $f$ is
integral, log étale, and universally surjective in the category
$\mathrm{lSch}$. This defines an _integral log étale topology_. Every integral
log étale morphism is Kummer by [37, Corollary I.4.3.10, Proposition IV.3.4.1]
(see also [37, p. 344]). Hence the integral log étale topology is coarser than
the Kummer log étale topology.

For a morphism of affine fs log schemes $X:=\operatorname{Spec}(A,M)\to
S:=\operatorname{Spec}(R,P)$, we set

$\operatorname{logHH}(X/S):=\operatorname{logHH}((A,M)/(R,P)).$

######  Proposition 7.11.

The presheaf
$\operatorname{logHH}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{D}(\mathrm{Ab}))$
satisfies integral log étale descent.

###### Proof.

Suppose $(X^{\prime}\to S^{\prime})\to(X\to S)$ is an integral log étale cover
in $\mathrm{Ar}_{\mathrm{lAff}}$. By Lemma 7.9, we have an equivalence

(7.1)
$\operatorname{logHH}(X^{\prime}/S^{\prime})\simeq\operatorname{logHH}(X^{\prime}/S).$

If $(\mathscr{X}\to\mathscr{S})\to(X\to S)$ is an integral log étale
hypercover in $\mathrm{Ar}_{\mathrm{lAff}}$, we need to show that the
canonical map

$\operatorname{logHH}(X/S)\to\lim_{i\in\Delta}\operatorname{logHH}(\mathscr{X}_{i}/{\mathscr{S}_{i}})$

is an equivalence. Thanks to (7.1), we are reduced to considering the
canonical map

$\operatorname{logHH}(X/S)\to\lim_{i\in\Delta}\operatorname{logHH}(\mathscr{X}_{i}/S).$

Suppose $S$ (resp. $X$, $\mathscr{X}_{i}$) is given by
$\operatorname{Spec}(A,M)$ (resp. $\operatorname{Spec}(B,N)$,
$\operatorname{Spec}(B_{i},N_{i})$). The underlying morphism of schemes of an
integral log étale morphism is flat by [37, Theorem IV.4.3.5]. Hence, by
faithfully flat descent, we obtain the exact sequence of $A$-modules

$0\to B\to B_{0}\to B_{1}\to\cdots.$

To conclude we apply $(-)\otimes_{A}\operatorname{logHH}(X/S)$ and appeal to
Lemma 7.10. ∎

######  Proposition 7.12.

There is an equivalence

$\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lAff}})\xrightarrow{\simeq}\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}}).$

Similar equivalences hold for
$\mathcal{S}\mathrm{hv}(-,\mathcal{S}\mathrm{pc})$ and
$\mathcal{S}\mathrm{hv}(-,\mathcal{D}(\mathrm{Ab}))$.

###### Proof.

Suppose $(X\to S)$ is an object of $\mathrm{Ar}_{\mathrm{lSch}}$. Choose a
strict Nisnevich cover $S^{\prime}\to S$ such that
$S^{\prime}\in\mathrm{lAff}$, and choose a strict Nisnevich cover
$X^{\prime}\to X\times_{S}S^{\prime}$ such that $X^{\prime}\in\mathrm{lAff}$.
Then the morphism $(X^{\prime}\to S^{\prime})\to(X\to S)$ is a strict
Nisnevich cover. Hence every object of $\mathrm{Ar}_{\mathrm{lSch}}$ has a
strict Nisnevich cover in $\mathrm{Ar}_{\mathrm{lAff}}$. Proposition 6.8
finishes the proof. ∎

######  Definition 7.13.

For $\mathcal{V}=\mathcal{S}\mathrm{pc}$ or $\mathcal{D}(\mathrm{Ab})$, the
inclusion functor
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{V})\to\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$
admits a left adjoint

$L_{t}\colon\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})\to\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{V}).$

This is called the _$t$ -localization functor_. We say that
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ satisfies _$t$
-descent_ if $\mathcal{F}$ is in the essential image of the inclusion functor
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{V})\to\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$.

######  Definition 7.14.

As a consequence of Propositions 7.11 and 7.12, we obtain a sheaf

$\operatorname{logHH}\in\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab})).$

This allows us to define $\operatorname{logHH}(X/S):=\operatorname{logHH}(X\to
S)\in\mathcal{D}(\mathrm{Ab})$ for every morphism $X\to S$ in $\mathrm{lSch}$.
We also set

$\operatorname{logHH}_{dNis}:=L_{dNis}\operatorname{logHH}\in\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$

and

$\operatorname{logHH}_{dNis}(X/S):=\operatorname{logHH}_{dNis}(X\to
S)\in\mathcal{D}(\mathrm{Ab}).$

###  7.15. $\Omega^{d}$ and $\mathbb{L}$ as presheaves on an arrow category

######  Definition 7.16.

For $i\geq 1$, we let
$\Omega^{i}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
be the presheaf given by

$\Omega^{i}(X\to S):=\Omega_{X/S}^{i}(X)$

concentrated in degree $0$. By convention,
$\mathcal{O}=\Omega^{0}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
is the presheaf given by $\mathcal{O}(X\to
S):=\Gamma(\underline{X},\mathcal{O}_{\underline{X}})$.

######  Proposition 7.17.

For $(X\to S)\in\mathrm{Ar}_{\mathrm{lSch}}$ and all $d\geq 0$, there are
equivalences

$L_{sNis}\Omega^{d}(X\to S)\simeq R\Gamma_{sNis}(X,\Omega_{X/S}^{d})\text{ and
}L_{dNis}\Omega^{d}(X\to S)\simeq R\Gamma_{dNis}(X,\Omega_{X/S}^{d}).$

###### Proof.

We first consider the strict Nisnevich case. Suppose $(X^{\prime}\to
S^{\prime})\to(X\to S)$ is a strict Nisnevich cover in
$\mathrm{Ar}_{\mathrm{lSch}}$. Since $\Omega_{S^{\prime}/S}^{1}=0$, we have an
isomorphism
$\Omega_{X^{\prime}/S}^{1}\xrightarrow{\simeq}\Omega_{X^{\prime}/S^{\prime}}^{1}$,
and hence

(7.2)
$\Omega_{X^{\prime}/S}^{d}\xrightarrow{\simeq}\Omega_{X^{\prime}/S^{\prime}}^{d}$

for every integer $d\geq 1$ (it holds trivially true for $d=0$).

Let $\mathrm{Hyp}_{(X\to S)}$ (resp. $\mathrm{Hyp}_{X}$) be the set of strict
Nisnevich hypercovers of $(X\to S)$ in $\mathrm{Ar}_{\mathrm{lSch}}$ (resp.
$X$ in $\mathrm{lSch}$). By appealing to (7.2) we have

$\displaystyle L_{sNis}\Omega^{d}(X\to S)$
$\displaystyle\simeq\mathop{\mathrm{colim}}_{(\mathscr{X}\to\mathscr{S})\in\mathrm{Hyp}_{(X\to
S)}}\lim_{i\in\Delta}\Omega^{d}(\mathscr{X}_{i}\to\mathscr{S}_{i})$
$\displaystyle\simeq\mathop{\mathrm{colim}}_{\mathscr{X}\in\mathrm{Hyp}_{X}}\lim_{i\in\Delta}\Omega^{d}(\mathscr{X}_{i}\to
S)\simeq R\Gamma_{sNis}(X,\Omega_{X/S}^{d}).$

The claim for the dividing Nisnevich case can be proven similarly. ∎

For a morphism $f\colon X\to S$ of fs log schemes, let $\mathbb{L}_{X/S}$
denote Gabber’s cotangent complex for log schemes defined in [39, §8.29]. This
complex lives in the $\infty$-category of chain complexes of quasi-coherent
$\mathcal{O}_{X}$-modules. If $f$ is a morphism of fs log schemes given by
$\operatorname{Spec}(g,g^{\flat})$ for some map $(g,g^{\flat})$ of discrete
log rings, then this is equivalent to the log cotangent complex in Remark 3.6.

######  Definition 7.18.

Let
$\mathbb{L}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
be the presheaf given by $(X\to S)\mapsto R\Gamma_{sNis}(X,\mathbb{L}_{X/S})$.
We set $\mathbb{L}^{dNis}:=L_{dNis}\mathbb{L}$, for the $dNis$-localization

$L_{dNis}\colon\mathrm{Psh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))\to\mathrm{Shv}_{dNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab})).$

We can similarly define $\bigwedge\nolimits^{d}\mathbb{L}$ and
$\bigwedge\nolimits^{d}\mathbb{L}^{dNis}$ for all integer $d$.

######  Proposition 7.19.

The presheaf
$\mathbb{L}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
satisfies integral log étale descent.

###### Proof.

Suppose $X^{\prime}\xrightarrow{f}S^{\prime}\xrightarrow{g}S$ are morphisms in
$\mathrm{lSch}$. If $g$ is an integral log étale cover, then
$\mathbb{L}_{S^{\prime}/S}\simeq 0$ by Proposition 4.6(2). The transitivity
triangle

$f^{*}\mathbb{L}_{S^{\prime}/S}\to\mathbb{L}_{X^{\prime}/S}\to\mathbb{L}_{X^{\prime}/S^{\prime}}\to
f^{*}\mathbb{L}_{S^{\prime}/S}[1]$

implies
$\mathbb{L}_{X^{\prime}/S}\xrightarrow{\simeq}\mathbb{L}_{X^{\prime}/S^{\prime}}$.
Argue as in the proof of Proposition 7.17 to finish the proof. ∎

######  Proposition 7.20.

For every integer $d\geq 0$,
$L_{sNis}\Omega^{d}\in\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
satisfies dividing Nisnevich descent.

###### Proof.

Let $(\mathscr{X}\to\mathscr{S})\to(X\to S)$ is a dividing Nisnevich
hypercover in $\mathrm{Ar}_{\mathrm{lSch}}$. We need to show that the
naturally induced map

$L_{sNis}\Omega^{d}(X\to
S)\to\mathop{\mathrm{colim}}_{(\mathscr{X}\to\mathscr{S})\in\mathrm{Hyp}_{(X\to
S)}^{dNis}}\lim_{i\in\Delta}L_{sNis}\Omega^{d}(\mathscr{X}_{i}\to\mathscr{S}_{i})$

is an equivalence, where $\mathrm{Hyp}_{(X\to S)}^{dNis}$ is the set of
dividing Nisnevich hypercovers of $(X\to S)$ in $\mathrm{Ar}_{\mathrm{lSch}}$.
There is an isomorphism
$\Omega_{\mathscr{X}_{j}/\mathscr{S}_{j}}^{1}\cong\Omega_{\mathscr{X}_{j}/S}^{1}$
for every integer $j\geq 0$ since $\mathscr{S}_{j}\to S$ is log étale. Hence
it suffices to show that the naturally induced homomorphism

(7.3)
$H_{sNis}^{i}(X,\Omega_{X/S}^{d})\to\mathbf{H}_{sNis}^{i}(\mathscr{X},\Omega_{\mathscr{X}/S}^{d})$

is an isomorphism for every integer $i\geq 0$. It remains to check that

$R\Gamma_{sNis}(-,\Omega_{-/S}^{d})\in\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{lSch}/S,\mathcal{D}(\mathrm{Ab}))$

satisfies dividing Nisnevich descent; this is done in Corollary 6.17. ∎

######  Definition 7.21.

Let $\mathrm{lSmAr}$ be the full subcategory of $\mathrm{Ar}_{\mathrm{lSch}}$
consisting of log smooth morphisms.

######  Lemma 7.22.

The inclusion functor

$\mathrm{lSmAr}\to\mathrm{Ar}_{\mathrm{lSch}}$

is a cocontinuous morphism of sites for the dividing Nisnevich topologies.

###### Proof.

Suppose $(Y^{\prime}\to X^{\prime})\to(Y\to X)$ is a dividing Nisnevich cover
in $\mathrm{Ar}_{\mathrm{lSch}}$. If $Y\to X$ is log smooth, then
$Y^{\prime}\to X^{\prime}$ is log smooth too by [37, Remark III.3.1.2]. ∎

######  Definition 7.23.

Let $f\colon X\to S$ be a morphism of fs log schemes. We say that $f$ is
_derived log smooth_ (resp.  _derived log étale_) if $\underline{f}$ is
locally of finite presentation and $\mathbb{L}_{X/S}$ is a finitely generated
locally free $\mathcal{O}_{X}$-module (resp. $\mathbb{L}_{X/S}\simeq 0$).

Observe that the conditions of being derived log smooth and derived log étale
are Zariski local on $X$ and $S$.

Let $(g,g^{\flat})\colon(R,P)\to(A,M)$ be a map of log rings. If
$(g,g^{\flat})$ is derived log smooth, then it is log smooth, so
$\mathbb{L}_{(A,M)/(R,P)}$ is a finitely generated projective $A$-module by
Proposition 4.5 and [37, Proposition IV.3.2.1]. Thus $(g,g^{\flat})$ is
derived log smooth if and only if $\operatorname{Spec}(g,g^{\flat})$ is
derived log smooth. We have a similar claim for the derived log étale case.
Hence the implications (4.1) and (4.2) are still valid for morphisms of fs log
schemes.

As in Proposition 4.3, one can show that the classes of derived log smooth and
derived log étale morphisms of fs log schemes are closed under compositions
and pullbacks.

######  Proposition 7.24.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$. Then
there is a canonical equivalence

(7.4) $\mathbb{L}^{dNis}(X\to S)\simeq R\Gamma_{dNis}(X,\Omega_{X/S}^{1}).$

In particular, if $f$ is log étale, then $\mathbb{L}^{dNis}(X\to S)\simeq 0$.

###### Proof.

Owing to Proposition 7.17 and Lemma 7.22, we need to show that the canonical
map

$\mathbb{L}^{dNis}\to L^{dNis}\Omega^{1}$

is an equivalence in
$\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{lSmAr},\mathcal{D}(\mathrm{Ab}))$. Let
$\mathrm{dlSmAr}$ be the full subcategory of $\mathrm{lSmAr}$ consisting of
derived log smooth morphisms. For every $(X\to S)\in\mathrm{lSmAr}$, we can
choose a dividing Nisnevich cover $S^{\prime}\to S$ such that the projection
$X\times_{S}S^{\prime}\to S^{\prime}$ is integral log smooth by [23, Theorem
1.1]. Together with Proposition 6.8, we have an equivalence

(7.5)
$\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{lSmAr},\mathcal{D}(\mathrm{Ab}))\simeq\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab})).$

Hence we may assume $f$ is derived log smooth. Now it suffices to show there
is a canonical equivalence

$\mathbb{L}_{X/S}\xrightarrow{\simeq}\Omega_{X/S}^{1}$

This question is Zariski local on $X$ and $S$, so we further reduce to the
case when $S,X\in\mathrm{lAff}$. In this case, Proposition 4.6(1) gives (7.4).
Suppose $f$ is log étale. For every log étale morphism $Y\to S$, we have
$\Omega_{Y/S}^{1}=0$. Hence the right-hand side of (7.4) vanishes. ∎

######  Remark 7.25.

The localization map $\mathbb{L}\to\mathbb{L}^{dNis}$ is not an equivalence
since there is a log étale morphism $f\colon X\to S$ in $\mathrm{lSch}$ such
that $\mathbb{L}_{X/S}$ is not contractible, see Example 4.9.

######  Proposition 7.26.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$. Then
there is a canonical equivalence

$\mathbb{L}^{dNis}(X\to S)\simeq R\Gamma_{sNis}(X,\Omega_{X/S}^{1}).$

###### Proof.

Due to Propositions 7.17 and 7.24, it only remains to check that
$R\Gamma_{sNis}(-,\Omega_{X/S}^{1})$ satisfies dividing Nisnevich descent.
This is done in Corollary 6.17. ∎

######  Example 7.27.

Let $f\colon X^{\prime}\to Y^{\prime}$ be the log étale map in Example 4.9
with non-vanishing log cotangent complex $\mathbb{L}_{X^{\prime}/Y^{\prime}}$.
Let $P$ (resp. $P_{1}$, $P_{2}$) be the submonoid of $\mathbb{Z}^{2}$
generated by $(2,0)$, $(1,1)$, and $(0,2)$ (resp. $(2,0)$ and $(-1,1)$,
$(0,2)$ and $(1,-1)$). We set $P_{12}:=P_{1}+P_{2}$,
$Y_{1}^{\prime}:=Y^{\prime}\times_{\mathbb{A}_{P}}\mathbb{A}_{P_{1}}$,
$Y_{2}^{\prime}:=Y^{\prime}\times_{\mathbb{A}_{P}}\mathbb{A}_{P_{2}}$, and
$Y_{12^{\prime}}:=Y^{\prime}\times_{\mathbb{A}_{P}}\mathbb{A}_{P_{12}}$. We
similarly define $X_{1}^{\prime}$, $X_{2}^{\prime}$, and $X_{12^{\prime}}$.
Then the glueing of $Y_{1}^{\prime}$ and $Y_{2}^{\prime}$ along
$Y_{12^{\prime}}$ is a dividing cover of $Y^{\prime}$. Furthermore, the
projections $X_{1}^{\prime}\to Y_{1}^{\prime}$, $X_{2}^{\prime}\to
Y_{2}^{\prime}$, and $X_{12^{\prime}}\to Y_{12^{\prime}}$ are integral. Hence

$\mathbb{L}(X_{1}^{\prime}\to Y_{1}^{\prime}),\mathbb{L}(X_{2}^{\prime}\to
Y_{2}^{\prime}),\mathbb{L}(X_{12}^{\prime}\to Y_{12}^{\prime})\simeq 0$

by Proposition 4.6(2). This means that the induced sequence

$\mathbb{L}(X^{\prime}\to Y^{\prime})\to\mathbb{L}(X_{1}^{\prime}\to
Y_{1}^{\prime})\oplus\mathbb{L}(X_{2}^{\prime}\to
Y_{2}^{\prime})\to\mathbb{L}(X_{12}^{\prime}\to Y_{12}^{\prime})$

is not a cofiber sequence, so $\mathbb{L}$ does not satisfy dividing Nisnevich
descent.

Theorem 5.24 shows that the induced sequence

$\operatorname{logHH}(X^{\prime}/Y^{\prime})\to\operatorname{logHH}(X_{1}^{\prime}/Y_{1}^{\prime})\oplus\operatorname{logHH}(X_{2}^{\prime}/Y_{2}^{\prime})\to\operatorname{logHH}(X_{12}^{\prime}/Y_{12}^{\prime})$

is not a cofiber sequence, so $\operatorname{logHH}$ does not satisfy dividing
Nisnevich descent either.

######  Remark 7.28 (Comparison to Olsson’s cotangent complex).

As explained in Section 3, we use Gabber’s log cotangent complex. It is a more
or less direct extension to the context of pre-log (derived) rings of the
classical construction due to Illusie and Quillen. There is another approach
to the deformation theory of log schemes: In [39, Definition 3.2], Olsson
defines, for every morphism $f\colon X\to S$ of fine log schemes, a projective
system $\mathbb{L}^{O}_{X/S}=(\mathbb{L}^{\geq-n}_{X/S})_{n}$ of essentially
constant ind-objects in $\mathcal{D}^{b}(\mathrm{Mod}_{\mathcal{O}_{X}})$. As
such, a direct comparison with $\mathbb{L}_{X/S}$ considered in this paper
(and in [39, §8]) is not immediate. We will make a condition comparison, in
the following sense.

Assume the existence of an object
$\mathbb{L}^{O}\in\mathcal{P}\mathrm{sh}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
with the following two formal properties:

  1. (i)

The projective system

$(\cdots\to\tau_{\geq-n-1}\mathbb{L}^{O}(X\to
S)\to\tau_{\geq-n}\mathbb{L}^{O}(X\to S)\to\cdots\to\tau_{\geq
0}\mathbb{L}^{O}(X\to S))$

is canonically equivalent to $\mathbb{L}_{X/S}^{O}$.

  2. (ii)

There exists a map $\mathbb{L}\to\mathbb{L}^{O}$ such that for all integers
$n\geq 0$, the canonical map

$\tau_{\geq-n}\mathbb{L}(X\to S)\to\tau_{\geq-n}\mathbb{L}^{O}(X\to S)$

is naturally induced by the map constructed in [39, §8.31].

In other words, $\mathbb{L}^{O}(-)$ is a presheaf on the arrow category
$\mathrm{Ar}_{\mathrm{lSch}}$ whose Postnikov tower is precisely the
projective system considered by Olsson.

We claim that under the above assumptions, the map
$\mathbb{L}\to\mathbb{L}^{O}$ becomes an equivalence after dividing Nisnevich
sheafification. Indeed, let $\mathrm{iAr}$ be the full subcategory of
$\mathrm{Ar}_{\mathrm{lSch}}$ consisting of the integral morphisms in
$\mathrm{lSch}$. If $(X\to S)$ is an object of $\mathrm{Ar}_{\mathrm{lSch}}$,
then there exists a dividing cover $(X^{\prime}\to S^{\prime})$ of $(X\to S)$
with $(X^{\prime}\to S^{\prime})\in\mathrm{iAr}$ by [23, Theorem 1.1].
Proposition 6.8 gives an equivalence

$\mathrm{Shv}_{dNis}(\mathrm{iAr},\mathcal{D}(\mathrm{Ab}))\xrightarrow{\simeq}\mathrm{Shv}_{dNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab})).$

To conclude, we observe that the restriction of $\mathbb{L}\to\mathbb{L}^{O}$
to $\mathrm{iAr}$ is an equivalence owing to [39, Corollary 8.34].

##  8\. Representability of $\operatorname{logHH}$ in logarithmic motivic
homotopy theory

Let us briefly recall the construction of the category of (effective) log
motives over an fs noetherian log scheme of finite Krull dimension
$S\in\mathrm{lSch}$. Let $\Lambda$ be a ring of coefficients and denote by
$\mathcal{D}(\Lambda)$ the $\infty$-category of complexes of
$\Lambda$-modules. We write $\overline{\Box}$ for the fs log scheme
$(\mathbb{P}^{1},\infty)$.

######  Definition 8.1.

(1) A complex
$C\in\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{lSm}/S,\mathcal{D}(\Lambda))$ is
called $\overline{\Box}$-local if for all $X\in\mathrm{lSm}/S$, the map
$R\Gamma_{dNis}(X,C)\to R\Gamma_{dNis}(X\times\overline{\Box},C)$ is an
equivalence in $\mathcal{D}(\Lambda).$

(2) $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S,\Lambda)$ is the full
$\infty$-subcategory of
$\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{lSm}/S,\mathcal{D}(\Lambda))$
consisting of $\overline{\Box}$-local objects. It is equivalent to the
underlying $\infty$-category of the model category of chain complexes of
presheaves of $\Lambda$-modules on $\mathrm{lSm}/S$ with the
$(\overline{\Box},dNis)$-local model structure.

The interested reader can verify that this definition agrees with the one
proposed in [9] or [8, Definition 2.9]. For $\Lambda=\mathbb{Z}$, we simply
write $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$. It is a monoidal, stable,
presentable $\infty$-category. For $X\in\mathrm{lSm}/S$, we denote by
$M_{S}(X)$ or simply by $M(X)$ the image in
$\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$ of the representable presheaf
$\mathbb{Z}(X)$ via the localization functor.

If we denote by $\mathbbm{1}=M(S)$ the $\otimes$-unit of
$\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$, we set

$\mathbbm{1}(1)=\mathrm{cofib}(M(S)\xrightarrow{i_{0}}M(\mathbb{P}_{S}^{1}))[-2],$

where $i_{0}\colon S\to\mathbb{P}^{1}_{S}$ is the $0$-section, and for $n\geq
1$ we write $\mathbbm{1}(n)$ for the $n$-fold tensor product of
$\mathbbm{1}(1)$ with itself. As for usual motives, this is called the $n$-th
Tate twist. If $Y\to X$ is a morphism in $\mathrm{lSm}/S$, we set

$M(X/Y):=\mathrm{cofib}(M(Y)\to M(X)).$

In [9], $\mathrm{log}\mathcal{DA}(S)$ is defined to be the
$\mathbb{P}^{1}$-stabilization

$\mathrm{log}\mathcal{DA}(S):=\mathrm{Stab}_{\mathbb{P}^{1}}(\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S))$

in the sense of [43, Theorem 2.14].

For $X\in\mathrm{lSm}/S$, we keep writing $M(X)$ for the object of
$\mathrm{log}\mathcal{DA}(S)$ representable by $X$.

###  8.2. $\overline{\Box}$-invariance of $\Omega^{d}$ and $\mathbb{L}$

This subsection aims to show that hypercohomology with coefficients in
$\Omega^{d}$ and $\mathbb{L}$ can be computed in terms of mapping spaces in
$\mathrm{log}\mathcal{DA}^{\mathrm{eff}}$. See also [11, §9] for related
results.

######  Proposition 8.3.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$. There is
a canonical quasi-isomorphism

(8.1) $\Omega_{X/S}^{d}\simeq
R_{Zar}p_{*}\Omega_{X\times\overline{\Box}/S}^{d}$

for every integer $d\geq 0$, where
$p\colon\underline{X}\times\mathbb{P}^{1}\to\underline{X}$ denotes the
projection.

###### Proof.

By [37, Proposition IV.1.2.15], there is a canonical isomorphism

$\Omega_{X\times\overline{\Box}/S}^{1}\cong
p^{*}\Omega_{X/S}^{1}\oplus\Omega_{X\times\overline{\Box}/X}^{1}.$

By taking exterior powers we obtain an isomorphism

$\Omega_{X\times\overline{\Box}/S}^{d}\cong\bigoplus_{j=0}^{d}p^{*}\Omega_{X/S}^{j}\otimes\Omega_{X\times\overline{\Box}/X}^{d-j}.$

Since $\Omega_{X/S}^{j}$ is locally free by [37, Proposition IV.3.2.1], the
projection formula gives a quasi-isomorphism

$R_{Zar}p_{*}\Omega_{X\times\overline{\Box}/S}^{d}\simeq\bigoplus_{j=0}^{d}\Omega_{X/S}^{j}\otimes
R_{Zar}p_{*}\Omega_{X\times\overline{\Box}/X}^{d-j}.$

It remains to check that the naturally induced morphism

(8.2) $\Omega_{X/X}^{d}\to R_{Zar}p_{*}\Omega_{X\times\overline{\Box}/X}^{d}$

is a quasi-isomorphism. By [37, Proposition IV.1.2.15], there is a canonical
isomorphism

$\Omega_{X\times\overline{\Box}/X}^{1}\cong\Omega_{\underline{X}\times\overline{\Box}/\underline{X}}^{1}.$

Hence we reduce to the case when $X=\underline{X}$. We may also assume that
$X$ is the spectrum of a local ring $A$. Elementary computations show that
there are isomorphisms

$\displaystyle\Omega_{X\times\overline{\Box}/X}^{1}(X\times\mathbb{A}^{1})\cong
A[t]dt,\quad\Omega_{X\times\overline{\Box}/X}^{1}(X\times(\mathbb{P}^{1}-0))\cong
A[1/t]dt/t,$
$\displaystyle\Omega_{X\times\overline{\Box}/X}^{1}(X\times\mathbb{G}_{m})\cong
A[t,1/t]dt.$

By computing the Čech cohomology of the Zariski cover
$\\{X\times\mathbb{A}^{1},X\times(\mathbb{P}^{1}-0)\to
X\times\mathbb{P}^{1}\\}$, we see that (8.2) is a quasi-isomorphism for $d=1$.
Moreover, $\Omega_{X\times\overline{\Box}/X}^{d}=0$ if $d>1$, so (8.2) is a
quasi-isomorphism for $d>1$. If $d=0$, then (8.2) becomes
$\mathcal{O}_{X}\simeq R_{Zar}p_{*}\mathcal{O}_{X\times\mathbb{P}^{1}}$. ∎

######  Proposition 8.4.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$ and
$d,i\geq 0$. There is a canonical isomorphism

(8.3) $H_{dNis}^{i}(X,\Omega_{X/S}^{d})\cong
H_{dNis}^{i}(X\times\overline{\Box},\Omega_{X\times\overline{\Box}/S}^{d}).$

###### Proof.

Combine (6.2), Corollary 6.17, and Proposition 8.3. ∎

######  Theorem 8.5.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$. Then

$L_{dNis}\Omega^{d}((-)\to S),\quad\mathbb{L}^{dNis}((-)\to
S)\in\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{lSm}/X,\mathcal{D}(\mathrm{Ab}))$

are in $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(X)$.

###### Proof.

Combine Propositions 7.17, 7.24, and 8.4. ∎

###  8.6. $\overline{\Box}$-invariance of $\operatorname{logHH}$

We are now ready to show $\overline{\Box}$-invariance and consequently
representability of log Hochschild homology in the setting of log motives.

From the convergence of the log André-Quillen spectral sequence in Theorem
5.17, we obtain an infinite sequence

(8.4)
$\cdots\to\mathrm{Fil}^{-1}\operatorname{logHH}\to\mathrm{Fil}^{0}\operatorname{logHH}:=\operatorname{logHH}$

in
$\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{D}(\mathrm{Ab}))$
such that

(8.5) $\lim_{q}\mathrm{Fil}^{d}\operatorname{logHH}\simeq 0$

and

(8.6)
$\mathrm{fib}(\mathrm{Fil}^{d-1}\operatorname{logHH}\to\mathrm{Fil}^{d}\operatorname{logHH})\simeq\bigwedge\nolimits^{-d}\mathbb{L}[d]$

for all integers $d$. We may view (8.4) as an infinite sequence in
$\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))$
due to Proposition 7.12.

######  Proposition 8.7.

Suppose $f\colon X\to S$ is a derived log smooth morphism in $\mathrm{lSch}$.
There is a naturally induced equivalence

$\operatorname{logHH}(X/S)\xrightarrow{\simeq}\operatorname{logHH}(X\times\overline{\Box}/S).$

###### Proof.

By restriction, we regard (8.4) as an infinite sequence in
$\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab}))$. As
observed in the proof of Corollary 5.18, we have an equivalence

(8.7)
$\mathrm{fib}(\mathrm{Fil}^{d-1}\operatorname{logHH}\to\mathrm{Fil}^{d}\operatorname{logHH})\simeq
L_{sNis}\Omega^{-d}[d]$

for all integer $d$. Our goal is to show $\operatorname{logHH}(X\to
S)\simeq\operatorname{logHH}(X\times\overline{\Box}\to S)$. By (8.5) and (8.7)
we are done if $L_{sNis}\Omega^{d}(X\to S)\simeq
L_{sNis}\Omega^{d}(X\times\overline{\Box}\to S)$. Propositions 7.17 and 8.3
finish the proof. ∎

######  Proposition 8.8.

Suppose $f\colon X\to S$ is a derived log smooth morphism in $\mathrm{lSch}$
(for example, $f$ is log smooth and integral). Then there is a canonical
equivalence

$\operatorname{logHH}(X/S)\simeq\operatorname{logHH}_{dNis}(X/S).$

###### Proof.

Owing to (7.5), it suffices to show

(8.8)
$\operatorname{logHH}\in\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab})).$

We have the infinite sequences (8.4) in
$\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab}))$. As
observed in the proof of Proposition 7.26, the graded pieces (8.7) are in
$\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab}))$. The
functor

$\mathcal{S}\mathrm{hv}_{sNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab}))\to\mathcal{S}\mathrm{hv}_{dNis}(\mathrm{dlSmAr},\mathcal{D}(\mathrm{Ab}))$

is a right adjoint and commutes with limits. By combining with (8.5), we
deduce (8.8). ∎

######  Proposition 8.9.

Suppose $f\colon X\to S$ is a log smooth morphism in $\mathrm{lSch}$. There is
a naturally induced equivalence

$\operatorname{logHH}_{dNis}^{S}(X)\xrightarrow{\simeq}\operatorname{logHH}_{dNis}^{S}(X\times\overline{\Box}).$

###### Proof.

The question is dividing Nisnevich local on both $X$ and $S$. Hence we may
assume $f$ is integral log smooth by [23, Theorem 1.1]. To conclude we combine
Propositions 8.7 and 8.8. ∎

This immediately implies the following fundamental result.

######  Theorem 8.10.

For $S\in\mathrm{lSch}$ we have

$\operatorname{logHH}_{dNis}(-/S)\in\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S).$

Suppose $S$ has a valuative log structure. Then every morphism $f\colon X\to
S$ of fs log schemes is integral by [37, Proposition I.4.6.3(5)]. Together
with Propositions 4.6(1) and 8.8, we deduce

(8.9)
$\operatorname{logHH}(-/S)\in\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S).$

###  8.11. Orientation of log Hochschild homology

In [9] we develop a notion of oriented cohomology theories for fs log schemes.
The main purpose of this subsection is to show that log Hochschild homology is
oriented. This observation is a crucial ingredient used for establishing the
generalized residue sequence in Theorem 9.5.

######  Proposition 8.12.

Suppose $S\in\mathrm{lSch}$ and $X,Y\in\mathrm{lSch}/S$. There exists a
canonical equivalence

$\operatorname{logHH}(X\times_{S}Y/S)\simeq\operatorname{logHH}(X/S)\otimes_{R\Gamma(S,\mathcal{O}_{\underline{S}})}\operatorname{logHH}(Y/S).$

###### Proof.

The question is strict Nisnevich local on $S$, $X$, and $Y$. Hence we reduce
to the case when $S$, $X$, and $Y$ are affine log schemes. Let $S={\rm
Spec}(R,P)$, $X={\rm Spec}(A,M)$ and $Y={\rm Spec}(B,N)$. By Proposition 5.6
and the corresponding fact for ordinary Hochschild homology (see for example
[13, §6]), it suffices to prove that

${\mathbb{Z}}[B^{\mathrm{rep}}_{P}(M\oplus_{P}N)]\simeq{\mathbb{Z}}[B^{\mathrm{rep}}_{P}(M)]\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[B^{\mathrm{rep}}_{P}(N)].$

This is readily seen using Lemma 5.10 twice. ∎

######  Proposition 8.13.

Suppose $X\to S$ is a morphism in $\mathrm{lSch}$. If
$\bigwedge\nolimits^{d}\mathbb{L}_{X/S}\simeq 0$ for $d\gg 0$, there exists a
second quadrant convergent spectral sequence

$E_{2}^{pq}:=H_{Zar}^{p}(X,\bigwedge\nolimits^{-q}\mathbb{L}_{X/S})\Rightarrow\pi_{-p-q}\operatorname{logHH}(X/S).$

###### Proof.

By restriction, we regard $\operatorname{logHH}(X/S)$ as an object of
$\mathcal{S}\mathrm{hv}_{Zar}(X_{Zar},\mathcal{D}(\mathrm{Ab}))$. From (8.4),
we obtain an infinite sequence

(8.10)
$\cdots\to\mathrm{Fil}^{-1}\operatorname{logHH}(X/S)\to\mathrm{Fil}^{0}\operatorname{logHH}(X/S):=\operatorname{logHH}(X/S)$

in $\mathcal{S}\mathrm{hv}_{Zar}(X_{Zar},\mathcal{D}(\mathrm{Ab}))$ such that

(8.11) $\lim_{d}\mathrm{Fil}^{d}\operatorname{logHH}(X/S)\simeq 0$

and

(8.12)
$\mathrm{fib}(\mathrm{Fil}^{d-1}\operatorname{logHH}(X/S)\to\mathrm{Fil}^{d}\operatorname{logHH}(X/S))\simeq\bigwedge\nolimits^{-d}\mathbb{L}_{X/S}[d]$

for all integers $d$. The vanishing condition on
$\bigwedge\nolimits^{d}\mathbb{L}_{X/S}$ implies that
$\mathrm{Fil}^{d}\operatorname{logHH}(X/S)\simeq 0$ for $d\ll 0$. Using [34,
Proposition 1.2.2.14] we obtain a convergent spectral sequence

$E^{\prime
1}_{pq}:=H_{Zar}^{-2p-q}(X,\bigwedge\nolimits^{-p}\mathbb{L}_{X/S})\Rightarrow\pi_{p+q}\operatorname{logHH}(X/S).$

To obtain the desired spectral sequence, we set $E_{r}^{pq}:=E^{\prime
r-1}_{q,-p-2q}$ for all $r\geq 2$. ∎

######  Proposition 8.14.

Suppose $S$ is an affine log scheme $\operatorname{Spec}(A,M)$. Then
$\operatorname{logHH}(\mathbb{P}_{S}^{n}/S)$ is concentrated in degree $0$,
and there is a canonical isomorphism of $A$-modules

$\pi_{0}(\operatorname{logHH}(\mathbb{P}_{S}^{n}/S))\cong A^{\oplus n+1}.$

###### Proof.

We have the direct computation

$H_{Zar}^{p}(\mathbb{P}_{S}^{n},\Omega_{\mathbb{P}_{S}^{n}}^{q})\cong\left\\{\begin{array}[]{ll}A&\text{if
}0\leq p=q\leq n,\\\ 0&\text{otherwise.}\end{array}\right.$

The convergent spectral sequence

$E_{2}^{pq}:=H_{Zar}^{p}(\mathbb{P}_{S}^{n},\Omega_{\mathbb{P}_{S}^{n}}^{-q})\Rightarrow\pi_{-p-q}\operatorname{logHH}(\mathbb{P}_{S}^{n}/S)$

obtained from Proposition 8.13 finishes the proof. ∎

For topological Hochschild homology of schemes, the following result is a
special case of the projective bundle formula [15, Theorem 1.5].

######  Proposition 8.15.

Suppose $S\in\mathrm{lSch}$ and $X\in\mathrm{lSch}/S$. There is a canonical
equivalence in $\mathcal{D}(\mathrm{Ab})$

$\operatorname{logHH}(X\times\mathbb{P}^{n}/S)\simeq\operatorname{logHH}(X/S)^{\oplus
n+1}.$

###### Proof.

Let $\mathcal{C}$ be the full subcategory of $\mathrm{Ar}_{\mathrm{lSch}}$
consisting of morphisms whose targets are in $\mathrm{lAff}$. As in
Proposition 7.12, there is an equivalence

$\mathrm{Shv}_{sNis}(\mathcal{C},\mathcal{D}(\mathrm{Ab}))\simeq\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab})).$

Hence it suffices to give a natural equivalence

$((X\to
S)\in\mathcal{C}\mapsto\operatorname{logHH}(X\times\mathbb{P}^{n}/S))\xrightarrow{\simeq}((X\to
S)\in\mathcal{C}\mapsto\operatorname{logHH}(X/S)^{\oplus n+1}).$

Due to Proposition 8.12, we reduce to constructing a natural equivalence

$(S\in\mathrm{lAff}\mapsto\operatorname{logHH}(S\times\mathbb{P}^{n}/S))\xrightarrow{\simeq}(S\in\mathrm{lAff}\mapsto\operatorname{logHH}(S/S)^{\oplus
n+1}).$

This follows from Proposition 8.14. ∎

Hence for $S\in\mathrm{Sch}$, we obtain an equivalence in
$\mathrm{log}\mathcal{DA}(S)$

(8.13)
$\operatorname{logHH}(-/S)\simeq\Omega_{\mathbb{P}^{1}}\operatorname{logHH}(-/S),$

where
$\Omega_{\mathbb{P}^{1}}:=\underline{\mathrm{Hom}}(M(\mathbb{P}^{1}/\mathrm{pt}),-)$,
and $\underline{\mathrm{Hom}}$ denotes the internal Hom functor.

######  Definition 8.16.

The logarithmic Hochchild homology of $S\in\mathrm{Sch}$ is given by

$\mathbf{logHH}(-/S):=(\operatorname{logHH}(-/S),\operatorname{logHH}(-/S),\ldots)\in\mathrm{log}\mathcal{DA}(S),$

where the bonding maps are given by (8.13).

For every map $(R,P)\to(A,P)$ of log rings,
$\operatorname{logHH}((A,M)/(R,P))$ is a derived ring. Hence
$\operatorname{logHH}$ is an $\mathbb{E}_{\infty}$-ring in
$\mathrm{Psh}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{D}(\mathrm{Ab}))$. The
localization functor is symmetric monoidal, so $\operatorname{logHH}$ is an
$\mathbb{E}_{\infty}$-ring in

$\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lAff}},\mathcal{D}(\mathrm{Ab}))\simeq\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab})).$

The restriction and localization functors

$\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lSch}},\mathcal{D}(\mathrm{Ab}))\to\mathrm{Shv}_{sNis}(\mathrm{lSm}/S,\mathcal{D}(\mathrm{Ab}))\to\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$

are symmetric monoidal too, so $\operatorname{logHH}$ is an
$\mathbb{E}_{\infty}$-ring in $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$.

By [9, Construction 4.3.9], one can show that $\mathbf{logHH}(-/S)$ is a
homotopy commutative monoid in $\mathrm{log}\mathcal{DA}(S)$, i.e., there are
multiplication and unit maps

$\mathbf{logHH}(-/S)\wedge\mathbf{logHH}(-/S)\to\mathbf{logHH}(-/S)\text{ and
}\mathbbm{1}\to\mathbf{logHH}(-/S)$

satisfying the associative, commutative, and unital axioms in the homotopy
category of $\mathrm{log}\mathcal{DA}(S)$. For all $X\in\mathrm{lSm}/S$, there
is a canonical equivalence

(8.14)
$\mathrm{map}_{\mathrm{log}\mathcal{DA}(S)}(M(X),\mathbf{logHH}(-/S))\simeq\operatorname{logHH}(X/S),$

where $\mathrm{map}_{\mathrm{log}\mathcal{DA}(S)}$ denotes the mapping
spectrum in $\mathrm{log}\mathcal{DA}(S)$, and we regard the right hand side
as a spectrum using the Dold-Kan correspondence in [34, §1.2.4]. Moreover, for
all $X\in\mathrm{lSm}/S$, there are canonical equivalences

(8.15)
$\begin{split}\mathbf{logHH}(-/S)=(\operatorname{logHH}(-/S),\operatorname{logHH}(-/S),\ldots)\simeq&(\Omega_{\mathbb{P}^{1}}\operatorname{logHH}(-/S),\Omega_{\mathbb{P}^{1}}\operatorname{logHH}(-/S),\ldots)\\\
\simeq&\mathbf{logHH}(-/S)(-1)[-2].\end{split}$

We note that (8.15) is analogous to the Bott periodicity witnessed in the
algebraic and complex $K$-theory spectra [54].

The next notion is a direct adaptation of [9, Definition 6.1.1] to
$\mathrm{log}\mathcal{DA}(S)$.

######  Definition 8.17.

Suppose $S\in\mathrm{Sch}$. A homotopy commutative monoid $\mathbb{E}$ in
$\mathrm{log}\mathcal{DA}(S)$ is called _oriented_ if there is a class

$c_{\infty}\in\mathrm{Hom}_{\mathrm{log}\mathcal{DA}(S)}(M(\mathbb{P}_{S}^{\infty}/\mathrm{pt}),\mathbb{E}(1)[2])$

whose restriction to $M(\mathbb{P}_{S}^{1}/\mathrm{pt})$ is given by

$M(\mathbb{P}_{S}^{1}/\mathrm{pt})\otimes\mathbbm{1}\colon
M(\mathbb{P}_{S}^{1}/\mathrm{pt})\to
M(\mathbb{P}_{S}^{1}/\mathrm{pt})\otimes\mathbb{E}\simeq\mathbb{E}(1)[2],$

where $\mathbbm{1}$ denotes the unit of $\mathbb{E}$, and
$M(\mathbb{P}_{S}^{\infty}/\mathrm{pt}):=\mathop{\mathrm{colim}}_{n\to\infty}M(\mathbb{P}_{S}^{n}/\mathrm{pt})$.

######  Theorem 8.18.

The homotopy commutative monoid
$\mathbf{logHH}(-/S)\in\mathrm{log}\mathcal{DA}(S)$ is oriented for all
$S\in\mathrm{Sch}$.

###### Proof.

By Proposition 8.15, there are isomorphisms

$\mathrm{Hom}_{\mathrm{log}\mathcal{DA}(S)}(M(\mathbb{P}_{S}^{\infty}/\mathrm{pt}),\mathbf{logHH}(-/S)(1)[2])\cong\lim_{n\to\infty}{\pi_{0}}\operatorname{logHH}(\mathbb{P}_{S}^{n}/S)\cong\prod_{n=1}^{\infty}\Gamma(S,\mathcal{O}_{S}).$

The element $(1,0,\ldots)$ of the above infinite product gives an orientation
class. ∎

##  9\. Generalized residue sequences for $\operatorname{logHH}$

Let $S=\operatorname{Spec}(A)$ be an affine scheme. In Section 8 we have seen
how

$\operatorname{logHH}(-/S)\in\mathrm{Alg}_{\mathbb{E}_{\infty}}(\mathrm{Shv}_{dNis}(\mathrm{SmAff}_{S},\mathcal{D}(\mathrm{Ab})))$

gives rise to a homotopy commutative monoid object $\mathbf{logHH}(-/S)$ in
$\mathrm{log}\mathcal{DA}(S)$. Combined with the formal properties of the
motivic category $\mathrm{log}\mathcal{DA}(S)$ we will deduce sequences
relating logarithmic and ordinary Hochschild homology. The fundamental
ingredient is the Gysin, or residue, sequence in
$\mathrm{log}\mathcal{DA}(S)$. Let us recall the statement from [11].

######  Definition 9.1 ([11, Definition 7.4.3]).

Suppose $S\in\mathrm{Sch}$, $X\in\mathrm{lSm}/S$, and $\mathcal{E}\to X$ is a
vector bundle (i.e., $\xi\colon\underline{\mathcal{E}}\to\underline{X}$ is a
vector bundle on the underlying scheme $\underline{X}$, and the log structure
on $\mathcal{E}$ is the pullback of the log structure on $X$ via $\xi$). The
_Thom motive of $\mathcal{E}$ over $X$_ is defined to be

$MTh(\mathcal{E}):=M(\mathcal{E}/(\mathrm{Bl}_{Z}\mathcal{E},E))\in\mathrm{log}\mathcal{DA}(S),$

where $Z$ is the zero section of $\mathcal{E}$, and $E$ is the exceptional
divisor.

######  Definition 9.2.

Suppose $S\in\mathrm{Sch}$, $X\in\mathrm{Sm}/S$, $Z_{1}+\cdots+Z_{r}$ is a
strict normal crossing divisor on $X$, and $Z$ is a smooth closed subscheme of
$X$ that is strict normal crossing with $Z_{1}+\cdots+Z_{r}$ in the sense of
[11, Definition 7.2.1]. Let $p\colon N_{Z}X\to Z$ be the projection, where
$N_{Z}X$ denotes the normal bundle of $Z$ in $X$. We set

$N_{Z}(X,Z_{1}+\cdots+Z_{r}):=N_{Z}X\times_{Z}(Z,Z\cap Z_{1}+\cdots+Z\cap
Z_{r}).$

Observe that $N_{Z}(X,Z_{1}+\cdots+Z_{r})$ is a vector bundle over $(Z,Z\cap
Z_{1}+\cdots+Z\cap Z_{r})$.

######  Theorem 9.3.

There exists a canonical cofiber sequence

$M(\mathrm{Bl}_{Z}X,\widetilde{Z}_{1}+\cdots+\widetilde{Z}_{r}+E)\xrightarrow{M(u)}M(X,Z_{1}+\cdots+Z_{r})\to
MTh(N_{Z}(X,Z_{1}+\cdots+Z_{r}))$

in $\mathrm{log}\mathcal{DA}(S)$, where $\widetilde{Z}_{i}$ is the strict
transform of $Z_{i}$ in $\mathrm{Bl}_{Z}X$ for $i=1,\ldots,r$, $E$ is the
exceptional divisor on $\mathrm{Bl}_{Z}X$, and
$u\colon(\mathrm{Bl}_{Z}X,\widetilde{Z}_{1}+\cdots+\widetilde{Z}_{r}+E)\to(X,Z_{1}+\cdots+Z_{r})$
is the obvious morphism.

###### Proof.

We refer to [11, Theorem 7.5.4]. ∎

######  Theorem 9.4.

Suppose $S\in\mathrm{Sch}$ and $\mathbb{E}$ is an oriented homotopy
commutative monoid in $\mathrm{log}\mathcal{DA}(S)$. For $X\in\mathrm{lSm}/S$,
a rank $d$ vector bundle $\mathcal{E}\to X$, and integers $p$ and $q$, there
is a canonical equivalence

(9.1)
$\mathrm{map}_{\mathrm{log}\mathcal{DA}(S)}(M(X),\mathbb{E}(q)[p])\simeq\mathrm{map}_{\mathrm{log}\mathcal{DA}(S)}(MTh(\mathcal{E}),\mathbb{E}(q+d)[p+2d]).$

###### Proof.

When $S$ is the spectrum of a perfect field $k$ with resolution of
singularities, and $\mathrm{log}\mathcal{DA}(S)$ is replaced by the version
with transfers $\mathrm{log}\mathcal{DM}(k)$, this is a consequence of [11,
Theorem 8.3.7]. Indeed, there is an isomorphism $MTh(\mathcal{E})\simeq
M(X)(d)[2d]$, and an orientation on $\mathbb{E}$ is not needed. The
equivalence (9.1) holds in greater generality: our proof of [9, Proposition
6.3.9] adapts to $\mathrm{log}\mathcal{DA}(S)$ under the assumption that
$\mathbb{E}$ is oriented. ∎

######  Theorem 9.5 (Generalized residue sequence).

With the notations of Definition 9.2 and Theorem 9.3, there exists a canonical
cofiber sequence

$\displaystyle\operatorname{logHH}((Z,Z\cap Z_{1}+\cdots+Z\cap Z_{r})/S)\to$
$\displaystyle\operatorname{logHH}((X,Z_{1}+\cdots+Z_{r})/S)$
$\displaystyle\xrightarrow{u^{*}}$
$\displaystyle\operatorname{logHH}((\mathrm{Bl}_{Z}X,\widetilde{Z}_{1}+\cdots+\widetilde{Z}_{r}+E)/S).$

The above cofiber sequence is called the _Gysin (or residue) sequence for
$\operatorname{logHH}$._

As a particular case, we obtain a canonical cofiber sequence

(9.2)
$\operatorname{HH}(Z/S)\to\operatorname{HH}(X/S)\to\operatorname{logHH}((\mathrm{Bl}_{Z}X,E)/S).$

###### Proof.

Apply Theorem 9.4 to the oriented homotopy commutative monoid
$\mathbf{logHH}(-/S)$ and use (8.14), (8.15), and Theorem 9.3. ∎

The interested reader can note that (9.2) is similar to the generalized Gysin
sequence available for the cohomology of reciprocity sheaves in [12, Theorem
7.16].

###  9.6. Comparison of cofiber sequences

Let $A$ be a discrete commutative ring and let $a\in A$ be a non-zero divisor.
By [45, Theorem 5.5, Example 5.7], there is a cofiber sequence

$\operatorname{THH}(A)\to\operatorname{THH}(A,\langle
a\rangle)\to\operatorname{THH}(A/(a))[1],$

where $\langle a\rangle\to(A,\cdot)$ is the pre-log structure generated by
$a$. Here $\operatorname{THH}(A,\langle a\rangle)$ denotes log topological
Hochschild homology in the sense of [45]. By cobase change along the
augmentation $\operatorname{THH}({\mathbb{Z}})\to{\mathbb{Z}}$, Proposition
5.6 and [31, Lemma 5.4] yield the cofiber sequence

(9.3) ${\rm HH}(A)\to{\rm logHH}(A,\langle a\rangle)\to{\rm HH}(A/(a))[1].$

We point out that the cofiber sequence (9.3) is a special case of Theorem 9.5.
Indeed, set $S={\rm Spec}({\mathbb{Z}})$ (with trivial log structure), $X={\rm
Spec}({\mathbb{Z}}[x])$ and $Z={\rm Spec}({\mathbb{Z}})$ in (9.2), so that the
resulting cofiber sequence reads

${\rm HH}({\mathbb{Z}}[x])\to{\rm logHH}({\mathbb{Z}}[x],\langle
x\rangle\times{\rm GL}_{1}({\mathbb{Z}}[x]))\to{\rm HH}({\mathbb{Z}})[1].$

Now we cobase-change along ${\rm HH}({\mathbb{Z}}[x])\to{\rm HH}(A)$ and apply
logification invariance of ${\rm logHH}$ to obtain a cofiber sequence of the
form (9.3). Logification invariance of ${\rm logHH}$ can be seen using cobase
change along $\operatorname{THH}({\mathbb{Z}})\to{\mathbb{Z}}$ and a similar
equivalence in log $\operatorname{THH}$, see [45, Theorem 4.24], or by using
our Quillen spectral sequence and the analogous statement for the log
cotangent complex in [39, Theorem 8.16].

When $a\in A$ is a non-zero divisor, the subscheme
$Z=\operatorname{Spec}(A/(a))$ is an effective Cartier divisor in $X$, and
clearly $\mathrm{Bl}_{Z}X=X$, i.e., the blow-up is irrelevant. It is then
clear by construction that the sequences (9.2) and (9.3) agree.

######  Remark 9.7.

Theorem 9.5 extends the results in [45] in two directions. First, it works for
log schemes (and not just for log rings), and the log structure is allowed to
be non-trivial in all the terms of the sequence. Second, it allows for a
higher-dimensional generalization of the residue sequence. Allowing for non-
affine log schemes is essential to that end.

To explain it better, let us consider the following situation. Let $R$ be a
noetherian commutative ring of finite Krull dimension, and let $A$ be a smooth
$R$-algebra. Let $f_{1},\ldots,f_{r}$ be a regular sequence in $A$, such that
the quotient $A/(f_{1},\ldots,f_{r})$ is a smooth complete intersection
$R$-algebra. Then, since the normal bundle of
$Z:=\operatorname{Spec}(A/(f_{1},\ldots,f_{r}))$ in
$X:=\operatorname{Spec}(A)$ is trivial, (9.2) reads as

$\operatorname{HH}(A/R)\to\operatorname{logHH}((\mathrm{Bl}_{(f_{1},\ldots,f_{r})}\operatorname{Spec}(A),E)/R)\to\operatorname{HH}((A/(f_{1},\ldots,f_{r}))/R)[1],$

where $E\cong\mathbb{P}(N_{Z}X)\cong\mathbb{P}_{Z}^{r-1}$. For $r=1$, using
the HKR Theorem, we obtain the classical residue sequence (see e.g., [18,
2.3.a])

$0\to\Omega^{n}_{A/R}\to\Omega^{n}_{A/R}(\log
f)\xrightarrow{Res}\Omega^{n-1}_{(A/f)/R}\to 0.$

Note that the map
$\pi_{n}\operatorname{HH}((A/f_{1})/R)\to\pi_{n}\operatorname{HH}(A/R)$ is
trivial for all $n\geq 0$ in this case since the differentials inject into the
log differentials.

We end with a discussion about residue sequences in log $\operatorname{THH}$,
and how this could potentially relate to the results of this paper:

######  Remark 9.8.

Quillen’s localization theorem [41, §5] produces a cofiber sequence

$K(\mathbb{Z}_{p})\to K(\mathbb{Q}_{p})\to K(\mathbb{F}_{p})[1]$

in algebraic $K$-theory, while Blumberg–Mandell [14, Localization Theorem]
establish a cofiber sequence

$K({\rm ku}_{p})\to K({\rm KU}_{p})\to K(\mathbb{Z}_{p})[1].$

Here ${\rm ku}_{p}$ and ${\rm KU}_{p}$ are the connective and periodic complex
($p$-complete) $K$-theory spectra. These examples answer the following
question in the affirmative for $n=0,1$: Is there a cofiber sequence

(9.4) $K(BP\langle n\rangle)\to K(BP\langle n\rangle[v_{n}^{-1}])\to
K(BP\langle n-1\rangle)[1]$

for the truncated Brown–Peterson spectrum $BP\langle n\rangle$? For $n\geq 2$,
such cofiber sequences were proven to _not_ exist in [1].

The search for cofiber sequences in log $\operatorname{THH}$ is often
motivated by the problem of finding close approximations of cofiber sequences
in algebraic $K$-theory that do not exist in ordinary Hochschild homology:
Such a perspective is appealing for the examples of [45].

We sketch a possible construction of a cofiber sequence

(9.5) ${\rm THH}(BP\langle n\rangle)\to{\rm logTHH}(BP\langle n\rangle,\langle
v_{n}\rangle_{*})\to{\rm THH}(BP\langle n-1\rangle)[1].$

This is a cofiber sequence in log ${\rm THH}$ involving $BP\langle n\rangle$
for which there is no corresponding $K$-theory sequence (9.4) when $n\geq 2$.
The truncated Brown–Peterson spectra $BP\langle n\rangle$ do in general not
admit the structure of ${\mathbb{E}}_{\infty}$-rings [28, Theorem 1.1.2], and
hence they do not fall under the framework of logarithmic ${\rm THH}$ as
developed by Rognes–Sagave–Schlichtkrull [45], [46]. We shall combine
groundbreaking work of Hahn–Wilson [20] with a construction of
Sagave–Schlichtkrull [48] to give an _ad hoc_ definition of ${\rm
logTHH}(BP\langle n\rangle,\langle v_{n}\rangle_{*})$ participating in cofiber
sequences (9.5).

We start with the cofiber sequence

${\rm THH}({\rm MUP}_{\geq 0})\to{\rm logTHH}({\rm MUP}_{\geq 0},V^{\cal
U}_{\geq 0})\to{\rm THH}({\rm MU})[1]$

of Sagave–Schlichtkrull [48, Example 8.6]. Here ${\rm MUP}_{\geq
0}=\oplus_{i\geq 0}{\rm MU}[2i]$ is the positively weighted part of the
periodic complex cobordism spectrum ${\rm MUP}$, equipped with a log structure
$V^{\cal U}_{\geq 0}$ localizing to ${\rm MUP}$. Subobjects of ${\rm
MUP}_{\geq 0}$ whose weights are multiples of $p^{n}-1$ are denoted by ${\rm
MU}[y]$ in Hahn–Wilson [20, §2.1]. We think of ${\rm MU}[y]$ as a polynomial
ring over ${\rm MU}$, and Sagave–Schlichtkrull’s argument still applies to
produce a cofiber sequence

(9.6) ${\rm THH}({\rm MU}[y])\to{\rm logTHH}({\rm MU}[y],\langle
y\rangle_{*})\to{\rm THH}({\rm MU})[1].$

As part of their construction of $\mathbb{E}_{3}$-${\rm MU}$-algebra
structures on $BP\langle n\rangle$, Hahn–Wilson [20] construct
$\mathbb{E}_{3}$-${\rm MU}[y]$-algebra structures on $BP\langle n\rangle$,
where $y$ acts by $v_{n}$. Applying a cobase change of (9.6) along the
resulting map ${\rm THH}({\rm MU}[y])\to{\rm THH}(BP\langle n\rangle)$ gives
(9.5), where ${\rm logTHH}(BP\langle n\rangle,\langle v_{n}\rangle_{*})$ is
defined as the cobase change of ${\rm logTHH}({\rm MU}[y],\langle
y\rangle_{*})$ along ${\rm THH}({\rm MU}[y])\to{\rm THH}(BP\langle n\rangle)$.
A version of this sequence will appear in the forthcoming work of
Ausoni–Bayındır–Moulinos. This further accentuates the difference between the
behavior of cofiber sequences in log ${\rm THH}$ and localization sequences in
algebraic $K$-theory.

One may attempt to define log structures on $BP\langle n\rangle$ generated by
all the $v_{i}$’s in the hope of obtaining a sequence in ${\rm logTHH}$
analogous to the $K$-theory sequence of Barwick–Lawson [4, Corollary 5.2]. By
[20, Remark 2.1.2], the approach sketched here does not apply to produce such
cofiber sequences in ${\rm logTHH}$. However, Theorem 9.5 suggests one way to
approach this problem: Replace $BP\langle n\rangle$ with an appropriate blow-
up in the context of spectral algebraic geometry and prove that it
participates in a ${\rm logTHH}$ residue sequence. In upcoming work, we hope
to address this question, as well as how the resulting sequence relates to
variants of algebraic $K$-theory.

Acknowledgements. The authors thank Bjørn Dundas, Elden Elmanto, Lars
Hesselholt, Teruhisa Koshikawa, Mauro Porta, John Rognes and Steffen Sagave
for conversations and comments on the subject of this paper.

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