# A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic
Hochschild homology

Federico Binda  Department of Mathematics “F. Enriques”, University of Milan,
Italy<EMAIL_ADDRESS>,  Tommy Lundemo  Department of Mathematics and
Informatics, University of Wuppertal, Germany<EMAIL_ADDRESS>,
Doosung Park  Department of Mathematics and Informatics, University of
Wuppertal, Germany<EMAIL_ADDRESS>and  Paul Arne Østvær  Department
of Mathematics “F. Enriques”, University of Milan, Italy & Department of
Mathematics, University of Oslo, Norway<EMAIL_ADDRESS>&
<EMAIL_ADDRESS>
###### Abstract.

This paper incorporates the theory of Hochschild homology into our program on
log motives. We discuss a geometric definition of logarithmic Hochschild
homology of derived pre-log rings and construct an André-Quillen type spectral
sequence. The latter degenerates for derived log smooth maps between discrete
pre-log rings. We employ this to show a logarithmic version of the Hochschild-
Kostant-Rosenberg theorem and that logarithmic Hochschild homology is
representable in the category of log motives. Among the applications, we
deduce a generalized residue sequence involving blow-ups of log schemes.

###### Key words and phrases:

Logarithmic Hochschild homology, dividing covers

###### 2020 Mathematics Subject Classification:

Primary: 13D03; Secondary: 14A21, 14F42, 19D55

The authors gratefully acknowledge the hospitality and support of the Centre
for Advanced Study at the Norwegian Academy of Science and Letters in Oslo,
Norway, which funded and hosted the research project “Motivic Geometry” during
the 2020/21 academic year, and the RCN Frontier Research Group Project no.
250399 “Motivic Hopf Equations” and no. 312472 “Equations in Motivic
Homotopy.” Binda would like to thank the Isaac Newton Institute for
Mathematical Sciences for support and hospitality during the program
“$K$-theory, algebraic cycles and motivic homotopy theory” in 2022 when part
of the work on this paper was undertaken, partially supported by the EPSRC
Grant EP/R014604/1 (UK) and by the PRIN “Geometric, Algebraic and Analytic
Methods in Arithmetic” at MUR (Italy). Lundemo was partially supported by the
NWO-grant 613.009.121. Park was partially supported by the research training
group GRK 2240 “Algebro-Geometric Methods in Algebra, Arithmetic and
Topology.” Østvær was partially supported by a Guest Professorship awarded by
The Radboud Excellence Initiative

###### Contents

  1. 1 Introduction
  2. 2 Affine derived logarithmic geometry
  3. 3 Log differentials and the log cotangent complex
  4. 4 Étale and smooth morphisms in derived log geometry
  5. 5 Logarithmic Hochschild homology of derived pre-log rings
  6. 6 A recollection of the dividing Nisnevich topology
  7. 7 Sheaves on arrow categories
  8. 8 Representability of $\operatorname{logHH}$ in logarithmic motivic homotopy theory
  9. 9 Generalized residue sequences for $\operatorname{logHH}$

##  1\. Introduction

Let $R$ be a commutative ring. If $A$ is a commutative $R$-algebra, the
Hochschild homology $\operatorname{HH}(A/R)$ of $A$ relative to $R$ admits a
filtration whose graded pieces are the derived exterior powers
$(\bigwedge\nolimits_{A}^{i}{\mathbb{L}}_{A/R})[i]$ of the cotangent complex
${\mathbb{L}}_{A/R}$ [36, Proposition IV.4.1]. If $A$ is smooth over $R$, this
gives an isomorphism of graded commutative rings

(1.1) $\Omega^{*}_{A/R}\xrightarrow{\cong}\operatorname{HH}_{*}(A/R).$

The isomorphism in (1.1) is commonly referred to as the
Hochschild–Kostant–Rosenberg (=HKR) theorem after [22]. The _HKR-filtration_
alluded to above is a motivic filtration in the sense of Bhatt–Morrow–Scholze
[7] (see also Mathew [35] and Raksit [42]), and it is a key input in many
classical $K$-theoretic computations involving trace methods.

One of the main differences between algebraic $K$-theory and Hochschild
homology is that the latter (as well as its topological counterpart) does not
enjoy a satisfactory localization property. For example, if $A$ is a discrete
valuation ring with finite residue field $k$ and fraction field $F$, a
classical result of Quillen [41] implies the existence of a cofiber sequence

$K(A)\to K(F)\to\Sigma K(k).$

Such a sequence does not exist if we replace $K$-theory with
$\operatorname{HH}$ or $\operatorname{THH}$, and the trace map from $K(F)$ to
$\operatorname{HH}(F)$ or to $\operatorname{THH}(F)$ carries much less
information than the one for $A$ or $k$. One of the initial motivations for
studying Hochschild homology of log rings, as introduced by Rognes [44], is
that it allows to fix part of this problem, since it participates in cofiber
sequences that are unavailable for ordinary Hochschild homology. For example,
there is a cofiber sequence

(1.2) $\operatorname{HH}(A)\to{\rm
logHH}(A,A-\\{0\\})\to\operatorname{HH}(k)[1]$

relating the Hochschild homology of $A$, its log Hochschild homology (for the
usual log structure) and the Hochschild homology of $k$, see e.g., [45,
Example 5.7].

With the goal of better understanding the term $\operatorname{logHH}$ and
studying its relationship with the motivic theory developed in [9], [10],
[11], in this paper we are interested in generalizing the HKR-theorem and
several of its surrounding statements to the context of log geometry.

Our first main result is a generalization of the HKR-theorem to pre-log rings.
We state the result in the context of derived pre-log rings: These are modeled
by simplicial pre-log rings in Sagave–Schürg–Vezzosi’s work on derived log
geometry [49].

######  Theorem 1.1 (Theorem 5.17).

For a morphism of derived pre-log rings $(R,P)\to(A,M)$, the log Hochschild
homology $\operatorname{logHH}((A,M)/(R,P))$ admits a descending separated
filtration with graded pieces the derived exterior powers
$(\bigwedge\nolimits_{A}^{i}{\mathbb{L}}_{(A,M)/(R,P)})[i]$ of Gabber’s
cotangent complex ${\mathbb{L}}_{(A,M)/(R,P)}$. If $(A,M)$ is discrete and
_derived log smooth_ over a discrete $(R,P)$-algebra, there is an isomorphism
of graded commutative rings

$\Omega^{*}_{(A,M)/(R,P)}\xrightarrow{\cong}\operatorname{logHH}_{*}((A,M)/(R,P))$

relating the log de Rham complex and log Hochschild homology.

Derived log smooth maps $(R,P)\to(A,M)$ of discrete pre-log rings are log
smooth in the sense of classical log geometry [24, §3], see [49, Theorem 6.4].
We will proceed to explain the meaning and necessity of the additional
adjective _derived_ in the statement of the logarithmic HKR-theorem, after
which we state generalizations and applications of Theorem 1.1.

######  Remark 1.2.

In the context of discrete pre-log rings, the log Hochschild homology in the
statement of Theorem 1.1 is equivalent to Rognes’ definition [44], see
Proposition 1.4, and also used by Krause–Nikolaus [27]. See also Leip [29,
Page 886] for a discussion of this definition of log ${\rm THH}$. The above
should not be confused with Hesselholt–Madsen’s [21] “relative term”
$\operatorname{THH}(A|K)$ defined in terms of Waldhausen categories. Olsson
[38] has proposed another definition involving his stack of log structures. As
we explain in Section 1.14, we hope the results herein yield a comparison
result relating Olsson’s definition of logarithmic Hochschild homology with
the one in this paper.

###  1.3. The logarithmic André–Quillen spectral sequence

One way to prove the HKR-theorem for ordinary Hochschild homology is to
exhibit $\operatorname{HH}_{*}(A/R)$ as the abutment of an _André–Quillen
spectral sequence_. Indeed, one readily shows that the HKR-theorem holds for
$A=R[x]$, from which it follows that it holds for any free commutative
$R$-algebra. Using this, results of Quillen [40, Theorem 8.1] yield a strongly
convergent spectral sequence

(1.3)
$E^{2}_{p,q}=\pi_{p}(\bigwedge\nolimits_{A}^{q}{\mathbb{L}}_{A/R})\implies\pi_{p+q}(A\otimes_{A\otimes_{R}^{\mathbb{L}}A}^{\mathbb{L}}A)$

relating the derived wedge powers of ${\mathbb{L}}_{A/R}$ and the derived
self-intersections of the (derived) “diagonal embedding”
$A\otimes_{R}^{\mathbb{L}}A\to A$. If $R\to A$ is smooth, then
${\mathbb{L}}_{A/R}\simeq\Omega^{1}_{A/R}[0]$. Since the derived tensor
product in (1.3) computes $\operatorname{HH}_{*}(A/R)$, this finishes the
proof.

To apply this proof strategy in the context of derived log rings, we define
log Hochschild homology ${\rm logHH}((A,M)/(R,P))$ to be the derived self-
intersections of a derived version of Kato–Saito’s log diagonal embedding [25,
§4]. We refer to Definition 5.3 for details of this construction. The fact
that our geometric description agrees with the direct generalization to
derived log rings of Rognes’ notion of log $\operatorname{HH}$ for discrete
log rings [44, Definition 3.21] is the content of the following result.

######  Proposition 1.4 (Proposition 5.6).

There is a natural equivalence between Rognes’ log Hochschild homology and
${\rm logHH}((A,M)/(R,P))$.

Our definition of log $\operatorname{HH}$ is inspired by the observation that
in the underived context, the conormal bundle of Kato–Saito’s log diagonal
coincides with the module of log Kähler differentials
$\Omega^{1}_{(A,M)/(R,P)}$ [25, Corollary 4.2.8(ii)]. Passing to the derived
context begets Gabber’s log cotangent complex ${\mathbb{L}}_{(A,M)/(R,P)}$
essentially by definition, see Proposition 3.17. Combining this observation
with Proposition 1.4, we can apply Quillen’s construction once again to obtain
a strongly convergent spectral sequence

(1.4)
$E^{2}_{p,q}=\pi_{p}(\bigwedge\nolimits_{A}^{q}{\mathbb{L}}_{(A,M)/(R,P)})\implies\pi_{p+q}\operatorname{logHH}((A,M)/(R,P))$

relating the derived wedge powers of Gabber’s log cotangent complex with log
Hochschild homology. This is the point of the proof where the adjective
“derived” in the statement of Theorem 1.1 comes into play: In general,
Gabber’s log cotangent complex is _not_ the log differentials concentrated in
degree zero for log smooth maps, see Example 4.9. Thus the log André–Quillen
spectral sequence generally fails to collapse for log smooth maps. For this
reason, it is necessary to impose additional flatness assumptions on the log
smooth map $(R,P)\to(A,M)$: It suffices to require that the map
${\mathbb{Z}}[P]\to{\mathbb{Z}}[M]$ is flat, a condition that holds for any
integral log smooth map.

###  1.5. Étale base change for log Hochschild homology

Using the André-Quillen spectral sequence, Weibel–Geller [56] proved the
canonical equivalence

$B\otimes_{A}\operatorname{HH}(A/R)\xrightarrow{\simeq}\operatorname{HH}(B/R)$

for étale maps $A\to B$ of commutative $R$-algebras. Their result follows from
the transitivity sequence for the cotangent complex combined with the
vanishing of the relative cotangent complex for étale morphisms. As Gabber’s
log cotangent complex enjoys a similar transitivity property, the spectral
sequence (1.4) yields:

######  Theorem 1.6 (Theorem 5.24).

Let $(R,P)$ be a derived pre-log ring. A map $(A,M)\to(B,N)$ of
$(R,P)$-algebras is derived log étale if and only if the canonical map

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

is an equivalence.

Theorem 1.6 applies for example to tamely ramified extensions of discrete
valuation rings (for which the classical result doesn’t hold). The adjective
“derived” is necessary here for the same reason as in Theorem 1.1: There exist
log étale maps for which Gabber’s log cotangent complex does not vanish, see
again Example 4.9.

######  Remark 1.7.

In this work we use Gabber’s cotangent complex for log schemes. Olsson has
proposed another definition [39]. Each has its own advantages and
disadvantages, and there is no choice of log cotangent complex satisfying all
desirable properties [39, §7]. The main reasons for our choice are the
following: 1) Gabber’s cotangent complex enjoys the same kind of transitivity
property that the classical cotangent complex has, and 2) Gabber’s cotangent
complex admits a straightforward adaptation to derived log rings.

###  1.8. Extension to fs log schemes

Hochschild homology of rings extends to schemes owing to the étale base change
theorem [56]. Likewise, log Hochschild homology of log rings extends to log
schemes by appeal to Theorem 1.6.

Several Grothendieck topologies are available in the category of log schemes.
We can broadly divide them into two classes: the _strict_ topologies, where
the coverings are defined on the underlying schemes, and the log structures
are simply pulled back; and the _log_ topologies, where a non-trivial
modification of the log structure happens along the covering. Among these, the
_dividing Nisnevich_ topology, which we review in Section 6, plays a crucial
role for the logarithmic motivic homotopy theory in [9], [10], [11].

While log Hochschild homology satisfies strict étale descent (in fact, it
satisfies _integral_ étale descent) thanks to Theorem 1.6, it does not satisfy
dividing Nisnevich descent; see Example 7.27. To remedy this, we need to
consider the dividing Nisnevich localization of log Hochschild homology, which
we now briefly discuss. In this process, we fix a base fs log scheme $S$ and
define $\operatorname{logHH}(X/S)$ for every morphism of fs log schemes $X\to
S$. The precise construction carried out in Sections 6 and 7 involves arrow
categories, and produces two sheaves of complexes, $\operatorname{logHH}(-/-)$
in $\mathrm{Shv}_{sNis}(\mathrm{Ar}_{\mathrm{lAff}})$ and
$\operatorname{logHH}^{-}_{dNis}(-)$ in
$\mathrm{Shv}_{dNis}(\mathrm{Ar}_{\mathrm{lAff}})$. The dividing Nisnevich
sheafification process is well-controlled: in fact, if $X\to S$ is integral
log smooth, Proposition 8.8 shows that the localization map
$\operatorname{logHH}(X/S)\rightarrow\operatorname{logHH}_{dNis}(X/S)$ is an
equivalence. Along the way, we obtain similar results for the dividing
Nisnevich localized log cotangent complex $\mathbb{L}^{dNis}$.

###  1.9. Representability of log Hochschild homology

We refer to [10] for an overview of log motives explaining our choice
$\overline{\Box}=(\mathbb{P}^{1},\infty)$ of the unit interval. For an fs log
scheme $S$, the $\infty$-category $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$
of effective motives without transfers is defined as the
$\overline{\Box}$-localization of the $\infty$-category of dividing Nisnevich
sheaves of chain complexes of abelian groups on $\mathrm{lSm}/S$, see also
[9]. Here $\mathrm{lSm}/S$ denotes the category of fs log schemes log smooth
over $S$. We claim that the dividing Nisnevich sheaf
$\operatorname{logHH}_{dNis}(-/S)$ is $\overline{\Box}$-invariant and hence
representable in $\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)$. Owing to the
logarithmic André–Quillen spectral sequence (1.4), it suffices to show that
there is a canonical equivalence

$\mathbb{L}^{dNis}(X\times\overline{\Box}\to S)\simeq\mathbb{L}^{dNis}(X\to
S)$

for all $X\in\mathrm{lSm}/S$. Proposition 7.26 yields a canonical equivalence

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

where $\underline{X}$ denotes the underlying scheme of $X$. Finally, by an
explicit calculation, we have

$R\Gamma_{Nis}(\underline{X},\Omega_{X/S}^{1})\simeq
R\Gamma_{Nis}(\underline{X\times\overline{\Box}},\Omega_{X\times\overline{\Box}/S}^{1}).$

This establishes a basic result relating logarithmic Hochschild homology and
log motives. We write $\mathrm{lSch}$ for the category of noetherian fs log
schemes of finite Krull dimensions and finite type morphisms.

######  Theorem 1.10 (Theorem 8.10).

Suppose that $S$ is a noetherian fs log scheme of finite Krull dimension. Then
we have

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

In particular, for every $X\in\mathrm{lSm}/S$, we have

$\operatorname{Map}_{\mathrm{log}\mathcal{DA}^{\mathrm{eff}}(S)}(M_{S}(X),\operatorname{logHH}_{dNis}(-/S))=\operatorname{logHH}_{dNis}(X/S).$

Note that if $S$ has a valuative log structure (for example, $S$ is the
spectrum of a DVR with canonical log structure, or $S$ has trivial log
structure), then every morphism $f\colon X\to S$ is integral, and thus the
dividing Nisnevich sheafification is redundant. See Section 8 for more
details.

###  1.11. Residue sequences in log Hochschild homology

Cofiber sequences for log Hochschild homology as in (1.2) have to this point
only been available for log structures generated by a single element. Theorem
1.10 allows for the results of [11] to be applied in the context of log
Hochschild homology. As we discuss in Section 9, this perspective shows that
the cofiber sequences in log $\operatorname{HH}$ should be thought of as
generalizations of residue sequences in log geometry, as opposed to
approximations of localization sequences in algebraic $K$-theory. For this
reason, it seems the term “ _residue sequence_ ” is more appropriate than “
_localization sequence_ ” in the context of cofiber sequences in log
(topological) Hochschild homology of [45].

Moreover, the representability result Theorem 1.10 allows us to apply the
results of [11] to generalize the residue sequence in log Hochschild homology
in the following way:

######  Theorem 1.12 (Theorem 9.5).

Fix a noetherian base $S$ of finite Krull dimension. If $X$ is a smooth
$S$-scheme and $Z\subset X$ is a smooth closed subscheme, then there is a
cofiber sequence

$\operatorname{HH}(X/S)\to\operatorname{logHH}((\mathrm{Bl}_{Z}X,E)/S)\to\operatorname{HH}(Z/S)[1].$

In fact, a more general result holds, when a non-trivial log structure on both
$X$ and $Z$ is allowed. See Theorem 9.5 for the exact statement. As we explain
in Remark 9.7, one special case of Theorem 1.12 is the cofiber sequence

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

for a regular sequence $(f_{1},\dots,f_{r})$ cutting a smooth subscheme in a
smooth commutative $R$-algebra $A$, where $R$ is a commutative ring of finite
Krull dimension. This recovers the discrete version of the
Rognes–Sagave–Schlichtkrull sequence for $r=1$. In other words, in order to
have cofiber sequences in log Hochschild homology for log structures generated
by several elements, we must leave the affine context behind and invoke blow-
ups.

In Remark 9.8 we discuss the existence of residue sequences in (log)
topological Hochschild homology

(1.5) ${\rm THH}(BP\langle n\rangle)\to{\rm logTHH}(BP\langle n\rangle)\to{\rm
THH}(BP\langle n-1\rangle)[1]$

involving the truncated Brown–Peterson spectrum $BP\langle n\rangle$ for which
there is provably no analogous localization sequence in algebraic $K$-theory.
In a slightly different setup, a version of this sequence will appear in the
forthcoming work of Ausoni–Bayındır–Moulinos. Using an appropriate notion of
blow-ups in spectral algebraic geometry, we suspect that Theorem 1.12 admits a
generalization to log topological Hochschild homology, and that this will give
rise to the generalization of the sequence (1.5). We refer to Remark 9.8 for
further discussion in this direction.

###  1.13. Analogous results for log ring spectra

Some of the results in this paper are algebraic analogues of results proved in
the context of log ring spectra in [31], and these results (appearing in
Sections $\S 2$ to $\S 5$) also appear in the second-named author’s thesis
[32]. At its time of writing, the second-named author was unaware of the
algebro-geometric log diagonal interpretation of these results. Versions of
Proposition 1.4 and Theorem 1.6 hold for log _topological_ Hochschild homology
of discrete pre-log rings in the sense of Krause–Nikolaus [27] and
Rognes–Sagave–Schlichtkrull [45], [46].

###  1.14. Upcoming work

The relationship between the circle action on log $\operatorname{HH}$ and the
log de Rham differential is not treated in the present paper. We intend to
prove a logarithmic analogue of the main result of [52], identifying log
Hochschild homology with derived log de Rham theory in characteristic zero.
Olsson [38] has proposed another definition of log Hochschild homology using
his approach to log geometry via algebraic stacks. We expect a comparison
theorem relating these definitions of log Hochschild homology, analogous to
[39, §8.31] for log cotangent complexes. We hope to return to these questions,
in addition to those raised in Section 1.11 and Remark 9.8.

###  1.15. Notation and conventions

We refer to Ogus [37] for standard conventions on monoids, (pre)-log rings and
schemes. If $A$ is a discrete commutative ring and $M$ is a discrete
commutative monoid, the triple $(A,M,\alpha)$ defines a pre-log ring if
$\alpha\colon M\to A$ is a homomorphism of monoids, where multiplication gives
the monoid operation on $A$. If no confusion arises, we will drop $\alpha$
from the notation and simply write $(A,M)$. A morphism of pre-log rings
$(f,f^{\flat})\colon(A,M)\to(B,N)$ is a pair of morphisms $f\colon A\to B$,
$f^{\flat}\colon M\to N$ such that the obvious square commutes. Since our
constructions are all implicitly derived, we will omit the symbol $\mathbb{L}$
on top of the tensor product in the main body of the text.

For a category $\mathcal{C}$, we employ the following notations.

$\mathrm{Sch}$ | noetherian schemes of finite Krull dimensions
---|---
$\mathrm{lSch}$ | noetherian fs log schemes of finite Krull dimensions
$\mathrm{Psh}(\mathcal{C})$ | presheaves on $\mathcal{C}$
$\mathrm{Shv}_{t}(\mathcal{C})$ |  $t$-sheaves on $\mathcal{C}$ with respect to the topology $t$
$\mathrm{sPsh}(\mathcal{C})$ | simplicial presheaves on $\mathcal{C}$
$\mathrm{sShv}_{t}(\mathcal{C})$ | simplicial $t$-sheaves on $\mathcal{C}$ with respect to the topology $t$
$\mathrm{Coh}(X)$ | coherent sheaves on $X$
$\mathrm{D}^{b}(\mathcal{A})$ | bounded derived category of an abelian category $\mathcal{A}$

##  2\. Affine derived logarithmic geometry

The definition of Hochschild homology, even in the classical case, is
inherently derived and better understood in the setting of derived algebraic
geometry. In order to pursue this point of view in the logarithmic context,
let us recall from [49, §2, 3] the notion of derived (pre-)log rings.

###  2.1. Logarithmic morphisms

We begin with the following basic definition.

######  Definition 2.2.

We denote by $\mathrm{sCMon}$ the category of simplicial commutative monoids.
By elementary model category techniques (see e.g. [49, Proposition 2.1]), this
admits a proper simplicial combinatorial model structure in which a map is a
fibration or a weak equivalence precisely when the underlying map of
simplicial sets is so in the (standard) Kan–Quillen model structure. This is
referred to as the _standard_ model structure on $\mathrm{sCMon}$. We write
${\rm dMon}$ for the associated $\infty$-category of simplicial commutative
monoids, which we shall refer to as the $\infty$-category of _derived
monoids_. We let $\oplus$ denote the coproduct in ${\rm dMon}$.

We now proceed to define and study derived analogues of logarithmic morphisms.
See [37, Chapter I.4.1] for the context of discrete commutative monoids.

######  Definition 2.3.

Let $M$ be a derived monoid. Its _space of units_ is the grouplike derived
monoid $\mathrm{GL}_{1}(M)$ of path components representing units in
$\pi_{0}(M)$.

This definition allows for the following generalization of [37, Definition
I.4.1.1(3)]:

######  Definition 2.4.

A map $f\colon P\to M$ of derived monoids is _logarithmic_ if the projection

$f^{-1}\mathrm{GL}_{1}(M):=P\times_{M}\mathrm{GL}_{1}(M)\to\mathrm{GL}_{1}(M)$

is an equivalence. If $f\colon P\to M$ is an arbitrary map of simplicial
commutative monoids, we define its _logification_ $f^{a}\colon P^{a}\to M$ by
the universal property of the cocartesian square

${f^{-1}{\rm GL}_{1}(M)}$${{\rm GL}_{1}(M)}$${P}$${P^{a}.}$

######  Proposition 2.5.

Let $f\colon P\to M$ be a map of derived monoids. Then the logification
$f^{a}\colon P^{a}\to M$ is the initial logarithmic map under $f$.

###### Proof.

By construction, the composite map $(f^{a})^{-1}{\rm
GL}_{1}(M)\xrightarrow{}P^{a}\xrightarrow{f^{a}}M$ is equivalent to the
inclusion ${\rm GL}_{1}(M)\to M$, so that $f^{a}$ is logarithmic. The
universal property of a pushout verifies the remaining claim. ∎

###  2.6. Pre-log simplicial rings

Let ${\rm dRing}$ denote the $\infty$-category of _derived_ (often called
_animated_) commutative rings. By definition, ${\rm dRing}$ is the underlying
$\infty$-category associated with the model structure on simplicial
commutative rings $\mathrm{sCRing}$ with fibrations and weak equivalences
defined on the level of simplicial sets.

######  Definition 2.7.

A _pre-log simplicial ring_ $(A,M,\alpha)$ is a simplicial object in the
category of pre-log rings. That is, a simplicial commutative ring $A$ equipped
with a simplicial commutative monoid $M$ and a map of simplicial monoids
$\alpha\colon M\to(A,\cdot)$.

By adjunction, $\alpha$ gives rise to a map
$\bar{\alpha}\colon{\mathbb{Z}}[M]\to A$ of simplicial commutative rings. We
let $\mathrm{sPreLog}$ denote the category of pre-log simplicial rings. By
[49, Proposition 3.3], it admits a _projective_ model structure. In this model
structure, $(f,f^{\flat})$ is a fibration if both $f$ and $f^{\flat}$ are Kan
fibrations of simplicial sets.

Let ${\rm dPreLog}$ denote the associated $\infty$-category of _derived pre-
log rings_. If no confusion arises, we will write $(A,M)$ for a pre-log
simplicial ring, omitting the structure map $\alpha$. The coproduct of derived
pre-log rings is defined using the underlying simplicial commutative rings and
simplicial commutative monoids; that is, $(A,M)\otimes(B,N):=(A\otimes
B,M\oplus N).$

######  Definition 2.8.

Let $(R,P,\beta)$ be a derived pre-log ring.

  1. (1)

$(R,P,\beta)$ is a _log derived ring_ if $\beta$ is logarithmic in the sense
of Definition 2.4.

  2. (2)

The _logification_ $(R,P^{a},\beta^{a})$ of $(R,P,\beta)$ is the logification
of $\beta$ in the sense of Definition 2.4.

######  Remark 2.9.

A log simplicial ring is not a synonym with a simplicial object in the
category of log rings, see [49, Remark 4.18]. We use the term _derived log
ring_ for a derived pre-log ring satisfying the condition of Definition
2.8(1).

There are Quillen pairs between model categories

(2.1)
$\mathrm{sCRing}\rightleftarrows\mathrm{sPreLog},\;\mathrm{sCMon}\rightleftarrows\mathrm{sCRing},\text{
and }\mathrm{sCMon}\rightleftarrows\mathrm{sPreLog}.$

The left adjoints in (2.1) send $A\in\mathrm{sCRing}$ to $(A,\\{1\\})$,
$M\in\mathrm{sCMon}$ to $\mathbb{Z}[M]$, and $M\in\mathrm{sCMon}$ to
$(\mathbb{Z}[M],M)$. On the underlying $\infty$-categories, the space of maps

${\rm Map}_{\rm dPreLog}((A,M),(B,N))$

from $(A,M)$ to $(B,N)$ in ${\rm dPreLog}$ is given by the pullback of

(2.2) ${\rm Map}_{\rm dMon}(M,N)\xrightarrow{}{\rm Map}_{\rm
dRing}({\mathbb{Z}}[M],B)\xleftarrow{}{\rm Map}_{\rm dRing}(A,B).$

The maps in (2.2) are naturally induced by the structure maps. Using the
$\infty$-category ${\rm dPreLog}$ as the basis of derived log geometry, we
make the following definitions:

######  Definition 2.10.

Let $(A,M)$ be a pre-log simplicial ring.

  1. (1)

The $\infty$-category of _affine pre-log derived schemes_ is the opposite of
the $\infty$-category of derived pre-log rings.

  2. (2)

Let $\operatorname{Spec}(A,M)$ denote the affine derived pre-log scheme
corresponding to $(A,M)$.

  3. (3)

For an affine derived pre-log scheme $X$, we let $\underline{X}$ denote the
underlying affine derived scheme of $X$. More precisely, if
$X=\operatorname{Spec}(A,M)$, then $\underline{X}:=\operatorname{Spec}(A)$.

###  2.11. Strict morphisms of pre-log simplicial rings

Following [49, §3.4], we recall the basic functoriality properties of pre-log
and log derived rings:

  1. (1)

Let $(R,P,\beta)$ be a derived pre-log ring and let $f\colon R\to A$ be a map
of derived commutative rings. The _inverse image pre-log structure_
$(A,f^{*}P)$ is the pre-log derived ring with a structure map
$P\xrightarrow{\beta}(R,\cdot)\xrightarrow{(f,\cdot)}(A,\cdot)$.

  2. (2)

Let $(A,M,\alpha)$ be a derived pre-log ring and let $f\colon R\to A$ be a map
of derived commutative rings. The _direct image pre-log struct_ is given by
the fiber product of derived monoids

${f^{*}M}$${(R,\cdot)}$${M}$${(A,\cdot).}$$\scriptstyle{f}$$\scriptstyle{\alpha}$

######  Definition 2.12.

A map $(R,P)\xrightarrow{(f,f^{\flat})}(A,M)$ of derived pre-log rings is
_strict_ if the canonical map $(A,(f^{*}P)^{a})\to(A,M^{a})$ is an
equivalence.

This notion of strictness differs slightly from the one in ordinary log
geometry, where one typically requires $(R,P)$ and $(A,M)$ to be log rings
already.

###  2.13. Repletions of derived commutative monoids

Next we discuss _repletions_ of derived monoids following Rognes [44] and
Sagave–Schürg–Vezzosi [49] in the derived context. An essentially identical
notion appearing earlier in Kato–Saito [25] traces back to exactifications of
closed immersions of log schemes: See Remark 2.16 for details.

Recall that a derived monoid is _grouplike_ if the commutative monoid
$\pi_{0}(M)$ is a group. The (de)looping adjunction $(B,\Omega)$ in the
category of derived commutative monoids is a Quillen pair for the standard
model structure and induces an adjunction on the underlying
$\infty$-categories. The unit of the adjunction

$\Omega B(-)\colon{\rm dMon}\to{\rm dMon},\quad M\mapsto\Omega BM,$

gives rise to the group completion construction $M\to\Omega BM$; if $M$ is
grouplike, this is an equivalence.

######  Definition 2.14.

The _group completion_ of a derived monoid $M$ is the grouplike derived monoid
$M^{\mathrm{gp}}:=\Omega BM$.

Recall the following definition from [44, Definition 3.6], [49, §2.2].

######  Definition 2.15.

Let $f\colon N\to M$ be a map of derived commutative rings. Then $f$ is

  1. (1)

_virtually surjective_ if
$\pi_{0}(f^{\mathrm{gp}})\colon\pi_{0}(N^{\mathrm{gp}})\to\pi_{0}(M^{\mathrm{gp}})$
is a surjection of abelian groups

  2. (2)

_exact_ if there is a naturally induced cartesian square of derived monoids

${N}$${N^{\mathrm{gp}}}$${M}$${M^{\mathrm{gp}}}$$\scriptstyle{f}$$\scriptstyle{f^{\mathrm{gp}}}$

  3. (3)

_replete_ if it is virtually surjective and exact.

The _exactification_ relative to the map $f$ is given by the pullback

$N^{\mathrm{ex}}:=M\times_{M^{\mathrm{gp}}}N^{\mathrm{gp}}.$

If $f$ is virtually surjective, we use the terminology _repletion_ instead of
_exactification_ and set $N^{\mathrm{rep}}:=N^{\mathrm{ex}}$ (see [44]). The
map $f$ admits a canonical factorization
$N\xrightarrow{}N^{\mathrm{rep}}\xrightarrow{f^{\mathrm{rep}}}M$.

######  Remark 2.16.

An analogous notion in the context of discrete integral commutative monoids is
called _exactification_ by Ogus [37, I.4.2.17]. More generally, a morphism of
discrete commutative monoids $P_{0}\to Q_{0}$ is exact [37, I.2.1.15] if
$P_{0}=Q_{0}\times_{Q_{0}^{\mathrm{gp}}}P_{0}^{\mathrm{gp}}$, where
$M_{0}^{\mathrm{gp}}$ is the classical group completion of the discrete
commutative monoid $M_{0}$. Under the additional assumption of virtual
surjectivity, repletion has been considered (with different terminology) by
Kato–Saito [25, Proposition 4.2.1].

######  Remark 2.17.

We briefly discuss various perspectives on the repletion of the addition map
${\mathbb{N}}^{2}\to{\mathbb{N}}$.

  1. (1)

As a map of discrete commutative monoids, ${\mathbb{N}}^{2}\to{\mathbb{N}}$ is
a Kan fibration. Hence its repletion is given by the ordinary pullback of
${\mathbb{N}}\xrightarrow{}{\mathbb{Z}}\xleftarrow{+}{\mathbb{Z}}^{2}$, which
we identify with ${\mathbb{N}}\oplus{\mathbb{Z}}$. The isomorphism can be
chosen so that the corresponding map from ${\mathbb{N}}^{2}$ to its repletion
sends $(m,n)$ to $(m+n,n)$.

  2. (2)

To give a geometric description of the repletion, recall that a closed
immersion of (discrete) log schemes is exact if and only if it is strict. As a
non-example, the diagonal map
$\Delta\colon\operatorname{Spec}(\mathbb{Z}[\mathbb{\mathbb{N}}])=\mathbb{A}_{\mathbb{N}}\to\mathbb{A}_{\mathbb{N}^{2}}=\operatorname{Spec}(\mathbb{Z}[\mathbb{\mathbb{N}}^{2}])$
for the addition $\mathbb{N}^{2}\to\mathbb{N}$ is clearly not strict (the
monoids have different $\mathbb{N}$-ranks). Hence it is not exact. On the
other hand, the repletion of $\Delta$ is constructed geometrically via the log
blow-up
$\mathrm{Bl}_{(0,0)}(\mathbb{A}_{\mathbb{N}^{2}})\to\mathbb{A}_{\mathbb{N}^{2}}$
equipped with its canonical log structure. The strict transform of the
diagonal $\Delta$ is a strict closed immersion; hence it is exact.

  3. (3)

More generally, any quasi-compact morphism of quasi-compact and fine log
schemes can be made exact by a suitable log blow-up by [37, Theorems
III.2.6.7, II.1.8.1].

In most of our examples, the map $f\colon N\to M$ admits a section and is
therefore virtually surjective. In this case, the repletion admits a
convenient description. The following is an analogue of [44, Lemma 3.11] and
[37, Proposition I.4.2.19] in the context of discrete commutative monoids, and
[31, Lemma 2.12] in the context of graded ${\mathbb{E}}_{\infty}$-spaces that
form the basis for spectral log geometry.

######  Proposition 2.18.

Let $N\xrightarrow{f}M$ be a map of derived commutative monoids, and assume
that it has a section $\eta\colon M\to N$. Then there is an equivalence

$N^{\mathrm{rep}}\xrightarrow{\simeq}M\oplus N^{\mathrm{gp}}/M^{\mathrm{gp}}$

over and under $M$, where $N^{\mathrm{gp}}/M^{\mathrm{gp}}$ denotes the
cofiber of $M^{\mathrm{gp}}\xrightarrow{\eta^{\mathrm{gp}}}N^{\mathrm{gp}}$.

###### Proof.

Consider the map of cofiber sequences of grouplike derived monoids

${M^{\mathrm{gp}}}$${N^{\mathrm{gp}}}$${N^{\mathrm{gp}}/M^{\mathrm{gp}}}$${M^{\mathrm{gp}}}$${M^{\mathrm{gp}}\oplus(N^{\mathrm{gp}}/M^{\mathrm{gp}})}$${N^{\mathrm{gp}}/M^{\mathrm{gp}}}$$\scriptstyle{=}$$\scriptstyle{=}$

The dashed arrow is induced by $f^{\mathrm{gp}}$ and the canonical map
$N^{\mathrm{gp}}\to N^{\mathrm{gp}}/M^{\mathrm{gp}}$ (recall that finite
products and finite coproducts of derived monoids coincide), and is an
equivalence since the outer vertical maps are. Consider now the commutative
cube of derived monoids

${M\oplus(N^{\mathrm{gp}}/M^{\mathrm{gp}})}$${M^{\mathrm{gp}}\oplus(N^{\mathrm{gp}}/M^{\mathrm{gp}})}$${N^{\mathrm{rep}}}$${N^{\mathrm{gp}}}$${M}$${M^{\mathrm{gp}}.}$${M}$${M^{\mathrm{gp}}}$$\scriptstyle{\simeq}$$\scriptstyle{=}$$\scriptstyle{f^{\mathrm{gp}}}$$\scriptstyle{=}$

The bottom face and the back vertical face are evidently cartesian, while the
front face is cartesian by definition. Hence the top face is cartesian, which
concludes the proof. ∎

The notion of repletion extends from derived monoids to derived pre-log rings
in the following manner:

######  Definition 2.19.

Let $(p,p^{\flat})\colon(B,N)\to(A,M)$ be a map of derived pre-log rings, and
assume that $p^{\flat}$ is virtually surjective. The _repletion_
$p^{\mathrm{rep}}$ of $p$ is the canonical map

$p^{\mathrm{rep}}\colon
B\otimes_{{\mathbb{Z}}[N]}{\mathbb{Z}}[N^{\mathrm{rep}}]\to
A\otimes_{{\mathbb{Z}}[M]}{\mathbb{Z}}[M]\cong A,$

where $N^{\mathrm{rep}}$ is the repletion of the map $p^{\flat}$.

If $(p,p^{\flat})$ corresponds to a map $Y\to X$ of affine derived pre-log
schemes, we shall write $Y\to X^{\mathrm{rep}}\to X$ for the corresponding
repletion (factoring the original map $Y\to X$). The following is a
consequence of Proposition 2.18:

######  Corollary 2.20.

Let $(p,p^{\flat})\colon(B,N)\to(A,M)$ be a morphism of derived pre-log rings
which admits a section. Then the repletion
$(p^{\mathrm{rep}},(p^{\flat})^{\mathrm{rep}})$ is strict. ∎

######  Remark 2.21.

Let $Y\to X$ be a closed immersion of affine derived pre-log schemes (i.e.,
the corresponding map of pre-log derived rings induces a surjection on
$\pi_{0}$), and assume that it has a section $X\to Y$. Then, by the previous
Corollary, the repletion $Y\to X^{\mathrm{rep}}$ is a strict (hence exact)
closed immersion.

##  3\. Log differentials and the log cotangent complex

Following [49], we interpret Gabber’s cotangent complex as the derived functor
of log Kähler differentials. We discuss log étale maps between affine pre-log
derived schemes and note that such maps have contractible cotangent complexes.
It turns out that _integral_ log étale maps of ordinary affine log schemes are
log étale in the derived sense.

###  3.1. The log differentials

Let
$(f_{0},f_{0}^{\flat})\colon(R_{0},P_{0},\beta_{0})\to(A_{0},M_{0},\alpha_{0})$
be a map of discrete pre-log rings. The log Kähler differentials
$\Omega^{1}_{(A_{0},M_{0})/(R_{0},P_{0})}$ are defined as the $A_{0}$-module

$(\Omega^{1}_{A_{0}/R_{0}}\oplus(A_{0}\otimes_{{\mathbb{Z}}}M_{0}^{\mathrm{gp}}))/\\!\\!\sim.$

The equivalence relation is $A_{0}$-linearly generated by
$(d\alpha_{0}(m),0)\sim(0,\alpha_{0}(m)\otimes\gamma_{0}(m))$ and
$0\sim(0,1\otimes\gamma(f_{0}^{\flat}(n)))$, where $\gamma_{0}\colon M_{0}\to
M_{0}^{\mathrm{gp}}$ is the canonical map from $M_{0}$ to its group
completion. It is common to write $d\hskip 0.70007pt\mathrm{log}(m)$ for the
element $(0,1\otimes\gamma_{0}(m))$, so that the first relation reads
$d\alpha_{0}(m)=\alpha_{0}(m)d\hskip 0.70007pt\mathrm{log}(m)$ and the second
relation reads $0=d\hskip 0.70007pt\mathrm{log}(f_{0}^{\flat}(n))$.

Let $\mathrm{PreLog}$ denote the category of ordinary (i.e., discrete) pre-log
rings. The module of log Kähler differentials corepresent logarithmic
derivations as in [37, IV.1.1.1]

${\rm Der}_{(R_{0},P_{0})}((A_{0},M_{0}),J_{0}):={\rm
Hom}_{\mathrm{PreLog}_{(R_{0},P_{0})//(A_{0},M_{0})}}((A_{0},M_{0}),(A_{0}\oplus
J_{0},M_{0}\oplus J_{0}))$

with values in some $A_{0}$-module $J_{0}$. Here the pre-log structure
$M_{0}\oplus J_{0}\to(A_{0}\oplus J_{0},\cdot)$ on the trivial square-zero
extension $A_{0}\oplus J_{0}$ is defined by
$(m,j)\mapsto(\alpha_{0}(m),\alpha_{0}(m)\cdot j)$. This is the coproduct of
the natural map

$M_{0}\xrightarrow{\alpha_{0}}(A_{0},\cdot)\xrightarrow{(d_{0},\cdot)}(A_{0}\oplus
J_{0},\cdot)$

and the inclusion of the units $J_{0}\cong 1+J_{0}\subset{\rm
GL}_{1}(A_{0}\oplus J_{0})$ in $A_{0}\oplus J_{0}$.

###  3.2. The log differentials via the log diagonal

Recall that the module of relative differentials $\Omega^{1}_{A_{0}/R_{0}}$
arises as the indecomposables of $A_{0}\otimes_{R_{0}}A_{0}\to A_{0}$, that
is, the conormal bundle of the diagonal map. If
$(R_{0},P_{0})\to(A_{0},M_{0})$ is a map of pre-log rings, this naturally
leads to the question of how the multiplication map

$(A_{0}\otimes_{R_{0}}A_{0},M_{0}\oplus_{P_{0}}M_{0})\to(A_{0},M_{0})$

relates to the log differentials. It is not clear what should be meant by the
module of indecomposables of this map; forming the indecomposables of the
underlying map of commutative rings merely recovers the ordinary module of
differentials $\Omega^{1}_{A_{0}/R_{0}}$.

However, after replacing the multiplication map with its repletion, we obtain
a map

$((A_{0}\otimes_{R_{0}}A_{0})\otimes_{\mathbb{Z}[M_{0}\oplus_{P_{0}}M_{0}]}\mathbb{Z}[(M_{0}\oplus_{P_{0}}M_{0})^{\rm
rep}],(M_{0}\oplus_{P_{0}}M_{0})^{\rm rep})\to(A_{0},M_{0})$

of pre-log rings. Moreover, this map is strict by Corollary 2.20. Hence it is
determined by the underlying map of commutative rings, and it makes sense to
define its module of indecomposables as that of the underlying map

(3.1)
$(A_{0}\otimes_{R_{0}}A_{0})\otimes_{{\mathbb{Z}}[M_{0}\oplus_{P_{0}}M_{0}]}{\mathbb{Z}}[(M_{0}\oplus_{P_{0}}M_{0})^{\rm
rep}]\to A_{0}$

of commutative rings. As a formal consequence of [25, Corollary 4.2.8(ii)]
(see also [37, Remark IV.1.1.8]), the module of indecomposables of (3.1) is
precisely the module of log differentials
$\Omega^{1}_{(A_{0},M_{0})/(R_{0},P_{0})}$. Indeed, the map (3.1) is a special
case of Kato–Saito’s _log diagonal_.

This implies that the log derivations ${\rm
Der}_{(R_{0},P_{0})}((A_{0},M_{0}),J_{0})$ is naturally isomorphic to the set
of augmented $A_{0}$-algebra maps

${\rm Hom}_{{\rm
CRing}_{A_{0}//A_{0}}}((A_{0}\otimes_{R_{0}}A_{0})\otimes_{{\mathbb{Z}}[M_{0}\oplus_{P_{0}}M_{0}]}{\mathbb{Z}}[(M_{0}\oplus_{P_{0}}M_{0})^{\mathrm{rep}}],A_{0}\oplus
J_{0}).$

This works as expected because log Kähler differentials are corepresented by
an $A_{0}$-module. This observation and its derived analogue serve as a
starting point for our approach to logarithmic Hochschild homology.

###  3.3. Gabber’s cotangent complex

We now discuss the Gabber cotangent complex in the context of derived pre-log
rings. In addition to establishing some properties which will be essential to
us, we explain how to obtain a description of it analogous to that of the log
differentials given in Section 3.2.

Let $(R,P)\to(A,M)$ be a morphism of derived pre-log rings, and let $J$ be a
connective $A$-module. Let $A\oplus J$ be the derived square-zero extension of
[34, Remark 7.3.4.15]. There is a canonical map $A\oplus J\to A$, informally
given by the projection onto the first component. It admits a canonical
section $s\colon A\to A\oplus J$. Let $(1+J)$ be the fiber of the canonical
projection ${\rm GL}_{1}(A\oplus J)\to{\rm GL}_{1}(A)$, and set $M\oplus
J:=M\oplus(1+J)$ in grouplike derived monoids.

We form the log square-zero extension associated with $J$ in the following
way.

######  Definition 3.4.

We let $(A\oplus J,M\oplus J)$ be the derived pre-log ring with a structure
map $M\oplus J\to(A\oplus J,\cdot)$ induced by the structure map
$M\xrightarrow{\alpha}(A,\cdot)\xrightarrow{s}(A\oplus J,\cdot)$ and the
inclusion $(1+J)\to{\rm GL}_{1}(A\oplus J)\to(A\oplus J,\cdot)$.

There is a canonical map $(A\oplus J,M\oplus J)\to(A,M)$, informally given by
the projection onto the first component. It admits a canonical section
$(A,M)\to(A\oplus J,M\oplus J)$. We define the space
$\mathrm{Der}_{(R,P)}((A,M),J)$ of $(R,P)$-linear derivations of $(A,M)$ with
values in $J$ as the fiber of the map

(3.2) $\operatorname{Map}_{{\rm dPreLog}_{(R,P)/}}((A,M),(A\oplus J,M\oplus
J))\to\operatorname{Map}_{{\rm dPreLog}_{(R,P)/}}((A,M),(A,M))$

induced by $(A\oplus J,M\oplus J)\to(A,M)$ at the point $\mathrm{id}_{(A,M)}$.
As in derived algebraic geometry [51, 1.4.1.14], we say that $(R,P)\to(A,M)$
has a cotangent complex (or that $(A,M)$ admits a cotangent complex over
$(R,P)$) if there exists a connective $A$-module co-representing the functor

$\operatorname{Mod}_{A}^{\rm cn}\to\mathcal{S},\quad
J\mapsto\mathrm{Der}_{(R,P)}((A,M),J).$

We note that such a log cotangent complex is unique up to equivalence in the
$\infty$-category $\operatorname{Mod}_{A}^{\rm cn}$ of connective $A$-modules.

Let $(R,P)$ be a simplicial pre-log ring. By extending the definition of the
log Kähler differentials degreewise, we obtain a functor

$\Omega^{1}_{(-,-)/(R,P)}\colon{\rm sPreLog}_{(R,P)/}\to{\rm sMod}_{R}$

from the category of pre-log derived $(R,P)$-algebras to derived $R$-modules.
As discussed in [49, §4.2], this is a left Quillen functor whose right adjoint
is given by the levelwise square-zero construction $J\mapsto(A\oplus J,M\oplus
J)$. This gives rise to an adjunction on the underlying $\infty$-categories;
we denote the left adjoint by ${\mathbb{L}}$.

######  Definition 3.5.

Let $(R,P)\to(A,M)$ be a map of derived pre-log rings. The _log cotangent
complex_ ${\mathbb{L}}_{(A,M)/(R,P)}$ is the derived $R$-module

${\mathbb{L}}_{(A,M)/(R,P)}:={\mathbb{L}}(A,M).$

We note that ${\mathbb{L}}_{(A,M)/(R,P)}$ has a natural $A$-module structure.
Following Illusie and Quillen, we write
$\bigwedge\nolimits^{i}_{A}{\mathbb{L}}_{(A,M)/(R,P)}$ for the $i$-th derived
exterior power of ${\mathbb{L}}_{(A,M)/(R,P)}$.

It is clear by construction that ${\mathbb{L}}_{(A,M)/(R,P)}$ co-represents
the functor of log-derivations. In particular, every morphism of pre-log
derived rings $(R,P)\to(A,M)$ admits a log cotangent complex.

######  Remark 3.6.

If $X\to S$ is a map of affine pre-log derived schemes, we write
${\mathbb{L}}_{X/S}$ for the log cotangent complex. Note that it is an object
of the $\infty$-category $\mathrm{QCoh}(\underline{X})\cong\mathrm{Mod}_{A}$
if $\underline{X}=\operatorname{Spec}(A)$ (see e.g., [33, 2.3.11]).

######  Remark 3.7.

The cotangent complex defined above is a model for Gabber’s cotangent complex,
defined as the derived $A$-module
${\mathbb{L}}_{(A,M)/(R,P)}:=A\otimes_{A_{\bullet}}\Omega^{1}_{(A_{\bullet},M_{\bullet})/(R,P)}$
for a free $(R,P)$-algebra resolution
$(A_{\bullet},M_{\bullet})\xrightarrow{\simeq}(A,M)$. This should not be
confused with Olsson’s log cotangent complex in [39, Definition 3.2]. Note
that the derived exterior powers can be modeled as
$\bigwedge\nolimits^{q}_{A}{\mathbb{L}}_{(A,M)/(R,P)}=A\otimes_{A_{\bullet}}\Omega^{q}_{(A_{\bullet},M_{\bullet})/(R,P)}$.

We now aim to describe some elementary properties of the log cotangent
complex. Using the definition of log derivations in (3.2) and the description
of mapping spaces in ${\rm dPreLog}$ (2.2), we learn that the log cotangent
complex can be realized as the pushout of the diagram

(3.3)
${\mathbb{L}}_{A/R}\xleftarrow{}A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{{\mathbb{Z}}[M]/{\mathbb{Z}}[P]}\xrightarrow{}A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)}$

of derived $A$-modules. The following describes the log cotangent complex in a
way that is reminiscent of the description of the spectral log cotangent
complex given by Rognes [44, §11] and Sagave [47] (therein called _logarithmic
${\rm TAQ}$_):

######  Proposition 3.8.

There is a canonical equivalence

${\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)}\simeq{\mathbb{Z}}[M]\otimes_{\mathbb{Z}}(M^{\mathrm{gp}}/P^{\mathrm{gp}}).$

In Proposition 3.8, we implicitly use [19, Appendix Q] (see also [49, Lemma
2.10]) to model the grouplike derived monoid $M^{\mathrm{gp}}/P^{\mathrm{gp}}$
by a derived abelian group, so that the tensor product
${\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}}(M^{\mathrm{gp}}/P^{\mathrm{gp}})$ is
defined.

###### Proof of Proposition 3.8.

We show that there is a natural equivalence

(3.4) ${\rm Map}_{{\rm
Mod}_{{\mathbb{Z}}[M]}}({\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)},J)\simeq{\rm
Map}_{{\rm
Mod}_{{\mathbb{Z}}[M]}}({\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}}(M^{\rm
gp}/P^{\rm gp}),J)$

of mapping spaces for any ${\mathbb{Z}}[M]$-module $J$. To see this, we first
use that the log cotangent complex
${\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)}$ corepresents log
derivations

${\rm Map}_{{\rm
dPreLog}_{({\mathbb{Z}}[P],P)//({\mathbb{Z}}[M],M)}}(({\mathbb{Z}}[M],M),({\mathbb{Z}}[M]\oplus
J,M\oplus J)).$

Moreover, by the description of mapping spaces in (2.2), this can be
simplified to

${\rm Map}_{{\rm dMon}_{P//M}}(M,M\oplus J)\simeq{\rm Map}_{{\rm
dMon}_{M//M}}(M\oplus_{P}M,M\oplus J).$

Now we observe that the map $M\oplus J\to M$ is replete, so that any map $K\to
M\oplus J$ of derived monoids over and under $M$ must factor uniquely through
the repletion $K^{\rm rep}$. Applying this to the case at hand, we find that
there is a natural equivalence

${\rm Map}_{{\rm dMon}_{M//M}}(M\oplus_{P}M,M\oplus J)\simeq{\rm Map}_{{\rm
dMon}_{M//M}}((M\oplus_{P}M)^{\rm rep},M\oplus J).$

We now apply Proposition 2.18 to infer an equivalence

(3.5) ${\rm Map}_{{\rm dMon}_{M//M}}((M\oplus_{P}M)^{\mathrm{rep}},M\oplus
J)\simeq{\rm Map}_{{\rm
dMon}_{M//M}}(M\oplus(M^{\mathrm{gp}}/P^{\mathrm{gp}}),M\oplus J).$

By restriction of scalars and pullback along $*\to M$, we obtain a natural
equivalence

${\rm Map}_{{\rm
dMon}_{M//M}}(M\oplus(M^{\mathrm{gp}}/P^{\mathrm{gp}}),M\oplus J)\simeq{\rm
Map}_{\rm dAb}(M^{\rm gp}/P^{\rm gp},J).$

By extension of scalars along ${\mathbb{Z}}\to{\mathbb{Z}}[M]$ we obtain the
equivalence (3.4), which concludes the proof. ∎

The following Corollary is of fundamental importance for us since it makes it
possible to infer properties of the log cotangent complex from its non-log
counterpart, together with some simple information from a derived abelian
group.

######  Corollary 3.9.

Let $(R,P)\to(A,M)$ be a map of derived pre-log rings. There is a cofiber
sequence of derived $A$-modules

$A\otimes_{\mathbb{Z}}(M^{\mathrm{gp}}/P^{\mathrm{gp}})\to{\mathbb{L}}_{(A,M)/(R,P)}\to{\mathbb{L}}_{A/R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]}.$

###### Proof.

Recall that all tensor products are implicitly derived. There is a pushout
square

${A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{{\mathbb{Z}}[M]/{\mathbb{Z}}[P]}}$${A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)}}$${{\mathbb{L}}_{A/R}}$${{\mathbb{L}}_{(A,M)/(R,P)}}$

of derived $A$-modules (3.3). By Proposition 3.8, there is an equivalence

$A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{({\mathbb{Z}}[M],M)/({\mathbb{Z}}[P],P)}\simeq
A\otimes_{\mathbb{Z}}(M^{\mathrm{gp}}/P^{\mathrm{gp}}).$

Thus it suffices to prove that the upper row in the diagram

(3.6)
${A\otimes_{{\mathbb{Z}}[M]}{\mathbb{L}}_{{\mathbb{Z}}[M]/{\mathbb{Z}}[P]}}$${{\mathbb{L}}_{A/R}}$${{\mathbb{L}}_{A/R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]}}$${A\otimes_{R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]}{\mathbb{L}}_{R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]/R}}$${{\mathbb{L}}_{A/R}}$${{\mathbb{L}}_{A/R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]}}$$\scriptstyle{=}$$\scriptstyle{=}$

is a cofiber sequence. Here the lower row is the transitivity sequence
associated to the composite $R\to R\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]\to
A$; thus, it is a cofiber sequence. The left-hand map is an equivalence by
derived base change for the classical cotangent complex applied to the pushout
of the diagram $R\xleftarrow{}{\mathbb{Z}}[P]\xrightarrow{}{\mathbb{Z}}[M]$,
from which we conclude the proof. ∎

The proof of Corollary 3.9 uses derived base change for the classical
cotangent complex [50, Tag 08QQ]. This suggests that some flatness hypothesis
is necessary to transport results from the derived setting to the discrete
case. In the log setting, this is guaranteed by the integrality condition: A
discrete commutative monoid $M_{0}$ is _integral_ if the canonical map
$M_{0}\to M_{0}^{\mathrm{gp}}$ is an injection, while a map $P_{0}\to M_{0}$
of discrete commutative monoids is _integral_ if its pushout along any map to
an integral monoid remains integral [37, Definition I.4.6.2(3)]. This holds
for any map of integral monoids $P_{0}\to M_{0}$ provided $P_{0}$ is valuative
[37, Proposition I.4.6.3(5)]. In particular, the canonical pre-log structure
on any discrete valuation ring is valuative.

######  Corollary 3.10.

Let $(R_{0},P_{0})\to(A_{0},M_{0})$ be a map of discrete pre-log rings. If
$P_{0}\to M_{0}$ is integral and $P_{0}^{\mathrm{gp}}\to M_{0}^{\mathrm{gp}}$
is injective, there is a cofiber sequence of derived $A_{0}$-modules

$A_{0}\otimes_{\mathbb{Z}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})\to{\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\to{\mathbb{L}}_{A_{0}/R_{0}\otimes_{{\mathbb{Z}}[P_{0}]}{\mathbb{Z}}[M_{0}]}.$

The integrality hypothesis is essential, see Example 4.9. Corollary 3.10 is
particularly convenient when relating the log cotangent complex with the
characterization of log étale and smooth maps in terms of charts [24, Theorem
3.5], see Proposition 4.6.

###### Proof.

The flat base change argument in the proof of Corollary 3.9 goes through since
the integrality hypothesis implies that
${\mathbb{Z}}[P_{0}]\to{\mathbb{Z}}[M_{0}]$ is flat by [37, Remark I.4.6.6].
Moreover, by [39, Lemma 8.18(ii)], there is an equivalence of derived
$A_{0}$-modules

$A_{0}\otimes_{{\mathbb{Z}}[M_{0}]}{\mathbb{L}}_{({\mathbb{Z}}[M_{0}],M_{0})/({\mathbb{Z}}[P_{0}],P_{0})}\simeq
A_{0}\otimes_{\mathbb{Z}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}).$

∎

Since the log cotangent complex corepresents (derived) log derivations, the
argument of [47, Corollary 6.7] yields

######  Lemma 3.11.

The logification construction induces equivalences of derived $A$-modules

${\mathbb{L}}_{(A,M)/(R,P)}\xrightarrow{\simeq}{\mathbb{L}}_{(A,M^{a})/(R,P)}\xrightarrow{\simeq}{\mathbb{L}}_{(A,M^{a})/(R,P^{a})}.$

We shall also have occasion to use that the log cotangent complex of discrete
pre-log rings is invariant under logification by mimicking Definition 2.4
using ordinary pullbacks and pushouts. This is the content of [39, Theorem
8.16].

######  Corollary 3.12.

Let $f\colon(R_{0},P_{0})\to(A_{0},M_{0})$ be a strict map of discrete pre-log
rings. Then
${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\simeq\mathbb{L}_{A_{0}/R_{0}}$,
where the latter denotes the classical relative cotangent complex. In
particular, if $f$ is strict étale, then
${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}$ is contractible.

###### Proof.

Since the construction is invariant under logification, we may assume that the
log structure on $(A_{0},M_{0})$ is induced by the inverse image pre-log
structure $P_{0}\to(R_{0},\cdot)\to(A_{0},\cdot)$. Now the equivalence
${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\cong\mathbb{L}_{A_{0}/R_{0}}$
follows immediately from Corollary 3.9. The last statement is a consequence of
[50, Tag 08R2]. ∎

Let $(R,P)\xrightarrow{(f,f^{\flat})}(A,M)\xrightarrow{(g,g^{\flat})}(B,N)$ be
maps of derived pre-log rings. According to [49, Proposition 4.12(i)], there
is a cofiber sequence

(3.7)
$B\otimes_{A}\mathbb{L}_{(A,M)/(R,P)}\to\mathbb{L}_{(B,N)/(R,P)}\to\mathbb{L}_{(B,N)/(A,M)}.$

By combining (3.7) with Corollary 3.12, we see that for every strict étale map
$(A_{0},M_{0})\to(B_{0},N_{0})$ of discrete pre-log rings, there is a natural
equivalence

(3.8)
$B_{0}\otimes_{A_{0}}\mathbb{L}_{(A_{0},M_{0})/(R_{0},P_{0})}\xrightarrow{\simeq}\mathbb{L}_{(B_{0},N_{0})/(R_{0},P_{0})}.$

###  3.13. Derived indecomposables

Let $A$ be a derived ring, and let $A\to B\to A$ be a derived augmented
$A$-algebra. We write $Q_{A}(B)$ for the derived $A$-module of indecomposables
in $B$. For example, $Q_{A}(-)$ can be realized as the functor
$Q_{A}(-)\colon{\rm dRing}_{A//A}\to{\rm dMod}_{A}$ of $\infty$-categories
associated to the left Quillen functor $Q_{A}(-)\colon{\rm
sCRing}_{A//A}\to{\rm sMod}_{A}$ obtained by taking indecomposables levelwise;
its right adjoint is determined by the trivial square-zero extension. If $A\to
C$ is a map of derived rings, there is a base-change formula

$Q_{C}(C\otimes_{A}B)\simeq C\otimes_{A}Q_{A}(B)$

which we will use freely.

###  3.14. The replete diagonal map

In what follows, we use the repletion to describe the log cotangent complex as
the conormal of the _replete_ diagonal.

######  Definition 3.15.

Suppose $f\colon X\to S$ is a map of affine pre-log derived schemes. The
_replete diagonal map_

$\Delta^{\mathrm{rep}}\colon X\to X\times_{S}^{\mathrm{rep}}X$

is the repletion of the diagonal map $\Delta\colon X\to X\times_{S}X$.

######  Remark 3.16.

If $f\colon X\to S$ is a map of ordinary affine log schemes, the replete
diagonal map defined above is a derived analogue of Kato-Saito’s log diagonal
map $X\to X\times_{S,[P]}^{\log}X$ [25, Corollary 4.2.8(i)]. See also [37,
Example III.2.3.6] for an equivalent construction.

The following gives a description of the log cotangent complex analogous to
that of the log differentials in Section 3.2:

######  Proposition 3.17.

Let $X\to S$ be a map of affine pre-log derived schemes. The cotangent complex
${\mathbb{L}}_{X/S}$ is equivalent to the conormal of
$\underline{\Delta^{\mathrm{rep}}}$, the map of derived schemes underlying the
replete diagonal $\Delta^{\mathrm{rep}}\colon X\to
X\times_{S}^{\mathrm{rep}}X$.

For a morphism of affine derived schemes arising from a map of derived
commutative rings admitting a section, we define its indecomposables as the
derived module obtained by applying the indecomposable functor levelwise. It
may be helpful to observe that the ordinary cotangent complex
${\mathbb{L}}_{A/R}$ arises as the indecomposables of $A\otimes_{R}A\to A$:
Indeed, the derived tensor product $A\otimes_{R}A$ can be computed via a
polynomial $R$-algebra resolution $A_{\bullet}\xrightarrow{\simeq}A$, and the
indecomposables of

$A\otimes_{R}A_{\bullet}\cong
A\otimes_{A_{\bullet}}(A_{\bullet}\otimes_{R}A_{\bullet})\to A$

is $A\otimes_{A_{\bullet}}\Omega^{1}_{A_{\bullet}/R}$, which is the definition
of the cotangent complex ${\mathbb{L}}_{A/R}$.

###### Proof of Proposition 3.17.

The replete diagonal corresponds to an augmentation

$((A\otimes_{R}A)\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}],(M\oplus_{P}M)^{\mathrm{rep}})\to(A,M)$

of derived pre-log rings. This evidently admits a section, so that Corollary
2.20 implies the map is strict. The module in question is therefore the
levelwise indecomposables of

(3.9)
$(A\otimes_{R}A)\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]\to
A.$

Since the ordinary cotangent complex $\mathbb{L}_{A/R}$ is the indecomposables
of $A\otimes_{R}A\to A$, and similarly for
$\mathbb{L}_{\mathbb{Z}[M]/\mathbb{Z}[P]}$, the indecomposables of (3.9) is
the pushout of the diagram

$\mathbb{L}_{A/R}\xleftarrow{}A\otimes_{\mathbb{Z}[M]}\mathbb{L}_{\mathbb{Z}[M]/\mathbb{Z}[P]}\to
Q_{A}(A\otimes_{\mathbb{Z}[M]}\mathbb{Z}[(M\oplus_{P}M)^{\rm rep}]),$

where $Q_{A}(-)\colon{\rm CAlg}_{A//A}\to{\rm Mod}_{A}$ is the indecomposables
functor. By Proposition 2.18, the right-hand $A$-module is equivalent to
$Q_{A}(A\otimes_{\mathbb{Z}}\mathbb{Z}[M^{\rm gp}/P^{\rm gp}])\simeq
A\otimes_{\mathbb{Z}}Q_{\mathbb{Z}}(\mathbb{Z}[M^{\rm gp}/P^{\rm gp}])$.
Since, for any group $G_{0}$, the module of indecomposables of the
augmentation $\mathbb{Z}[G_{0}]\to\mathbb{Z}$ is isomorphic to abelianization
of $G_{0}$, this further identifies
$A\otimes_{\mathbb{Z}}Q_{\mathbb{Z}}(\mathbb{Z}[M^{\rm gp}/P^{\rm gp}])$ with
$A\otimes_{\mathbb{Z}}(M^{\rm gp}/P^{\rm gp})$. Now Proposition 3.8 and (3.3)
apply to conclude the proof. ∎

######  Remark 3.18.

Proposition 3.17 gives rise to a model-independent construction of the log
cotangent complex. Indeed, since we have described
${\mathbb{L}}_{(A,M)/(R,P)}$ as the indecomposables of a certain augmented
$A$-algebra, which can be used to describe the
${\mathbb{E}}_{\infty}$-cotangent complex, we may now mimic the approach of
[34, 7.3.5]. We refer to [31, §9] for details.

##  4\. Étale and smooth morphisms in derived log geometry

We shall use the following notion of formal étaleness in the context of
derived log geometry from [49, Definition 5.3]:

######  Definition 4.1.

A map $(f,f^{\flat})\colon(R,P)\to(A,M)$ of derived pre-log rings is _derived
formally log étale_ if the log cotangent complex ${\mathbb{L}}_{(A,M)/(R,P)}$
is contractible.

Following [49], we say that a map $(f,f^{\flat})$ is derived log étale if it
is formally derived log étale and if $f$ is homotopically finitely presented
in the sense of [51, Definition 1.2.3.1]. By [49, Theorem 5.6], a derived log
étale map $(f,f^{\flat})$ such that $\pi_{0}(f^{\flat})$ is finitely presented
induces a log étale map $(\pi_{0}(f),\pi_{0}(f^{\flat}))$ of discrete log
rings in the sense that $(\pi_{0}(f),\pi_{0}(f^{\flat}))$ lifts uniquely
against log square-zero extensions [24]. In particular, [24, Theorem 3.5]
shows that, locally on charts, $(\pi_{0}(f),\pi_{0}(f^{\flat}))$ satisfies the
following properties:

  1. (1)

$\pi_{0}(R)\otimes_{{\mathbb{Z}}[\pi_{0}(P)]}{\mathbb{Z}}[\pi_{0}(M)]\to\pi_{0}(A)$
is étale;

  2. (2)

$\pi_{0}(M)^{\mathrm{gp}}/\pi_{0}(P)^{\mathrm{gp}}$ is finite of order
invertible in $\pi_{0}(A)$.

In future work, we intend to develop the deformation theory of derived log
schemes. We prove a derived analogue of [24, Theorem 3.5], obtaining several
equivalent notions of log étaleness in the context of derived log geometry.

######  Definition 4.2.

A map $(f,f^{\flat})\colon(R,P)\to(A,M)$ of derived pre-log rings is _derived
formally log smooth_ if the log cotangent complex ${\mathbb{L}}_{(A,M)/(R,P)}$
is a projective $A$-module, i.e., a direct summand of a free $A$-module.

Our definition of derived formal log smoothness is equivalent to [49,
Definition 6.3] by [34, Propositions 7.2.2.6, 7.2.2.7], [51, 1.2.8.3]. One can
define a map of derived pre-log rings to be _derived log smooth_ if the
underlying map of derived commutative rings is homotopically finitely
presented. By [49, Theorem 6.4], this yields a log smooth map of log rings on
$\pi_{0}$.

######  Proposition 4.3.

The classes of derived formally log étale morphisms and derived formally log
smooth morphisms of derived pre-log rings are closed under compositions and
pushouts.

###### Proof.

Let $(R,P)\xrightarrow{(f,f^{\flat})}(A,M)\xrightarrow{(g,g^{\flat})}(B,N)$ be
maps of derived pre-log rings, and consider the transitivity sequence (3.7).
If $(f,f^{\flat})$ and $(g,g^{\flat})$ are derived formally log smooth, then
$B\otimes_{A}\mathbb{L}_{(A,M)/(R,P)}$ and $\mathbb{L}_{(B,N)/(A,M)}$ are
projective $B$-modules. Hence $\mathbb{L}_{(B,N)/(R,P)}$ is a projective
$B$-module by [34, Proposition 7.2.2.6]. It follows that
$(gf,g^{\flat}f^{\flat})$ is derived formally log smooth. Let

${(R,P)}$${(A,M)}$${(R^{\prime},P^{\prime})}$${(A^{\prime},M^{\prime})}$$\scriptstyle{(f,f^{\flat})}$$\scriptstyle{(f^{\prime},f^{\prime\flat})}$

be a pushout square of derived pre-log rings. By [49, Proposition 4.12(ii)],
we have

$A^{\prime}\otimes_{A}\mathbb{L}_{(A,M)/(R,P)}\simeq\mathbb{L}_{(A^{\prime},M^{\prime})/(R^{\prime},P^{\prime})}.$

The assumption on $(f,f^{\flat})$ implies that $\mathbb{L}_{(A,M)/(R,P)}$ is a
projective $A$-module. Therefore,
$\mathbb{L}_{(A^{\prime},M^{\prime})/(R^{\prime},P^{\prime})}$ is a projective
$A^{\prime}$-module, i.e., $(f^{\prime},f^{\prime\flat})$ is derived formally
log smooth.

The claim for derived formally log étale maps is shown similarly. ∎

###  4.4. Derived formally log étale maps of discrete log rings

Example 4.9 below shows that log étale maps of ordinary log rings need not
give rise to derived formally log étale maps of affine pre-log derived
schemes. This is in contrast to ordinary derived algebraic geometry and
closely related to the fact that the log cotangent complex may fail to be
contractible for a given log étale map. However, a derived formally log étale
map has a vanishing cotangent complex by definition. Derived formally log
smooth maps enjoy the property of being concentrated as the log differentials
in degree zero:

######  Proposition 4.5.

If $(R_{0},P_{0})\xrightarrow{(f_{0},f^{\flat}_{0})}(A_{0},M_{0})$ is a
derived formally log smooth map of discrete pre-log rings, there is a
canonical equivalence

${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\xrightarrow{\simeq}\pi_{0}({\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})})\cong\Omega^{1}_{(A_{0},M_{0})/(R_{0},P_{0})}.$

###### Proof.

Since ${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}$ is a direct summand of a
free $A_{0}$-module and $A_{0}$ is discrete,
${\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}$ is concentrated in degree $0$.
This implies the claim. ∎

As suggested by the application of base change in the proof of Corollary 3.9
and the resulting integrality hypothesis in Corollary 3.10, flatness
hypotheses are necessary when passing from ordinary to derived log geometry.
Corollary 3.10 is particularly convenient when using the characterization of
log étale and log smooth morphisms in terms of charts, for which we need to
impose an additional finiteness hypothesis:

######  Proposition 4.6.

Let $(R_{0},P_{0})\xrightarrow{(f_{0},f^{\flat}_{0})}(A_{0},M_{0})$ be an
integral map of discrete fine pre-log rings.

  1. (1)

If $(f_{0},f_{0}^{\flat})$ is log smooth, then it is derived log smooth.

  2. (2)

If $(f_{0},f_{0}^{\flat})$ is log étale, then it is derived log étale.

###### Proof.

We give a proof of (1), since (2) is similar (in fact, easier). Using the base
change property for strict étale maps (3.8), one readily verifies that the
claims can be checked strict étale locally on $(A_{0},M_{0})$. Indeed, suppose
$(A_{0},M_{0})\to(B_{0},N_{0})$ is a strict étale cover such that
$(R_{0},P_{0})\to(B_{0},N_{0})$ is derived log smooth. Then
$B_{0}\otimes_{A_{0}}\mathbb{L}_{(A_{0},M_{0})/(R_{0},P_{0})}\simeq\mathbb{L}_{(B_{0},N_{0})/(R_{0},P_{0})}$
is a finitely generated projective $B_{i}$-module. By fppf descent [50, 03DX],
we deduce that $\mathbb{L}_{(A_{0},M_{0})/(R_{0},P_{0})}$ is a finitely
generated projective $A_{0}$-module as well.

Since the log cotangent complex is invariant under logification [39, Theorem
8.16], it suffices to check the statement on charts. Since the monoids are
assumed to be fine, by appealing to [24, Theorem 3.5] or [37, Theorem
IV.3.3.1] we may assume

  1. (1)

$R_{0}\otimes_{{\mathbb{Z}}[P_{0}]}{\mathbb{Z}}[M_{0}]\to A_{0}$ is smooth,
and

  2. (2)

the induced map $P_{0}^{\mathrm{gp}}\to M_{0}^{\mathrm{gp}}$ is injective, and
the torsion part of $M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}$ is of finite
order invertible in $A_{0}$.

After a further localization, [37, Theorem IV.3.3.1] allows us to assume that
$P_{0}\to M_{0}$ remains integral. Hence Corollary 3.10 yields a cofiber
sequence

$A_{0}\otimes_{\mathbb{Z}}^{\mathbb{L}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})\to{\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\to{\mathbb{L}}_{A_{0}/R_{0}\otimes_{{\mathbb{Z}}[P_{0}]}{\mathbb{Z}}[M_{0}]}.$

The right-hand module is concentrated in degree zero as the differentials by
(1), while the left-hand module is concentrated in degree zero as the free
$A_{0}$-module of rank that of the free part of
$M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}$ by (2). Hence the middle term is
concentrated in degree zero as the log differentials. ∎

In summary, there are implications for morphisms of discrete fine pre-log
rings:

(4.1) $\displaystyle\text{(integral log smooth)}\Rightarrow\text{(derived log
smooth)}\Rightarrow\text{(log smooth),}$ (4.2) $\displaystyle\text{(integral
log \'{e}tale)}\Rightarrow\text{(derived log \'{e}tale)}\Rightarrow\text{(log
\'{e}tale).}$

Here the first implications in both (4.1) and (4.2) follow from Proposition
4.6, while second implications follow from [49, Theorem 5.6 and Theorem 6.4].

######  Example 4.7.

Recall that the replete diagonal participates in a factorization

$X\xrightarrow{\Delta^{\mathrm{rep}}}X\times_{S}^{\mathrm{rep}}X\to
X\times_{S}X$

of the diagonal map associated to a map $X\to S$ of affine derived pre-log
schemes. The second map in this composite is an example of a log étale map
according to Corollary 3.9. This is analogous to the log diagonal studied by
Kato–Saito [25, §4].

######  Example 4.8.

Let $A_{0}$ be a ring, and let $\theta\colon P_{0}\to M_{0}$ be a (not
necessarily integral) map of finitely generated monoids. We assume that the
kernel and the torsion part of the cokernel of $\theta^{\mathrm{gp}}$ are
finite groups with order invertible in $A_{0}$. This implies that the tensor
product
$A_{0}\otimes_{\mathbb{Z}}^{\mathbb{L}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})$
is a free $A_{0}$-module concentrated in degree zero. Moreover, it does not
make a difference whether we consider the ordinary or derived quotient
$M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}$ after inducing up to $A_{0}$. Hence
Corollary 3.9 implies that
${\mathbb{L}}_{(A_{0}[M_{0}],M_{0})/(A_{0}[P_{0}],P_{0})}$ is a free
$A_{0}$-module concentrated in degree zero. This means that the naturally
induced map $f\colon(A_{0}[P_{0}],P_{0})\to(A_{0}[M_{0}],M_{0})$ is derived
log smooth even if we do not assume that $\theta$ is integral. If we further
assume that the cokernel of $\theta^{\mathrm{gp}}$ is a finite group whose
order is invertible in $A_{0}$, the same argument shows there is a canonical
equivalence

$A_{0}\otimes_{\mathbb{Z}}^{\mathbb{L}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})\xrightarrow{\simeq}\mathbb{L}_{(A_{0}[M_{0}],M_{0})/(A_{0}[P_{0}],P_{0})}$

with contractible source, so that $f$ is derived log étale.

######  Example 4.9.

Let $k$ be a field of characteristic $0$, and let $P$ be the submonoid of
$\mathbb{N}^{2}$ generated by $(2,0)$, $(1,1)$, and $(0,2)$. The inclusion
$P\to\mathbb{N}^{2}$ naturally induces

$f\colon(k[P],P)\to(k[\mathbb{N}^{2}],\mathbb{N}^{2}).$

Example 4.8 shows that $f$ is derived log étale. The homomorphism $u\colon
k[P]\to k$ sending all elements of $P$ to $0$ gives the pushout map

$f^{\prime}\colon(k,P)\to(k\otimes_{k[P]}k[\mathbb{N}^{2}],\mathbb{N}^{2}).$

If $\mathbb{L}_{(k\otimes_{k[P]}k[\mathbb{N}^{2}],\mathbb{N}^{2})/(k,P)}$ is
contractible, the argument of Bauer in [39, §7.2] would imply there is a
naturally induced equivalence

$(k\otimes_{k[P]}k[\mathbb{N}^{2}])\otimes_{k}\mathbb{L}_{(k,P)/(k[P],P)}\xrightarrow{\simeq}\mathbb{L}_{(k\otimes_{k[P]}k[\mathbb{N}^{2}],\mathbb{N}^{2})/(k[\mathbb{N}^{2}],\mathbb{N}^{2})}$

This is false, however, as observed in [39, Example 7.3]. Thus $f^{\prime}$ is
_not_ derived log étale even though $f^{\prime}$ is log étale. The reason for
this discrepancy is that $f^{\prime}$ is the pullback of $f$ but not the
homotopy pullback of $f$, i.e.,
$k\otimes_{k[P]}k[\mathbb{N}^{2}]\not\simeq\pi_{0}(k\otimes_{k[P]}k[\mathbb{N}^{2}])$.

##  5\. Logarithmic Hochschild homology of derived pre-log rings

After a brief review of Hochschild homology, we define log Hochschild homology
of derived pre-log rings and note that it recovers Rognes’ definition for
discrete pre-log rings. Throughout, all tensor products are implicitly derived
unless explicitly stated otherwise.

###  5.1. Hochschild homology

Let $R\to A$ be a map of derived rings. The _Hochschild homology_
$\operatorname{HH}(A/R)$ of $A$ relative to $R$ is the colimit
$S^{1}\otimes_{R}A$ of the constant diagram $A$ from $S^{1}$ to derived
commutative $R$-algebras. The maps $*\to S^{1}\to*$ exhibit $A\to{\rm
HH}(A/R)\to A$ as a derived augmented commutative $A$-algebra, while the
equivalence $S^{1}\simeq*\sqcup_{*\sqcup*}*$ gives an equivalence
$\operatorname{HH}(A/R)\simeq A\otimes_{A\otimes_{R}A}A$. If
$\underline{X}\to\underline{S}$ denotes the corresponding map of affine
derived schemes, we see that

$\operatorname{HH}({\underline{X}}/{\underline{S}})\simeq{\underline{X}}\times_{\underline{X}\times_{\underline{S}}\underline{X}}\underline{X},$

the derived self-intersection of the derived diagonal
$\Delta\colon{\underline{X}}\to{\underline{X}}\times_{\underline{S}}\underline{X}$.

If $R_{0}\to A_{0}$ is a map of discrete commutative rings, we note there are
isomorphisms (see e.g., [30, Corollary 1.18, Proposition 1.1.10])

(5.1) $\pi_{0}\operatorname{HH}(A_{0}/R_{0})\cong
A_{0},\qquad\pi_{1}\operatorname{HH}(A_{0}/R_{0})\cong\Omega^{1}_{A_{0}/R_{0}}.$

###  5.2. Log Hochschild homology

Inspired by the description of Hochschild homology as the derived self-
intersections of the diagonal and the relationship between the replete
diagonal and the log cotangent complex described in Proposition 3.17, we
propose the following definition (the reader is invited to compare with [44,
§13]):

######  Definition 5.3.

Let $(R,P)\to(A,M)$ be a map of derived pre-log rings. The _log Hochschild
homology_ ${\rm logHH}((A,M)/(R,P))$ is the derived commutative ring

$A\otimes_{(A\otimes_{R}A)_{M\oplus_{P}M}^{\mathrm{rep}}}A=A\otimes_{(A\otimes_{R}A)\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]}A.$

Let $X\to S$ be the map of affine pre-log derived schemes corresponding to
$(R,P)\to(A,M)$. Then log Hochschild homology ${\rm logHH}(X/S)$ is by
definition the affine derived scheme

$\underline{X\times_{X\times_{S}X}^{\mathrm{rep}}X}$

underlying the derived self-intersections
$X\times_{X\times_{S}X}^{\mathrm{rep}}X$ of the replete diagonal $X\to
X\times_{S}^{\mathrm{rep}}X$.

######  Proposition 5.4.

Let $(R,P)\to(A,M)\to(B,N)$ be maps of derived pre-log rings. There is an
equivalence of derived commutative rings

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

###### Proof.

It suffices to prove that

(5.2)
$A\otimes_{(A\otimes_{R}A)^{\mathrm{rep}}_{M\oplus_{P}M}}(B\otimes_{R}B)^{\mathrm{rep}}_{N\oplus_{P}N}\simeq(B\otimes_{A}B)^{\mathrm{rep}}_{N\oplus_{M}N}.$

Since $A\otimes_{A\otimes_{R}A}(B\otimes_{R}B)\simeq B\otimes_{A}B$ and
${\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[N\oplus_{P}N]\simeq{\mathbb{Z}}[N\oplus_{M}N]$,
(5.2) follows from the equivalences

$\displaystyle{\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]}{\mathbb{Z}}[(N\oplus_{P}N)^{\mathrm{rep}}]$
$\displaystyle\simeq{\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}[M\oplus(M^{\mathrm{gp}}/P^{\mathrm{gp}})]}{\mathbb{Z}}[N\oplus(N^{\mathrm{gp}}/P^{\mathrm{gp}})]$
$\displaystyle\simeq{\mathbb{Z}}[N\oplus(N^{\mathrm{gp}}/M^{\mathrm{gp}})]$
$\displaystyle\simeq{\mathbb{Z}}[(N\oplus_{M}N)^{\mathrm{rep}}]$

provided by Proposition 2.18. ∎

###  5.5. Comparison with Rognes’ definition of log $\operatorname{HH}$

In [44, Definition 3.21], Rognes introduced absolute log Hochschild homology
of discrete log rings. A direct generalization goes as follows: for a map
$(R,P)\to(A,M)$ of derived pre-log rings, let $S^{1}\oplus_{P}M$ denote the
tensor with $S^{1}$ in the $\infty$-category of derived commutative monoids
under $P$. Let $(S^{1}\oplus_{P}M)^{\mathrm{rep}}$ denote the repletion of the
augmentation $S^{1}\oplus_{P}M\to M$, and consider the pushout of the diagram

(5.3)
$S^{1}\otimes_{R}A\xleftarrow{}S^{1}\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]\simeq{\mathbb{Z}}[S^{1}\oplus_{P}M]\xrightarrow{}{\mathbb{Z}}[(S^{1}\oplus_{P}M)^{\mathrm{rep}}]$

in the $\infty$-category of derived commutative rings.

######  Proposition 5.6.

Let $(R,P)\to(A,M)$ be a map of pre-log derived rings. The pushout of the
diagram (5.3) is naturally equivalent to ${\rm logHH}((A,M)/(R,P))$.

###### Proof.

There are natural equivalences

$S^{1}\otimes_{R}A\simeq A\otimes_{A\otimes_{R}A}A\quad\text{and}\quad
S^{1}\otimes_{{\mathbb{Z}}[P]}{\mathbb{Z}}[M]\simeq{\mathbb{Z}}[M]\oplus_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[M].$

Moreover, we see that

${\mathbb{Z}}[(S^{1}\oplus_{P}M)^{\mathrm{rep}}]\simeq{\mathbb{Z}}[M]\otimes_{{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]}{\mathbb{Z}}[M]$

by applying Proposition 2.18 twice and the equivalence $B^{\rm
cy}_{P^{\mathrm{gp}}}(M^{\mathrm{gp}})/M^{\mathrm{gp}}\simeq
B(M^{\mathrm{gp}}/P^{\mathrm{gp}})$: Indeed, there are equivalences of derived
monoids

$B^{\rm cy}_{P}(M)^{\mathrm{rep}}\simeq M\oplus
B(M^{\mathrm{gp}}/P^{\mathrm{gp}})\simeq
M\oplus_{M\oplus(M^{\mathrm{gp}}/P^{\mathrm{gp}})}M\simeq
M\oplus_{(M\oplus_{P}M)^{\mathrm{rep}}}M.$

For the second equivalence we use that $B(-)$ is the suspension functor on the
category of derived monoids. Hence the result follows from commuting colimits
in the diagram

${A}$${A\otimes_{R}A}$${A}$${{\mathbb{Z}}[M]}$${{\mathbb{Z}}[M\oplus_{P}M]}$${{\mathbb{Z}}[M]}$${{\mathbb{Z}}[M]}$${{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]}$${{\mathbb{Z}}[M]}$$\scriptstyle{=}$$\scriptstyle{=}$

of derived rings: Forming the horizontal pushouts first gives Rognes’
definition, while forming the vertical pushouts first gives our formulation. ∎

###  5.7. Free pre-log algebras

Following Bhatt [6, §6] and Olsson [39, §8.2], we discuss free pre-log
algebras. Let $(R_{0},P_{0})$ be a discrete pre-log ring. The forgetful
functor

${\rm PreLog}_{(R_{0},P_{0})/}\to{\rm Set}\times{\rm Set}$

from pre-log $(R_{0},P_{0})$-algebras (sending $(A_{0},M_{0})$ to
$(M_{0},A_{0})$) admits a left adjoint $F_{(R_{0},P_{0})}$ given by

(5.4) $F_{(R_{0},P_{0})}(X,Y)=(R_{0}\langle X\sqcup
Y\rangle,P_{0}\oplus\langle X\rangle).$

The underlying commutative ring of $F_{(R_{0},P_{0})}(X,Y)$ is the polynomial
ring $R_{0}\langle X\sqcup Y\rangle$ (we use the bracket notation to avoid
confusion with the monoid ring construction). Moreover, its pre-log structure
is induced by the pre-log structure on $R_{0}$ and the canonical map from the
free commutative monoid on $X$ to $(R_{0}\langle X\sqcup Y\rangle,\cdot)$.
Notice that a free pre-log ring, viewed as a constant simplicial pre-log ring,
is cofibrant in the projective model structure.

######  Example 5.8.

Let $X=\\{x_{1},\dots,x_{r}\\},Y=\\{y_{1},\dots,y_{m}\\}$ and let $P_{0}$ be
the trivial monoid. Then the free pre-log ring (5.4) is the pre-log ring

$(R_{0}[x_{1},\dots,x_{r},y_{1},\dots,y_{m}],{\mathbb{N}}^{r})$

with the structure map $e_{i}\mapsto x_{i}$, where
$e_{i}=(0,\dots,1,\dots,0)$.

By definition, the log Kähler differentials of the canonical map
$(R_{0},P_{0})\to F_{(R_{0},P_{0})}(X,Y)$ are given by

(5.5) $\Omega^{1}_{F_{(R_{0},P_{0})}(X,Y)/(R_{0},P_{0})}=R_{0}\langle X\sqcup
Y\rangle\otimes_{\mathbb{Z}}(\bigoplus_{x\in X}{\mathbb{Z}}\\{d\hskip
0.70007pt\mathrm{log}(x)\\}\oplus\bigoplus_{y\in Y}{\mathbb{Z}}\\{{\rm
d}y\\}).$

###  5.9. Log Hochschild homology of free pre-log algebras

Let $(R_{0},P_{0})$ be a discrete pre-log ring and let $(A_{0},M_{0})$ be a
free pre-log algebra over $(R_{0},P_{0})$. Since $(A_{0},M_{0})$ is cofibrant
over $(R_{0},P_{0})$, no replacement is necessary when defining log Hochschild
homology ${\rm logHH}((A_{0},M_{0})/(R_{0},P_{0}))$. In particular, by virtue
of Proposition 5.6, ${\rm logHH}((A_{0},M_{0})/(R_{0},P_{0}))$ is naturally
equivalent to the point-set pushout of the diagram

(5.6) ${\rm HH}(A_{0}/R_{0})\xleftarrow{}{\mathbb{Z}}[B^{\rm
cy}_{P_{0}}(M_{0})]\xrightarrow{}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0})]$

of simplicial commutative rings. Here $B^{\rm cy}_{P_{0}}(M_{0})$ is the
(underived) cyclic bar construction of $M_{0}$ relative to $P_{0}$ and
$B^{\mathrm{rep}}_{P_{0}}(M_{0})$ is the (discrete) pullback of
$M_{0}\xrightarrow{}M_{0}^{\mathrm{gp}}\xleftarrow{}B^{\rm
cy}_{P_{0}^{\mathrm{gp}}}(M_{0}^{\mathrm{gp}})$.

We will consider these (underived) constructions for any map of discrete
commutative monoids $P_{0}\to M_{0}$ and apply the following results when
$M_{0}$ is free over $P_{0}$:

######  Lemma 5.10.

Let $P_{0}\to M_{0}$ be a map of discrete commutative monoids. There is an
isomorphism of derived commutative monoids

$M_{0}\oplus
B(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})\xrightarrow{\cong}B^{\mathrm{rep}}_{P_{0}}(M_{0}).$

In simplicial degree $q$, it is given by
$(m,[g_{1}],\dots,[g_{q}])\mapsto(m,(\gamma_{0}(m)(g_{1}\cdots
g_{q})^{-1},g_{1},\dots,g_{q})$.

###### Proof.

The proposed map is well-defined, as for different representatives of the
elements in $M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}$, the inversion used in
the map will cancel their difference. An inverse sends a $q$-simplex
$(m,(g_{0},\dots,g_{q}))$ to the element $(m,[g_{1}],\dots,[g_{q}])$. It is
straight-forward to check that this map respects the simplicial structure
maps.∎

######  Corollary 5.11.

Let $P_{0}\to M_{0}$ be a map of discrete commutative monoids. There are
isomorphisms of ${\mathbb{Z}}[M_{0}]$-modules

$\pi_{0}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0})]\cong{\mathbb{Z}}[M_{0}]\quad\text{
and
}\quad\pi_{1}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0})]\cong{\mathbb{Z}}[M_{0}]\otimes_{\mathbb{Z}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}).$

###### Proof.

By Lemma 5.10, the replete bar construction $B^{\mathrm{rep}}_{P_{0}}(M_{0})$
is levelwise isomorphic to $M_{0}\oplus
B(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})$. Hence the Moore complex
associated to ${\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0})]$ takes the form

$\cdots\xrightarrow{}{\mathbb{Z}}[M_{0}]\otimes_{\mathbb{Z}}{\mathbb{Z}}[M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}]^{\otimes
2}\xrightarrow{\partial_{2}}{\mathbb{Z}}[M_{0}]\otimes_{\mathbb{Z}}{\mathbb{Z}}[M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}]\xrightarrow{\partial_{1}=0}{\mathbb{Z}}[M_{0}]\to
0.$

This immediately gives the desired identification in degree $0$. In degree
$1$, we notice that the map

$({\mathbb{Z}}[M_{0}]\otimes_{{\mathbb{Z}}}{{\mathbb{Z}}[M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}}]})/\\!\\!\sim\xrightarrow{}{\mathbb{Z}}[M_{0}]\otimes_{{\mathbb{Z}}}(M_{0}^{\mathrm{gp}}/P_{0}^{\mathrm{gp}})$

sending $[m\otimes g]$ to $m\otimes g$ is a well-defined isomorphism, where
$\sim$ is the equivalence relation ${\mathbb{Z}}[M_{0}]$-linearly generated by
$m\otimes g_{2}-m\otimes g_{1}g_{2}+m\otimes g_{1}$. The source of this map is
by definition $H_{1}$ of the Moore complex of the replete bar construction. ∎

The following description of the log Hochschild homology of a free
$(R_{0},P_{0})$-algebra, which follows from Proposition 5.4, will prove
useful:

######  Lemma 5.12.

Let $(R_{0},P_{0})$ be a discrete pre-log ring. There is a natural equivalence
of derived commutative rings

${\rm logHH}((F_{(R_{0},P_{0})}(X,Y))/(R_{0},P_{0}))\simeq
R_{0}\otimes_{{\mathbb{Z}}}\operatorname{HH}({\mathbb{Z}}\langle
Y\rangle)\otimes_{{\mathbb{Z}}}{\rm logHH}(({\mathbb{Z}}\langle
X\rangle,\langle X\rangle)/({\mathbb{Z}},\\{1\\})).$

###  5.13. Recovering the log Kähler differentials from log Hochschild
homology

Let
$(f_{0},f_{0}^{\flat})\colon(R_{0},P_{0},\beta_{0})\to(A_{0},M_{0},\alpha_{0})$
be a map of discrete pre-log rings. Recall the definition of the log Kähler
differentials from Section 3.1. Lemma 5.12 gives the following:

######  Corollary 5.14.

Let $F_{(R_{0},P_{0})}(X,Y)$ be a free pre-log algebra. There is an
isomorphism of $R_{0}\langle X\sqcup Y\rangle$-modules

$\Phi_{1}\colon\Omega^{1}_{F_{(R_{0},P_{0})}(X,Y)/(R_{0},P_{0})}\to\pi_{1}{\rm
logHH}((F_{(R_{0},P_{0})}(X,Y))/(R_{0},P_{0})).$

###### Proof.

Since $F_{(R_{0},P_{0})}(X,Y)$ is free, we can model log Hochschild homology
by the pushout of (5.6). Apply Lemma 5.12 and consider the convergent ${\rm
Tor}$-spectral sequence

$E^{2}_{p,q}:={\rm
Tor}_{p}^{{\mathbb{Z}}}(\pi_{*}\operatorname{HH}^{R_{0}}(R_{0}\langle
Y\rangle),\pi_{*}{\mathbb{Z}}[B^{\mathrm{rep}}\langle
X\rangle])_{q}\implies\pi_{p+q}{\rm
logHH}((F_{(R_{0},P_{0})}(X,Y))/(R_{0},P_{0})).$

By Corollary 5.11, we obtain $E^{2}_{p,0}$ for $p>0$ and
$E^{2}_{0,1}\cong\Omega^{1}_{F_{(R_{0},P_{0})}(X,Y)/(R_{0},P_{0})}$. ∎

From this, we deduce the following more general result:

######  Proposition 5.15.

Let $(R_{0},P_{0})\to(A_{0},M_{0})$ be a map of discrete pre-log rings. There
is an isomorphism of $A_{0}$-modules

$\Phi_{1}\colon\Omega^{1}_{(A_{0},M_{0})/(R_{0},P_{0})}\to\pi_{1}\operatorname{logHH}((A_{0},M_{0})/(R_{0},P_{0})).$

###### Proof.

In view of Proposition 5.6, we model log Hochschild homology by taking a
cofibrant replacement $(A_{0}^{\rm cof},M_{0}^{\rm
cof})\xrightarrow{\simeq}(A_{0},M_{0})$, where $(A_{0}^{\rm cof},M_{0}^{\rm
cof})$ is a levelwise free simplicial $(R_{0},P_{0})$-algebra and applying the
construction (5.6) levelwise. As the resulting object
$\operatorname{logHH}((A_{0},M_{0})/(R_{0},P_{0}))$ is the diagonal of a
bisimplicial commutative ring, there is a convergent spectral sequence

$E^{2}_{p,q}:=\pi_{p}(\pi_{q}(B^{\rm cy}_{R_{0}}(A_{0}^{\rm
cof})\otimes_{{\mathbb{Z}}[B^{\rm cy}_{P_{0}}(M_{0}^{\rm
cof})]}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0}^{\rm
cof})]))\implies\pi_{p+q}\operatorname{logHH}((A_{0},M_{0})/(R_{0},P_{0})).$

Since $\pi_{0}{\mathbb{Z}}[B^{\rm cy}_{P_{0}}(M_{0}^{\rm
cof})]\cong\pi_{0}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0}^{\rm
cof})]\cong{\mathbb{Z}}[M_{0}^{\rm cof}]$ by Corollary 5.11, we have

$\pi_{0}(B^{\rm cy}_{R_{0}}(A_{0}^{\rm cof})\otimes_{{\mathbb{Z}}[B^{\rm
cy}_{P_{0}}(M_{0}^{\rm cof})]}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0}^{\rm
cof})])\cong A_{0}^{\rm cof}.$

Thus $E_{p,0}=0$ for all $p>0$. Let us compute $E^{2}_{0,1}$. Since
$(A_{0}^{\rm cof},M_{0}^{\rm cof})$ is levelwise free, Corollary 5.14 applies
to obtain

$\pi_{1}(B^{\rm cy}_{R_{0}}(A_{0}^{\rm cof})\otimes_{{\mathbb{Z}}[B^{\rm
cy}_{P_{0}}(M_{0}^{\rm cof})]}{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0}^{\rm
cof})]\cong\Omega^{1}_{(A^{\rm cof}_{0},M_{0}^{\rm
cof})/(R_{0},P_{0})}\xleftarrow{\simeq}{\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}.$

Consequently,
$E^{2}_{0,1}\cong\pi_{0}{\mathbb{L}}_{(A_{0},M_{0})/(R_{0},P_{0})}\cong\Omega^{1}_{(A_{0},M_{0})/(R_{0},P_{0})}$
[39, Lemma 8.9], as desired. ∎

Keeping track of the maps involved in constructing the map $\Phi_{1}$, we find
that it admits the explicit description

$\Phi_{1}(da)=[1\otimes a],\qquad\Phi_{1}(d\hskip
0.70007pt\mathrm{log}(m))=[1\otimes\gamma_{0}(m)].$

By abuse of notation we identify $[a\otimes b]$ in
$\pi_{1}\operatorname{HH}(A_{0}/R_{0})$ and $[1\otimes\gamma_{0}(m)]$ in
$\pi_{1}{{\mathbb{Z}}[B^{\mathrm{rep}}_{P_{0}}(M_{0})]}$ with their images in
$\pi_{1}\operatorname{logHH}((A_{0},M_{0})/(R_{0},P_{0}))$. The map $\Phi_{1}$
in Proposition 5.15 and the universal property of the exterior algebra give
rise to a map of graded commutative rings

(5.7)
$\Phi_{*}\colon\Omega^{*}_{(A_{0},M_{0})/(R_{0},P_{0})}\to\pi_{*}\operatorname{logHH}((A_{0},M_{0})/(R_{0},P_{0})).$

###  5.16. The log ${\rm HKR}$-filtration

We are ready to construct a logarithmic version of the ${\rm HKR}$-filtration
[36, Proposition IV.4.1]:

######  Theorem 5.17.

Let $(R,P)\to(A,M)$ be a map of derived pre-log rings. Then the relative log
Hochschild homology $\operatorname{logHH}((A,M)/(R,P))$ admits a descending
separated filtration with graded pieces of the form
$(\bigwedge\nolimits_{A}^{n}{\mathbb{L}}_{(A,M)/(R,P)})[n]$. In particular,
there is a strongly convergent spectral sequence

$E^{2}_{p,q}:=\pi_{p}(\bigwedge\nolimits\nolimits_{A}^{q}{\mathbb{L}}_{(A,M)/(R,P)})\implies\pi_{p+q}\operatorname{logHH}((A,M)/(R,P))$

relating the derived wedge powers of the log cotangent complex and log
Hochschild homology.

Theorem 5.17 is proven at the end of this section. As a corollary, we obtain a
version of the ${\rm HKR}$-theorem for log smooth log rings:

######  Corollary 5.18.

Let $(R_{0},P_{0})\to(A_{0},M_{0})$ be a derived formally log smooth map of
discrete pre-log rings. Then the map defined in (5.7) is an isomorphism of
graded commutative rings.

###### Proof.

By Proposition 4.5, the spectral sequence of Theorem 5.17 collapses to give
the desired isomorphism. ∎

We first establish the ${\rm HKR}$-theorem for free pre-log algebras and use
this to construct the logarithmic ${\rm HKR}$-filtration.

###  5.19. The log ${\rm HKR}$-theorem for free pre-log algebras

The general log ${\rm HKR}$-theorem for free pre-log algebras will reduce to
the following:

######  Lemma 5.20.

The map

$\Omega^{*}_{F_{(R_{0},P_{0})}(\\{x\\},\emptyset)/(R_{0},P_{0})}\xrightarrow{}\pi_{*}{\rm
logHH}(F_{(R_{0},P_{0})}(\\{x\\},\emptyset)/(R_{0},P_{0}))$

defined in (5.7) is an isomorphism of graded rings.

Stated differently, the log ${\rm HKR}$-theorem holds for free pre-log
algebras $(R_{0}\langle x\rangle,\langle x\rangle)$ on a single generator.

###### Proof.

By Lemma 5.10 and Lemma 5.12, there is an equivalence

${\rm logHH}(F_{(R_{0},P_{0})/(R_{0},P_{0})}(\\{x\\},\emptyset))\simeq
R_{0}\otimes_{{\mathbb{Z}}}{\rm logHH}({\mathbb{Z}}[x],\langle x\rangle)\cong
R_{0}[x]\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[B\langle x\rangle^{\mathrm{gp}}].$

The homotopy groups of ${\mathbb{Z}}[B\langle x\rangle^{\mathrm{gp}}]$ are the
${\rm Tor}$-groups ${\rm
Tor}^{{\mathbb{Z}}[x,x^{-1}]}_{*}({\mathbb{Z}},{\mathbb{Z}})$. Indeed, observe
that $B\langle x\rangle^{\mathrm{gp}}$ is the suspension of $\langle
x\rangle^{\mathrm{gp}}$ in the category of derived abelian groups. Hence,
since ${\mathbb{Z}}[-]$ commutes with homotopy pushouts, the derived
commutative ring ${\mathbb{Z}}[B\langle x\rangle^{\mathrm{gp}}]$ is equivalent
to the derived tensor product ${\mathbb{Z}}\otimes_{{\mathbb{Z}}[\langle
x\rangle^{\mathrm{gp}}]}^{\mathbb{L}}{\mathbb{Z}}$. Using the free resolution

$0\to{\mathbb{Z}}[x,x^{-1}]\xrightarrow{\cdot(x-1)}{\mathbb{Z}}[x,x^{-1}]\xrightarrow{}{\mathbb{Z}}\to
0,$

we find the ${\rm Tor}$-groups to be copies of $\mathbb{Z}$ concentrated in
degrees 0 and 1

so that the graded ring $\pi_{*}{\rm
logHH}(F_{(R_{0},P_{0})}(\\{x\\},\emptyset)/(R_{0},P_{0}))$ is concentrated as
$R_{0}[x]$ in degrees $0$ and $1$. By Proposition 5.15, there is a preferred
isomorphism

$\Phi_{1}\colon\Omega^{1}_{F_{(R_{0},P_{0})}(\langle
x\rangle,\emptyset)/(R_{0},P_{0})}\xrightarrow{\cong}\pi_{1}{\rm
logHH}(F_{(R_{0},P_{0})}(\langle x\rangle,\emptyset)/(R_{0},P_{0})).$

Since $\pi_{*}\operatorname{logHH}(F_{(R_{0},P_{0})}(\langle
x\rangle,\emptyset)/(R_{0},P_{0}))$ is concentrated in degrees $0$ and $1$,
this suffices to conclude that the map is an isomorphism. ∎

By commuting colimits, Lemma 5.12 implies the log ${\rm HKR}$-theorem for any
free pre-log ring $F_{(R,P)}(X,\emptyset)$. Combining this with the ordinary
${\rm HKR}$-theorem, we obtain:

######  Corollary 5.21.

Let $F_{(R_{0},P_{0})}(X,Y)$ be a free $(R_{0},P_{0})$-algebra. Then the map
(5.7) is an isomorphism of strictly commutative graded rings

$\Phi_{*}\colon\Omega^{*}_{F_{(R_{0},P_{0})}(X,Y)/(R_{0},P_{0})}\xrightarrow{\cong}\pi_{*}{\rm
logHH}(F_{(R_{0},P_{0})}(X,Y)/(R_{0},P_{0})).$

We can now deduce the log André–Quillen spectral sequence:

###### Proof of Theorem 5.17.

On free pre-log algebras, it is clear by Corollary 5.21 that the Whitehead
filtration of ${\rm logHH}((A,M)/(R,P))$ gives the result. Since both the log
cotangent complex and log Hochschild homology are in general left Kan extended
from free pre-log algebras, this also gives the filtration in general. Hence
we obtain a conditionally convergent spectral sequence

$E^{2}_{p,q}:=\pi_{p}(\bigwedge\nolimits\nolimits_{A}^{q}{\mathbb{L}}_{(A,M)/(R,P)})\implies\pi_{p+q}\operatorname{logHH}((A,M)/(R,P))$

concentrated in the first quadrant. Therefore it is strongly convergent, see
e.g. [16, Theorem 7.1] and the subsequent remark. ∎

######  Remark 5.22.

From a more classical perspective, Corollary 5.21 allows us to use the results
of Quillen [40] to obtain the log HKR-theorem. One key point is that Corollary
5.21 implies the simplicial augmentation ideal of

$(A\otimes_{R}A)_{(M\oplus_{P}M)}^{\mathrm{rep}}\to A$

is levelwise _quasi-regular_ in the sense of [40, Definition 6.10]. The
definition of $\operatorname{logHH}$ as derived self-intersections makes [40,
Theorem 6.3] applicable and the resulting spectral sequence converges. We also
observe that the spectral sequence can be obtained from the totalization of a
double complex, see e.g., [40, §8].

###  5.23. Base change for log Hochschild homology

The following is a generalization of Weibel–Geller’s [56] étale base change
formula for Hochschild homology:

######  Theorem 5.24.

Let $(R,P)$ be a derived pre-log ring. A map $(A,M)\to(B,N)$ of pre-log
$(R,P)$-algebras is derived formally log étale if and only if the canonical
map

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

is an equivalence.

######  Remark 5.25.

If $(R_{0},P_{0})\to(A_{0},M_{0})$ is log étale and integral, then
$\mathbb{L}_{(A_{0},M_{0})/(R_{0},P_{0})}$ is contractible by Proposition
4.6(2). On the other hand, Example 4.9 shows the base change formula does not
hold for a general log étale map of discrete commutative rings.

The next result makes use of the indecomposables functor discussed in Section
3.13.

######  Lemma 5.26.

Let $(R,P)\to(A,M)$ be a map of derived pre-log rings. There is a natural
equivalence of derived $A$-modules

$Q_{A}(\operatorname{logHH}((A,M)/(R,P)))\simeq{\mathbb{L}}_{(A,M)/(R,P)}[1].$

###### Proof.

Recall $\operatorname{logHH}((A,M)/(R,P))$ is the suspension of the augmented
simplicial commutative $A$-algebra
$(A\otimes_{R}A)\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}]$.
Since $Q_{A}$ is left Quillen, we obtain

$Q_{A}(\operatorname{logHH}((A,M)/(R,P)))\simeq
Q_{A}((A\otimes_{R}A)\otimes_{{\mathbb{Z}}[M\oplus_{P}M]}{\mathbb{Z}}[(M\oplus_{P}M)^{\mathrm{rep}}])[1].$

The result now follows from Proposition 3.17. ∎

###### Proof of Theorem 5.24.

If the base change map is an equivalence, then the induced map of
indecomposables is also an equivalence. By Lemma 5.26, this holds if and only
if there is an equivalence

(5.8)
$B\otimes_{A}{\mathbb{L}}_{(A,M)/(R,P)}[1]\to{\mathbb{L}}_{(B,N)/(R,P)}[1],$

implying that the cofiber ${\mathbb{L}}_{(B,N)/(A,M)}[1]$ is contractible.
Conversely, if ${\mathbb{L}}_{(B,N)/(A,M)}$ is contractible, the map (5.8) is
an equivalence, and so we obtain the base change formula for log Hochschild
homology by comparing the Quillen spectral sequences converging to
$B\otimes_{R}{\rm logHH}((A,M)/(R,P))$ and ${\rm logHH}((B,N)/(R,P))$. ∎

######  Corollary 5.27.

Let $(R,P)\to(A,M)$ be a derived formally log étale map of derived pre-log
rings. Then the unit map $A\to\operatorname{logHH}((A,M)/(R,P))$ is an
equivalence.

###### Proof.

This follows from Proposition 5.4 and Theorem 5.24. ∎

######  Proposition 5.28.

Let $R\to A$ be a tamely ramified finite extension of discrete valuation rings
with perfect residue fields, and let
$(f,f^{\flat})\colon(R,R-\\{0\\})\to(A,A-\\{0\\})$ be the induced map of log
rings. Then there is a canonical equivalence

$A\otimes_{R}\operatorname{logHH}(R,R-\\{0\\})\to\operatorname{logHH}(A,A-\\{0\\}).$

###### Proof.

Owing to [37, Proposition IV.3.1.19], $(f,f^{\flat})$ is log étale.
Furthermore, $(f,f^{\flat})$ is integral since $R-\\{0\\}$ is a valuative
monoid [37, Proposition I.4.6.3(5)]. Proposition 4.6(2) shows that
$(f,f^{\flat})$ is derived log étale, and we conclude by Theorem 5.24. ∎

##  6\. A recollection of the dividing Nisnevich topology

We want to understand the global properties of logarithmic Hochschild homology
and the André-Quillen spectral sequence. To that end, we consider the dividing
Nisnevich topology on log schemes introduced in [11]. For the reader’s
convenience we recall some of its basic properties. From now on, all schemes
and rings are underived.

###  6.1. Dividing Nisnevich cd-structure

See Section 1.15 for our notation on schemes and log schemes. If
$S\in\mathrm{lSch}$, let $\mathrm{lSm}/S$ be the category of log smooth
morphisms $X\to S$ in $\mathrm{lSch}$. Likewise, if $S\in\mathrm{Sch}$, we let
$\mathrm{SmlSm}/S$ be the category of log smooth morphisms $X\to S$ in
$\mathrm{lSch}$ such that $\underline{X}\to S$ is smooth ($\underline{X}$ is
the underlying scheme of $X$).

######  Definition 6.2.

The _Zariski topology on $\mathrm{lSch}$_ (resp.  _strict étale topology on
$\mathrm{lSch}$_) is the Grothendieck topology generated by strict morphisms
$X\to S$ in $\mathrm{lSch}$ such that $\underline{X}\to\underline{S}$ is a
Zariski cover (resp. a strict étale cover).

######  Definition 6.3.

A morphism $f\colon X\to S$ in $\mathrm{lSch}$ is called a _dividing cover_ if
it is a surjective proper log étale monomorphism. For details, we refer to
[11, §A.11].

We refer to [55] for Voevodsky’s notion of cd-structures. The following cd-
structures are defined in [11].

######  Definition 6.4.

  1. (i)

The _strict Nisnevich cd-structure_ on $\mathrm{lSch}$ is the collection of
cartesian squares

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such that the underlying square $\underline{Q}$ of schemes is a Nisnevich
distinguished square and $f$, $f^{\prime}$, $g$, and $g^{\prime}$ are strict
maps.

  2. (ii)

The _dividing cd-structure_ on $\mathrm{lSch}$ is the collection of cartesian
squares of the form (6.1) such that $X^{\prime}=S^{\prime}=\emptyset$ and $f$
is a dividing cover.

  3. (iii)

The _dividing Nisnevich cd-structure_ on $\mathrm{lSch}$ is the union of the
strict Nisnevich and dividing cd-structures.

The topology associated with the strict Nisnevich (resp. dividing, dividing
Nisnevich) cd-structure is called the _strict Nisnevich_ (resp.  _dividing_ ,
_dividing Nisnevich_) topology.

###  6.5. Comparison of sheaves

Let us review the comparison lemma [2, Théorème III.4.1] and its
$\infty$-categorical generalization. As usual, $\Delta$ denotes the simplex
category. We write $\mathcal{S}\mathrm{pc}$ for the $\infty$-category of
spaces, and let $\mathcal{D}(\mathrm{Ab})$ denote the derived
$\infty$-category of abelian groups $\mathrm{Ab}$.

######  Definition 6.6.

Suppose $\mathcal{V}$ is an $\infty$-category admitting colimits and
$\mathcal{C}$ is a category. Let
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ be the $\infty$-category
$\mathrm{Fun}(\mathrm{N}(\mathcal{C})^{op},\mathcal{V})$, where
$\mathrm{N}(\mathcal{C})$ denotes the nerve of $\mathcal{C}$.

Suppose $\mathscr{X}$ is a simplicial object in $\mathcal{C}$. For
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$, we define

$\lvert\mathcal{F}(\mathscr{X})\rvert:=\mathop{\mathrm{colim}}_{n\in\Delta}\mathcal{F}(\mathscr{X}_{n}).$

If $t$ is a topology on $\mathcal{C}$, let
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{V})$ be the $\infty$-category
of hypercomplete sheaves, i.e., the full subcategory of
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ consisting of $\mathcal{F}$
such that the naturally induced map

$\mathcal{F}(X)\to\lvert\mathcal{F}(\mathscr{X})\rvert$

is an equivalence for every $t$-hypercover $\mathscr{X}\to X$.

######  Example 6.7.

The $\infty$-category
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{S}\mathrm{pc})$ is equivalent
to the underlying $\infty$-category of Jardine’s $t$-local projective model
structure on $\mathrm{sPsh}(\mathcal{C})$, and likewise for
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{D}(\mathrm{Ab}))$ and chain
complexes of presheaves of abelian groups
$\mathrm{Ch}(\mathrm{Psh}(\mathcal{C},\mathbb{Z}))$).

Suppose $\alpha\colon\mathcal{C}\to\mathcal{D}$ is a fully faithful functor of
categories and $t$ is a topology on $\mathcal{D}$. Consider the topology on
$\mathcal{C}$ induced by $t$. Owing to [2, Proposition III.1.2] the functor

$\alpha^{*}\colon\mathrm{Shv}_{t}(\mathcal{D})\to\mathrm{Shv}_{t}(\mathcal{C})$

sending $\mathcal{F}\in\mathrm{Shv}_{t}(\mathcal{D})$ to
$(X\in\mathcal{C})\mapsto\mathcal{F}(\alpha(X))$ admits a left adjoint

$\alpha_{!}^{t}\colon\mathrm{Shv}_{t}(\mathcal{C})\to\mathrm{Shv}_{t}(\mathcal{D}).$

Similarly, we write $\alpha_{!}^{t}$ for the left adjoints of the functors

$\alpha^{*}\colon\mathcal{S}\mathrm{hv}_{t}(\mathcal{D},\mathcal{S}\mathrm{pc})\to\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{S}\mathrm{pc})\text{
and
}\alpha^{*}\colon\mathcal{S}\mathrm{hv}_{t}(\mathcal{D},\mathcal{D}(\mathrm{Ab}))\to\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{D}(\mathrm{Ab})).$

######  Proposition 6.8.

Suppose every object of $\mathcal{D}$ admits a $t$-cover in $\mathcal{C}$.
Then there are equivalences

$\displaystyle\alpha^{*}\colon$
$\displaystyle\mathrm{Shv}_{t}(\mathcal{D})\xrightarrow{\simeq}\mathrm{Shv}_{t}(\mathcal{C}),$
$\displaystyle\alpha^{*}\colon$
$\displaystyle\mathcal{S}\mathrm{hv}_{t}(\mathcal{D},\mathcal{S}\mathrm{pc})\xrightarrow{\simeq}\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{S}\mathrm{pc}),$
$\displaystyle\alpha^{*}\colon$
$\displaystyle\mathcal{S}\mathrm{hv}_{t}(\mathcal{D},\mathcal{D}(\mathrm{Ab}))\xrightarrow{\simeq}\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{D}(\mathrm{Ab})).$

###### Proof.

The first $\alpha^{*}$ is an equivalence by the implication
(i)$\Rightarrow$(ii) in [2, Théorème III.4.1]. For the second one, consider
the Joyal $t$-local model structure on the category of simplicial sheaves
$\mathrm{sShv}_{t}(\mathcal{C})\simeq\mathrm{sShv}_{t}(\mathcal{D})$. Since we
have left Quillen equivalences

$\mathrm{sPsh}_{t}(\mathcal{C})\to\mathrm{sShv}_{t}(\mathcal{C})\text{ and
}\mathrm{sPsh}_{t}(\mathcal{D})\to\mathrm{sShv}_{t}(\mathcal{D}),$

we deduce the second equivalence. For the third one, use [3, Proposition
4.4.56]. ∎

######  Definition 6.9.

For a fan $\Sigma$, let $\mathbb{T}_{\Sigma}$ be the fs log scheme $Z$ whose
underlying scheme is the toric variety associated with $\Sigma$ and whose log
structure is the compactifying log structure associated with the torus
embedding.

For an fs monoid $P$ such that $P^{\mathrm{gp}}$ is torsion-free, let
$\operatorname{Spec}(P)$ denote the associated fan whose only maximal cone is
the dual monoid of $P$.

Let $\mathrm{lFan}$ be the full subcategory of $\mathrm{lSch}$ consisting of
disjoint unions $\amalg_{i\in I}X_{i}$ such that each $X_{i}$ admits a strict
morphism $X\to\mathbb{T}_{\Sigma_{i}}$ with a fan $\Sigma_{i}$. Let
$\eta\colon\mathrm{lFan}\to\mathrm{lSch}$ be the inclusion functor.

######  Lemma 6.10.

If $S\in\mathrm{lSch}$, there is a naturally induced equivalence

$\eta^{*}\colon\mathrm{Shv}_{sNis}(\mathrm{lSch}/S)\xrightarrow{\simeq}\mathrm{Shv}_{sNis}(\mathrm{lFan}/S),$

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

###### Proof.

Every object of $\mathrm{lSch}/S$ admits a strict Nisnevich cover in
$\mathrm{lFan}/S$ according to [37, Proposition II.2.3.7]. Proposition 6.8
finishes the proof. ∎

###  6.11. Dividing Nisnevich descent property

We refer to [11, Definitions 3.3.22, 3.4.2] and [55] for the definitions of
quasi-bounded, squareable, bounded, complete, and regular cd-structures.

######  Proposition 6.12.

Suppose $P$ is a quasi-bounded, regular, and squareable cd-structure on a
category $\mathcal{C}$ with an initial object. Let $t_{P}$ be the topology on
$\mathcal{C}$ associated with $P$. Then
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{S}\mathrm{pc})$
satisfies $t_{P}$-descent if and only if $\mathcal{F}(Q)$ is cocartesian for
every $P$-distinguished square $Q$.

###### Proof.

This is shown in [55, Proposition 3.8(2), (3)] if $P$ is a bounded, complete,
and regular cd-structure. The same proof works in over setting by replacing
[55, Lemma 3.5] with [11, Lemma 3.4.10]. ∎

######  Lemma 6.13.

Suppose $\mathcal{V}$ is an $\infty$-category admitting colimits and
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathrm{lSch}/S,\mathcal{V})$, where
$S\in\mathrm{lSch}$. Then $\mathcal{F}$ satisfies dividing descent if and only
if $\mathcal{F}(X)\to\mathcal{F}(Y)$ is an equivalence for every dividing
cover $Y\to X$ in $\mathrm{lSch}/S$. The same result holds for
$\mathrm{lFan}/S$.

###### Proof.

We focus on $\mathrm{lSch}/S$ since the proofs are similar. Suppose
$\mathcal{F}$ satisfies dividing descent. For a dividing cover $f\colon Y\to
X$ in $\mathrm{lSch}/S$, the Čech nerve associated with $f$ is isomorphic to
the constant simplicial scheme $Y$. The above shows that
$\mathcal{F}(X)\to\mathcal{F}(Y)$ is an equivalence. Conversely, suppose
$\mathcal{F}(X)\to\mathcal{F}(Y)$ is an equivalence for every dividing cover
$Y\to X$ in $\mathrm{lSch}/S$. If $\mathscr{X}\to X$ is a dividing hypercover,
then $\mathscr{X}_{i}\to X$ is a dividing cover for every integer $i\geq 0$.
Hence $\mathcal{F}(X)\to\lvert\mathcal{F}(\mathscr{X})\rvert$ is an
equivalence, i.e., $\mathcal{F}$ satisfies dividing descent. ∎

######  Proposition 6.14.

Suppose $\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathrm{lSch}/S,\mathcal{V})$,
where $S\in\mathrm{lSch}$, and $\mathcal{V}=\mathcal{S}\mathrm{pc}$ or
$\mathcal{D}(\mathrm{Ab})$. Then $\mathcal{F}$ satisfies dividing Nisnevich
descent if and only if $\mathcal{F}$ satisfies both strict Nisnevich descent
and dividing descent.

###### Proof.

Due to Proposition 6.12 and [11, Proposition 3.3.30], $\mathcal{F}$ satisfies
$t$-descent if and only if $\mathcal{F}(Q)$ is cocartesian for every
$t$-distinguished square $Q$ for $t=sNis,div,dNis$ when
$\mathcal{V}=\mathcal{S}\mathrm{pc}$. The same holds by [11, Proposition
3.4.11] when $\mathcal{V}=\mathcal{D}(\mathrm{Ab})$. This implies the claim
since the union of the strict Nisnevich cd-structure and the dividing cd-
structure is the dividing Nisnevich cd-structure. ∎

###  6.15. Dividing invariance of Hodge cohomology

Owing to [11, (9.1.1)], for all $X\in\mathrm{Sch}$, quasi-coherent sheaf
$\mathcal{F}$ on $X$, and integer $i$, we have isomorphisms

(6.2) $H_{Zar}^{i}(X,\mathcal{F})\cong H_{sNis}^{i}(X,\mathcal{F})\cong
H_{s{\acute{e}t}}^{i}(X,\mathcal{F}).$

If $X\in\mathrm{lSch}$, $t=Zar,sNis,s{\acute{e}t}$, we have a canonical
isomorphism

$H_{t}^{i}(\underline{X},\mathcal{F})\cong H_{t}^{i}(X,\mathcal{F}).$

In fact, cohomology groups of quasi-coherent modules are invariant under
dividing covers.

######  Theorem 6.16.

Let $f\colon Y\to X$ be a dividing cover in $\mathrm{lSch}$. Then the functor

$\underline{f}^{*}\colon\mathrm{D}^{b}(\mathrm{Coh}(\underline{X}))\to\mathrm{D}^{b}(\mathrm{Coh}(\underline{Y}))$

of the bounded derived categories of coherent sheaves naturally induced by the
underlying morphism $\underline{f}\colon\underline{Y}\to\underline{X}$ is
fully faithful. In particular, the naturally induced homomorphism

(6.3)
$\mathbf{H}_{Zar}^{i}(\underline{X},\mathcal{F})\to\mathbf{H}_{Zar}^{i}(\underline{Y},f^{*}\mathcal{F})$

is an isomorphism for all
$\mathcal{F}\in\mathrm{D}^{b}(\mathrm{Coh}(\underline{X}))$ and integers $i$.

We prove Theorem 6.16 at the end of this section. It improves on [11,
Proposition 9.2.4], where we assumed that $X$ and $Y$ are log smooth over a
field.

######  Corollary 6.17.

Suppose $S\in\mathrm{lSch}$. Then

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

satisfies dividing Nisnevich descent.

###### Proof.

Owing to (6.2), it suffices to show that the naturally induced homomorphism

$f^{*}\colon H_{Zar}^{i}(X,\Omega_{X/S}^{d})\to
H_{Zar}^{i}(Y,\Omega_{Y/S}^{d})$

is an isomorphism for all dividing covers $Y\to X$ in $\mathrm{lSch}/S$ and
integers $d,i\geq 0$.

We may assume $d>0$ due to Theorem 6.16. Corollary IV.3.2.4(1) in [37] shows
there is an isomorphism $f^{*}\Omega_{X/S}^{1}\cong\Omega_{Y/S}^{1}$. Hence we
have $f^{*}\Omega_{X/S}^{d}\cong\Omega_{Y/S}^{d}$. To conclude, we apply
Theorem 6.16 to see that the homomorphism

$\mathrm{Hom}_{\mathrm{D}^{b}(\mathrm{Coh}(\underline{X}))}(\mathcal{O}_{X},\Omega_{X/S}^{d}[i])\to\mathrm{Hom}_{\mathrm{D}^{b}(\mathrm{Coh}(\underline{Y}))}(\mathcal{O}_{Y},\Omega_{Y/S}^{d}[i])$

is an isomorphism for all integers $i$. ∎

A fan $\Sigma$ has a lattice $N_{\Sigma}$ and a dual lattice $M_{\Sigma}$. We
let $N_{\Sigma,\mathbb{R}}$ be shorthand for the tensor product
$N_{\Sigma}\otimes\mathbb{R}$. Let $\lvert\Sigma\rvert$ denote the support of
$\Sigma$, which is a closed subset of $N_{\Sigma,\mathbb{R}}$.

######  Proposition 6.18.

Let $A$ be a commutative ring. For a fan $\Sigma$ we set
$X:=\operatorname{Spec}(A)\times\mathbb{T}_{\Sigma}$ and
$Z_{m}:=\\{x\in\Sigma:\langle m,x\rangle\geq 0\\}$ for $m\in M_{\Sigma}$. If
$i\geq 0$, there is a canonical isomorphism

$H_{Zar}^{i}(X,\mathcal{O}_{X})\cong\bigoplus_{m\in
M_{\Sigma}}H_{Z_{m}}^{i}(\lvert\Sigma\rvert,A).$

###### Proof.

If $A$ is a field, this is a special case of [17, Theorem 7.2]. We argue
similarly for a general commutative ring $A$. First, consider the case when
$\Sigma$ consists of a single cone $\sigma$ with dual monoid $P$. Then there
is an isomorphism $X\simeq\operatorname{Spec}(A)\times\mathbb{A}_{P}$. Since
$\underline{X}$ is affine, we have the vanishing
$H_{Zar}^{i}(X,\mathcal{O}_{X})\cong 0$ for $i>0$ and also
$H_{Zar}^{0}(X,\mathcal{O}_{X})\cong A[P]$. On the other hand, there is a long
exact sequence

$\cdots\to H^{i-1}(\lvert\Sigma\rvert-Z_{m},A)\to
H^{i}_{Z_{m}}(\lvert\Sigma\rvert,A)\to H^{i}(\lvert\Sigma\rvert,A)\to
H^{i}(\lvert\Sigma\rvert-Z_{m},A)\to\cdots.$

Since $|\Sigma|$ is convex, it is contractible. Furthermore, $|\Sigma|-Z_{m}$
is contractible (resp. empty) if and only if $m\in P$ (resp. $m\notin P$).
When combined with the above, we deduce

$H_{Z_{m}}^{i}(\lvert\Sigma\rvert,A)\cong\left\\{\begin{array}[]{ll}A&\text{if
}i=0\text{ and }m\in P,\\\ 0&\text{otherwise}.\end{array}\right.$

For general $\Sigma$, let $\sigma_{1},\ldots,\sigma_{n}$ be the maximal cones
of $\Sigma$, and let $P_{1},\ldots,P_{n}$ be their dual cones. We set
$U_{i}:=\operatorname{Spec}(A)\times\mathbb{A}_{P_{i}}$ for simplicity of
notation. The family $\\{U_{1},\ldots,U_{n}\\}$ is a Zariski covering of $X$,
and the family $\\{\lvert\sigma_{1}\rvert,\ldots,\lvert\sigma_{n}\rvert\\}$ is
a closed covering of $\lvert\Sigma\rvert$. For all integers $1\leq
i_{1}<\ldots<i_{r}\leq n$, the intersection of
$\sigma_{i_{1}},\ldots,\sigma_{i_{r}}$ is a cone. What we have established
above for the case of cones produces an isomorphism of Čech complexes

${\cdots}$${\bigoplus_{1\leq i,j\leq n}\Gamma(U_{i}\cap
U_{j},\mathcal{O}_{U_{i}\cap U_{j}})}$${\bigoplus_{1\leq i\leq
n}\Gamma(U_{i},\mathcal{O}_{U_{i}})}$${\cdots}$${\bigoplus_{1\leq i,j\leq
n}\bigoplus_{m\in
M_{\Sigma}}\Gamma_{Z_{m}\cap\lvert\sigma_{i}\cap\sigma_{j}\rvert}(|\sigma_{i}\cap\sigma_{j}|,A)}$${\bigoplus_{1\leq
i\leq n}\bigoplus_{m\in
M_{\Sigma}}\Gamma_{Z_{m}\cap\lvert\sigma_{i}\rvert}(\lvert\sigma_{i}\rvert,A).}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Taking the $i$-th cohomology groups of the horizontal rows finishes the proof.
∎

######  Proposition 6.19.

Suppose $S$ is a scheme and $\theta\colon\Sigma^{\prime}\to\Sigma$ is a
subdivision of fans. Let $f\colon S\times\mathbb{T}_{\Sigma^{\prime}}\to
S\times\mathbb{T}_{\Sigma}$ be the naturally induced morphism. Then the
homomorphism

$f^{*}\colon
H_{Zar}^{i}(S\times\mathbb{T}_{\Sigma},\mathcal{O}_{S\times\mathbb{T}_{\Sigma}})\to
H_{Zar}^{i}(S\times\mathbb{T}_{\Sigma^{\prime}},\mathcal{O}_{S\times\mathbb{T}_{\Sigma^{\prime}}})$

is an isomorphism for all integers $i$.

###### Proof.

The question is Zariski local on $S$ so we may assume it is affine.
Proposition 6.18 finishes the proof since
$\lvert\Sigma\rvert=\lvert\Sigma^{\prime}\rvert$ ∎

######  Proposition 6.20.

There is a canonical equivalence

$\mathcal{O}_{S\times\mathbb{T}_{\Sigma}}\xrightarrow{ad}R_{Zar}f_{*}f^{*}\mathcal{O}_{S\times\mathbb{T}_{\Sigma}}.$

###### Proof.

We may assume $S$ is affine since the assertion is Zariski local. Let $\sigma$
be a cone of $\Sigma$, and let $\Delta^{\prime}$ be the subfan of
$\Sigma^{\prime}$ consisting of cones $\sigma^{\prime}\in\Sigma^{\prime}$ such
that $\theta(\sigma^{\prime})\subset\sigma$. For all integers $i$, the
restriction

$R_{Zar}^{i}f_{*}f^{*}\mathcal{O}_{S\times\mathbb{T}_{\Sigma}}|_{S\times\mathbb{T}_{\sigma}}$

is the sheaf associated to the module
$H_{Zar}^{i}(S\times\mathbb{T}_{\Delta^{\prime}},\mathcal{O}_{S\times\mathbb{T}_{\Delta^{\prime}}})$.
Since the $\mathbb{T}_{\sigma}$’s, for all cones $\sigma$ of $\Sigma$, form a
Zariski cover of $\mathbb{T}_{\Sigma}$, we only need to show there is a
naturally induced isomorphism

$H_{Zar}^{i}(S\times\mathbb{T}_{\sigma},\mathcal{O}_{S\times\mathbb{T}_{\sigma}})\to
H_{Zar}^{i}(S\times\mathbb{T}_{\Delta^{\prime}},\mathcal{O}_{S\times\mathbb{T}_{\Delta^{\prime}}}).$

This claim is a special case of Proposition 6.19. ∎

###### Proof of Theorem 6.16.

The question is Zariski local on $X$. We may assume $X$ has an fs chart $P$
with $P$ sharp, see [37, Proposition II.2.3.7]. By [11, Proposition A.11.5],
there is a toric subdivision $\Sigma\to\operatorname{Spec}(P)$ and a naturally
induced isomorphism

$Y\times_{\mathbb{A}_{P}}\mathbb{T}_{\Sigma}\xrightarrow{\cong}X\times_{\mathbb{A}_{P}}\mathbb{T}_{\Sigma}.$

Due to [11, Proposition A.11.3], there is a finite Zariski cover
$\\{U_{i}\\}_{i\in I}$ of $Y$ such that each $U_{i}\to X$ admits a chart $P\to
Q_{i}$ for which $P^{\mathrm{gp}}\to Q_{i}^{\mathrm{gp}}$ is an isomorphism.
Then
$\Delta_{i}:=\Sigma\times_{\operatorname{Spec}(P)}\operatorname{Spec}(Q_{i})$
is a subdivision of $\operatorname{Spec}(Q_{i})$. We need to show the claim
for the projections
$U_{i}\times_{\mathbb{A}_{Q_{i}}}\mathbb{T}_{\Delta_{i}}\to U_{i}$ and
$X\times_{\mathbb{A}_{P}}\mathbb{T}_{\Sigma}\to X$. These two cases have only
notational differences, so we focus on the second case. In the said case,
there is a cartesian square of fs log schemes

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