2.1. Dieudonné algebras, saturation and V-completion

First recall the décalage operator.Footnote ⁴

Definition 2.1. Let R be a ring and $\mu \in R$ a nonzero divisor. Let $(M^{\ast }, d)$ be a cochain complex of $\mu $ -torsion-free R-modules; then $(\eta _\mu M)^{\ast } \subset M^*[\frac {1}{\mu }]$ is defined to be the subcomplex given by

$$ \begin{align*}(\eta_\mu M)^i = \{ x \in \mu^i M^i : dx \in \mu^{i+1} M^{i+1} \}.\end{align*} $$

$\eta _\mu $ kills $\mu $ -torsion in the cohomology. More precisely, one has

$$ \begin{align*}H^i(\eta_\mu M^*) \cong H^i(M^*)/H^i(M^*)[\mu].\end{align*} $$

2.1.1. Dieudonné complexes and Dieudonné algebras

Definition 2.2. A Dieudonné algebra is a triple $(A^{\ast }, d, F)$ where

• • $(A^{\ast } = \underset {i \ge 0}{\bigoplus } A^i, d)$ is a cdga concentrated in nonnegative degrees

• • $F: A^{\ast } \rightarrow A^{\ast }$ is a graded algebra map satisfying the conditions $d F (x) = p F (dx)$ for all $x \in A^*$ , and $F (x) \equiv x^p \mod p $ for all $x \in A^0$ .

More generally, a Dieudonné complex is a triple $(M^{\ast }, d, F)$ consisting of a cochain complex $(M^*, d)$ together with a Frobenius map $F: M^\ast \rightarrow M^\ast $ satisfying $dF = p Fd$ .

Definition 2.3. Let $M^{\ast } =(M^{\ast }, d, F)$ be a p-torsion-free Dieudonné complex; then F determines a map of cochain complexes $\phi _F: M^{\ast } \rightarrow \eta _p M^{\ast }$ by sending $x \mapsto p^n F(x)$ for $x \in M^n$ . A Dieudonné complex $M^{\ast }$ is saturated if it is p-torsion free and $\phi _F$ is an isomorphism.

Remark 2.4. From the definition above, it follows by applying iterations of $\eta _p$ (see Subsection 2.1.3) that, for a saturated Dieudonné complex, we have

$$ \begin{align*}d^{-1} (p^n M^{i+1}) = F^n (M^i)\end{align*} $$

for all $n \ge 1$ . As remarked in the Introduction, this is one of the essential features of de Rham–Witt complexes constructed in [Reference Illusie10] (and its logarithmic variant in [Reference Hyodo and Kato9]).

Notation 2.5. A Dieudonné algebra is saturated if its underlying Dieudonné complex is saturated. The category of Dieudonné algebras (respectively the full subcategory spanned by saturated algebras) is denoted by $\text {DA}$ (respectively $\text {DA}_{\mathfrak {sat}}$ ). Morphisms between two Dieudonné algebras are morphisms between cdgas compatible with the Frobenius maps.

2.1.2. Verschiebung

Consider a saturated Dieudonné complex $M^{\ast }$ . For each $i \in \mathbb {Z}$ , the composed map

$$ \begin{align*}\phi_F: M^i \xrightarrow{\: \: F \: \: } \{x \in M^i: dx \in p M^{i+1}\} \xrightarrow{ \times p^i} (\eta_p M)^i\end{align*} $$

is an isomorphism; hence, F is injective and $F(M^{\ast })$ contains $p M^{\ast }$ . Therefore, for each $x \in M^i$ , there is a unique element $V x \in M^i$ such that $F(Vx) = px$ . This defines the Verschiebung map

$$ \begin{align*}V: M^i \longrightarrow M^i.\end{align*} $$

It is straightforward to check that

$$ \begin{align*}F V = V F = p, \quad F d V = d, \quad V d = p \; d V, \quad x Vy = V(Fx \cdot y).\end{align*} $$

The saturated condition imposes strong restrictions on differential, Frobenius and Verschiebung.

Lemma 2.6. Suppose that $M^\ast $ is saturated; then for any $r \ge 1$ , we have

1. (1) $d^{-1} (V^r M^{i+1}) = F^r (M^i) = d^{-1} (p^r M^{i+1})$ .

2. (2) If $F x \in V^r M^i$ , then $x \in V^r M^i$ .

Proof. For (1) it suffices to show that the inclusion

(2) $$ \begin{align} d^{-1} (p^r M^{i+1}) \subset d^{-1} (V^r M^{i+1}) \end{align} $$

is an equality. To this end, suppose that $x \in M^i$ is an element satisfying $d x = V^r y$ for some $y \in M^{i+1}$ ; then we have

$$ \begin{align*}d (F^r x) = p^r F^r dx = p^{2r} y.\end{align*} $$

Thus, $F^r x = F^{2r} z$ for some $z \in M^i$ by the saturated assumption. Because F is injective, we know that $x = F^r z$ and hence $dx \in p^r M^{i+1}$ . For (2), suppose that $Fx = V^r y$ for some $y \in M^i$ ; then

$$ \begin{align*}dy = F^{r} dV^r y = p F^{r+1} d x,\end{align*} $$

so $y = F z$ for some $z \in M^{i}$ ; thus, $F x = F V^r z$ and it follows that $x = V^r z$ .

2.1.3. Saturation

Suppose that $A^{\ast } \in \text {AD}$ is a p-torsion-free Dieudonné algebra; then $(\eta _p A)^{\ast }$ with its inherited differential and Frobenius structure is again a Dieudonné algebra. The only nontautological point to check is that for any $x \in (\eta _p A)^0$ , $F(x) = x^p + p y$ for some $y \in (\eta _p A)^0$ (not just in $A^0$ ). The map $\phi _F: A^\ast \rightarrow (\eta _p A)^{\ast }$ is a morphism of Dieudonné algebras.

The inclusion functor $\text {AD}_{\mathfrak {sat}} \hookrightarrow \text {AD}$ admits a left adjoint, $A^{\ast } \mapsto A_{\mathfrak {sat}}^{\ast } $ , which is called the saturation of $A^*$ in [Reference Bhatt, Lurie and Mathew2] and is described as follows. Suppose that $A^*$ is p-torsion free (replacing $A^{\ast } \in \text {AD}$ by its p-torsion-free quotient if necessary); we then define $ A_{\mathfrak {sat}}^{\ast } $ to be the direct limit of

$$ \begin{align*}A^{\ast} \xrightarrow{\phi_F} (\eta_p A)^{\ast} \xrightarrow{\eta_p(\phi_F)} (\eta_p \eta_p A)^{\ast} \xrightarrow{\eta_p^2(\phi_F)} (\eta_p^3 A)^{\ast} \rightarrow \cdots\end{align*} $$

By the discussion above, the saturation $A^{\ast }_{\mathfrak {sat}}$ inherits the structure of a Dieudonné algebra, and the natural map $A^{\ast } \rightarrow A^{\ast }_{\mathfrak {sat}}$ is a morphism of Dieudonné algebras.

2.1.4. Strict Dieudonné algebras and V-completion

Let $M^{\ast }$ be a saturated Dieudonné complex. The V-filtrationFootnote ⁵ of $M^\ast $ is given by

(3) $$ \begin{align} \text{Fil}^r_V M^i = V^r M^i + d V^r M^{i - 1}. \end{align} $$

For each $r \ge 1$ , we form the quotient

$$ \begin{align*}W_r (M^{\ast}):= M^{\ast}/(V^r M^{\ast} + d V^{r} M^{\ast}).\end{align*} $$

If $A^\ast $ is a saturated Dieudonné algebra, then each $W_r (A^\ast )$ is a cdga because $\text {Fil}^r_V = V^r A^{\ast } + d V^{r} A^{\ast }$ is a differential graded ideal. The V-completion of $M^{\ast }$ is defined to be the limit

$$ \begin{align*}W (M^{\ast}) := \varprojlim W_r M^{\ast}\end{align*} $$

along the natural projection maps $R: W_r M^{\ast } \rightarrow W_{r-1} M^{\ast }$ . It is easy to check that the Frobenius and Verschiebung maps F and V on $M^{\ast }$ induce maps

• • $F: W_r (M^{\ast }) \rightarrow W_{r-1} (M^{\ast })$ and

• • $V: W_r (M^{\ast }) \rightarrow W_{r+1} (M^{\ast })$

on the quotients.

Definition 2.7. A saturated Dieudonné algebra (or, more generally, a saturated Dieudonné complex) $A^{\ast }$ is strict if it is complete with respect to the V-filtration; in other words, if the canonical map $ A^{\ast } \longrightarrow W(A^{\ast })$ is an isomorphism. The full subcategory of $\text {DA}_{\mathfrak {sat}}$ spanned by strict algebras is denoted by $\text {DA}_{\mathfrak {str}}$ .

Remark 2.8. Let $A^*$ be a saturated Dieudonné algebra.

• • $A^0/V A^0$ is a reduced $\mathbb {F}_p$ -algebra by Lemma 3.6.1 of [Reference Bhatt, Lurie and Mathew2].

• • The V-completion $W(A^*)$ becomes a Dieudonné algebra with F the inverse limit of $F: W_r (A^{\ast }) \rightarrow W_{r-1} (A^{\ast })$ . It is in fact still saturated. One needs to check that (i) $W(A^*)$ is p-torsion free; (ii) if $x \in A^i := (W(A^{\ast }))^i$ is an element with $d x \in p W^{i+1}$ , then $x \in \operatorname {\mathrm {im}} (F)$ ; and (iii) the inverse limit F on $W(A^*)$ satisfies $F(x) \equiv x^p \mod p$ . The details are left to the reader (see [Reference Bhatt, Lurie and Mathew2, Section 2.6 & 3.5]).

Remark 2.9. The V-completion $W(A^*)$ for $A^* \in \text {DA}_{\mathfrak {sat}}$ is strict. More precisely, for each $r \ge 1$ , the canonical map $W_r (A^*) \rightarrow W_r (W(A^*))$ is an isomorphism (by [Reference Bhatt, Lurie and Mathew2] Proposition 2.7.5). The completion functor $A^{\ast } \mapsto W(A^{\ast })$ provides a left adjoint of the inclusion $\text {DA}_{\mathfrak {str}} \hookrightarrow \text {DA}_{\mathfrak {sat}}$ .

Remark 2.10. For a strict Dieudonné algebra $A^*$ , each $A^i$ is p-complete. Moreover, $A^0$ is the ring of Witt vectors of $W_1(A)^0 = A^0/V A^0$ . In other words, there is a canonical isomorphism $\mu : A^0 \rightarrow W(A^0/VA^0)$ such that the composition $A^0 \rightarrow W(A^0/VA^0) \xrightarrow {\text {can}} A^0/V A^0$ is the projection map. The Frobenius F on $A^0$ corresponds to the Witt vector Frobenius under $\mu $ .

2.2. Saturated Dieudonné complexes

Let $A^\ast $ be a p-torsion-free Dieudonné algebra (or, more generally, a Dieudonné complex). Consider the cochain complex $(H^*(A^*/p A^*), \beta )$ where $\beta $ is the Bockstein differential induced from

$$ \begin{align*}0 \rightarrow A^*/p \xrightarrow{\:p\:} A^*/p^{2} \rightarrow A^*/p \rightarrow 0.\end{align*} $$

We have the following commutative diagram, where each arrow is a map of cochain complexes:

Here $\gamma $ is defined by sending $x \in (\eta _p A)^i$ to $x/p^i$ . Suppose that $A^*$ is in addition saturated; then the composition F in the diagram factors through the quotient cochain complex $W_1(A^*)$ , as in the diagram below:

We record a corollary for our discussion later on Nygaard filtrations.

Corollary 2.12. Let $A^\ast $ be a saturated Dieudonné algebra or complex; then the iterated Frobenius $F^r$ induces an isomorphism

$$ \begin{align*}F^r: W_r(A^i) \;\xrightarrow{\: {}_{\sim} \:} \; H^i(W_r (A^\ast)).\end{align*} $$

Proof. We have the following diagram:

The right-most arrow on cohomology (induced by q) is an isomorphism by Proposition 2.11 part (3), so the top arrow is an isomorphism by Proposition 2.11 part (2).

Remark 2.13. In fact, the map $F^r$ above can be upgraded to an isomorphism

$$ \begin{align*}F^r: W_r(A^\ast) \;\xrightarrow{\: {}_{\sim} \:} \; H^\ast (W_r (A^\ast))\end{align*} $$

of cochain complexes, where the target is equipped with the ‘Bockstein differential’ induced from the cofibre sequence $W_r A^* \xrightarrow {p^r} W_{2r} A^* \rightarrow W_r A^*.$

2.3. Dieudonné complexes of Cartier type

Definition 2.14. We say that a Dieudonné algebra (or, more generally, a Dieudonné complex) $\Omega ^*$ is of Cartier type (or satisfies the Cartier criterion) if

$$ \begin{align*}F: (\Omega^*/p, d) \rightarrow (H^*(\Omega^*/p), \beta)\end{align*} $$

is an isomorphism of cochain complexes.

In the case of the above definition, we denote the map $F^{-1}$ by C, because it is evidently related to the Cartier isomorphism (see Subsection A.3.2).

Next we consider the saturation of a Dieudonné complex of Cartier type.

Corollary 2.16. Let $\Omega ^*$ be a p-torsion-free Dieudonné algebra or complex of Cartier type; then $\phi _F: \Omega ^*/p \rightarrow \eta _p \Omega ^*/p$ is a quasi-isomorphism. Moreover, we have a quasi-isomorphism

$$ \begin{align*}\Omega^{\ast}/p \longrightarrow \Omega^\ast_{\mathfrak{ sat}} /p\end{align*} $$

2.4. Structure of the V-filtration

In this subsection we let $M^\ast $ be a saturated Dieudonné complex and analyze the structure of its V-filtration $\text {Fil}_V^\ast $ defined in (3). We begin by considering the following maps:

(4) $$ \begin{align} & \beta: \text{Fil}_V^n (M^i) \rightarrow M^{i- 1}/(V, F^n) \end{align} $$

(5) $$ \begin{align} & \beta': \text{Fil}_V^n (M^i) \rightarrow M^i/(V, F^n d) \end{align} $$

defined by

$$ \begin{align*}\beta(V^n x + d V^n y) = y, \qquad \beta' (V^n x + d V^n y ) = x.\end{align*} $$

First we check that these maps are well defined. Suppose that $V^n x + d V^n y = V^n x' + d V^n y'$ ; then

$$ \begin{align*}d (y - y') = F^n d V^n (y- y') = F^n (V^n x' - V^n x) = p^n (x' - x).\end{align*} $$

By Remark 2.4 we know that $y - y' = F^n z$ for some $z \in M^{i-1}$ , so

$$ \begin{align*}\beta(V^n x + d V^n y) = \beta (V^n x' + d V^n y') \in M^{i-1}/(V, F^n).\end{align*} $$

On the other hand, because $y -y' = F^n z$ , we further know that $p^n (x' - x) = d F^n z = p^n F^n dz,$ so $x' - x = F^n dz$ . Thus, we also have $\beta '(V^n x + d V^n y) = \beta '(V^n x' + d V^n y')$ .

The following proposition echoes [Reference Illusie10, Corollary I.3.9].

Proposition 2.18. Let $M^*$ be a saturated Dieudonné complex as above. The nth-graded piece of the V-filtration sits in the following short exact sequences:

(6)

such that the composition $\beta \circ dV^n$ (respectively $\beta ' \circ V^n$ ) is the natural projection. Note that when $n = 0$ , we set $F^{n-1} dM^{i - 1} = 0$ in the horizontal diagram.

Proof. We check that the horizontal sequence is exact and leave the vertical sequence to the reader. Clearly, the map $\beta $ is surjective and factors through

$$ \begin{align*}\beta: \text{gr}_V^n = (V^n M^i + d V^n M^{i-1})/(V^{n+1} M^i + d V^{n+1} M^{i-1}) \longrightarrow M^{i-1}/(V, F^n ).\end{align*} $$

First we check exactness in the middle. Suppose that $\beta (V^n x + d V^n y) = 0$ ; then $y = Vz + F^n w$ for some $z, w \in M^{i-1}$ ; thus,

$$ \begin{align*}V^n x + d V^n y = V^n (x + F^n dw) + d V^{n+1}z,\end{align*} $$

which lies in $\operatorname {\mathrm {im}} (V^n) \subset \text {gr}_V^n M^i$ . It remains to show injectivity of the first map. To this end, suppose that $V^n x = V^{n+1} y + d V^{n+1} z$ ; then

$$ \begin{align*}dz = F^{n+1} d V^{n+1} z = F^{n+1} (V^n x - V^{n+1} y) = p^n F( x - V y).\end{align*} $$

Let $s = x - Vy$ ; then u satisfies $p^n F s = d z$ ; thus, by Lemma 2.20 we have $s = F^{n-1} d t$ for some $t \in M^{i-1}$ , so $x = Vy + F^{n-1} dt$ and we are done.

In view of the lemma above, it is convenient to introduce the following filtrations on  $M^*$ .

Definition 2.19. Let $M^\ast $ be saturated as above. For $n \ge 1$ , we define subgroups of $M^i$  by

1. (1) $B_n (M^i) := F^{n-1} d M^{i-1}$

2. (2) $Z_n (M^i) := F^n M^i$ .

From the definition we have a chain of inclusions

$$ \begin{align*}0 \subset B_1 (M^i) \subset B_2 (M^i) \subset \cdots \subset B_{n} (M^i) &\subset B_{n+1} (M^i) \subset \cdots \\&\subset Z_n (M^i) \subset \cdots \subset Z_2 \subset Z_1 (M^i) \subset M^i. \end{align*} $$

The following simple lemma is used in the proof of the proposition above.

Proof. (2) Is a restatement of Lemma 2.6 part (1). For (1), first note that

$$ \begin{align*}B_n (M^i) \subset \{x \in M^i \: |\: p^{n-1} x = d y \} \subset \{x \in M^i \: |\: p^{n} F x = d y \}.\end{align*} $$

Now suppose that $x \in M^i$ and $p^n F x = d y$ , so $y = F^n z$ for some $z \in M^{i-1}$ . This implies that $p^n F x = p^n F^n dz$ ; thus, $x = F^{n-1} dz$ .

Notation 2.21. Next consider the image of $B_n$ and $Z_n$ under the projection $M^\ast \rightarrow W_1 (M^\ast ) = M^\ast /(V+ dV)$ , which we denote by

$$ \begin{align*}B_n(W_1M^i) = F^{n-1} d M^{i-1}/ (dV, V \cap B_n), \quad \text{and } \: Z_n(W_1 M^i) = F^n M^i/(dV, V \cap Z_n).\end{align*} $$

(Note that $d V (M^{i-1}) \subset B_n (M^i) \subset Z_n (M^i).$ )

Proof. (1) Is tautological and (2) follows from Lemma 2.20. For (3), we know directly from construction that the Frobenius F on $B_n (W_1M^i)$ and $Z_n(W_1 M^i)$ are both surjective. To check injectivity (in both cases), it suffices to show that if $x \in M^i$ satisfies $Fx = d y + V z$ , then $x \in \text {Fil}_V^1 = (V + dV)$ . To this end, we simply observe that

$$ \begin{align*}Vz = Fx - dy = Fx - F d V y = F(x - dV y).\end{align*} $$

By Lemma 2.6, this implies that $x - dVy \in \operatorname {\mathrm {im}} V$ ; the injectivity thus follows.

As a corollary, we have the following equivalent form of Proposition 2.18.

Corollary 2.23. The graded pieces of the V-filtration of $M^\ast $ are extensions of the following quotients of $W_1 (M^*)$ :

(7) $$ \begin{align} 0 \: \longrightarrow W_1 M^i/B_n (W_1M^i) \xrightarrow{\:\: V^n} & \text{ gr}_V^n M^i \xrightarrow{\:\: \beta \: } W_1 M^{i-1} /Z_n (W_1 M^{i-1})\rightarrow 0 \end{align} $$

(8) $$ \begin{align} 0 \rightarrow W_1 M^{i-1}/Z_{n+1} (W_1M^{i-1}) \xrightarrow{d V^n} & \text{ gr}_V^n M^i \xrightarrow{\: \: \beta'} W_1 M^i /B_{n+1} (W_1 M^i) \longrightarrow \: 0. \end{align} $$

Remark 2.24. Suppose that $A^*$ is a saturated Dieudonné algebra; then the exact sequences (7) and (8) are exact sequences of $W_1 A^0 = A^0/V A^0$ -modules, where the $W_1 A^0$ -structure is given as follows (we only focus on the first sequence (7) and leave the second to the reader):

• • The action of $A^0$ on $A^i$ makes $W_1 A^i$ into a $W_1 A^0$ -module because $Vx \cdot y = V(x \cdot Fy)$ , though this is not the $W_1 A^0$ -structure we use. Instead, we twist by $(n+1)$ -iterates of Frobenius and consider $W_1 A^i$ as a $W_1 A^0$ -module via

  $$ \begin{align*}F^{n+1}: W_1A^0 \longrightarrow W_1 A^0.\end{align*} $$
  In other words, $a \cdot x := F^{n+1}(a) x$ for all $a \in W_1 A^0, x \in W_1 A^i$ .

• • The subgroups $B_n (W_1 A^i)$ (respectively $Z_n (W_1 A^i)$ ) are naturally $W_1 A^0$ -submodules via the $W_1 A^0$ -action described above, because

  $$ \begin{align*}a \cdot F^{n-1} dx = F^{n-1} (F^2(a) dx) = F^{n-1} d (F^2 (a)x) - p^2 F^{n-1} \big(( F^2 da) x\big),\end{align*} $$
  which lies in $F^{n-1} d A^i $ modulo V. The quotients $W_1 A^i/B_n (W_1 A^i)$ (respectively $W_1 A^i/Z_n (W_1 A^i)$ ) are thus naturally $W_1 A^0$ -modules with the $F^{n+1}$ -twisted structure.

• • Finally, the middle term $\text {gr}_V^n A^i$ is a $W_1 A^0$ -module via the following Frobenius map:

  $$ \begin{align*}F: W_1 A^0 = A^0/V A^0 \longrightarrow A^0/p A^0.\end{align*} $$
  First note that $\text {Fil}_V^n = V^n A^i + d V^n A^{i-1}$ is a $A^0$ -submodule of $A^i$ . This gives a natural $A^0/p A^0$ -module structure on $\text {gr}_V^n A^i$ , which thus carries a $W_1 A^0$ via Frobenius.

The sequences in Corollary 2.23 are short exact sequences of $W_1 A^0$ -modules with the module structure described above. In other words, we have

$$ \begin{align*} V^n (F^{n+1}(a) x) & = F(a) V^n(x) \quad \text{and }\\ \beta (F(a) dV^n y) & = \beta(d V^n (F^{n+1} (a) y) - V^n (F^{n+1}(da) y) ) = F^{n+1} (a) \beta(d V^n y) \end{align*} $$

for all $a \in W_1 A^0, y \in A^{i-1}$ .

3.1. Log Dieudonné algebras

Notation. We often suppress notations and simply write $A^*$ for a log Dieudonné algebra.

Notation. We denote the category of log Dieudonné algebras by $\text {DA}_{/ \underline {W}}^{\log }$ , and we often suppress notations to write $\text {DA}^{\log } = \text {DA}_{/ \underline {W}}^{\log }$ when $\underline {W}$ is understood.

3.2. p-Compatible log Dieudonné algebras

Let $\mathrm {DA}^{\log ,p}$ denote the full subcategory of $\mathrm {DA}^{\log }$ spanned by p-compatible objects. In this article we are mostly interested in p-compatible log Dieudonné algebras (see Proposition 4.5).

3.3. Saturated log Dieudonné algebras

In this subsection and the next we discuss saturation and V-completion in the logarithmic setting.

Both $\eta _p (\alpha )$ and $\eta _p (\delta )$ are well defined by the relations $d F = p F d$ and $\delta F = p F \delta $ . It is straightforward to verify that $\eta _p ({A}^*)$ is in fact a log Dieudonné $\underline {W}$ -algebra, with its $\underline {W}$ -structure given by $W \rightarrow A^0 \rightarrow (\eta _p A)^0$ and $N \rightarrow L$ . Now let $\text {DA}^{\log }_{p\text {-tf}}$ be the full subcategory of $\mathrm {DA}^{\log }$ spanned by p-torsion-free complexes; then $\eta _p$ induces a logarithmic décalage operator $ \text {DA}^{\log }_{p\text {-tf}} \longrightarrow \text {DA}^{\log }_{p\text {-tf}}$ . For a morphism $f = (f, \psi ): {A}^* \rightarrow {B}^*$ between log Dieudonné algebras, one obtains a morphism $\eta _p (f): \eta _p ({A}^*) \rightarrow \eta _p({B}^*)$ in a functorial way. The operator $\eta _p$ clearly preserves the p-compatible objects. Moreover, for a p-torsion-free object ${A}^*$ in the category $\mathrm {DA}^{\log }$ (respectively in $\mathrm {DA}^{\log ,p}$ ), we have a natural map

$$ \begin{align*}{\phi}_F = (\phi_F, \text{id}): {A}^* \longrightarrow \eta_p ({A}^*)\end{align*} $$

of log Dieudonné algebras, where $\phi _F$ is given by the map $p^i F$ on $A^i$ and the identity on the monoid L.

Notation. The full subcategory of $\mathrm {DA}^{\log }$ spanned by saturated objects (respectively p-compatible saturated objects) is denoted by $\mathrm {DA}_{\mathfrak {sat}}^{\log }$ (respectively $\mathrm {DA}_{\mathfrak {sat}}^{\log ,p}$ ).

Because the log structure is irrelevant in the condition of being saturated, the construction of the saturation of a log Dieudonné algebra is essentially the same as in the nonlogarithmic setup. First we replace $\underline {A}^* = (A^*, L)$ by $(A^*/A^*[p^\infty ], L)$ to reduce to the p-torsion-free case while keeping the monoid L. For a p-torsion-free log Dieudonné algebra $ A^*$ , we apply the log décalage operator $\eta _p$ to the morphism $\phi _F = (\phi _F, \text {id}): {A}^* \rightarrow \eta _p ( A^*)$ repeatedly and obtain

$$ \begin{align*}A^*_{\mathfrak{sat}} := \text{colim} \big( A^* \xrightarrow{\phi_F} \eta_p( A^*) \xrightarrow{\phi_F} \eta_p \eta_p( A^*) \rightarrow \cdots \: \big).\end{align*} $$

It is a log Dieudonné algebra with monoid maps $\alpha : L \rightarrow A^0 \rightarrow A_{\mathfrak {sat}}^0$ and $\delta : L \rightarrow A^1 \rightarrow A_{\mathfrak {sat}}^1$ .

3.4. Strict log Dieudonné algebras and V-completion

Let $A^* \in \mathrm {DA}^{\log }$ be a saturated object. In Subsection 2.1 we formed the inverse limit

$$ \begin{align*}W(A^*) = \varprojlim W_r(A^*)\end{align*} $$

along the restriction maps $R: W_r(A^*) \rightarrow W_{r-1} (A^*),$ where $W_r (A^*) = A^*/\text {Fil}^r$ is the quotient of $A^*$ by the differential graded ideal $ \text {Fil}^r = V^r (A^*) + \text {d} V^r (A^*) \subset A^*$ (which defines the V-filtration on $A^*$ ). The Dieudonné algebra $W(A^*)$ inherits a log structure $L \rightarrow A^0 \rightarrow W(A)^0$ and a log derivation $L \rightarrow A^1 \rightarrow W(A)^1$ from $A^*$ , which makes it a log Dieudonné algebra. The canonical map $\rho : A^* \rightarrow W(A^*)$ of saturated Dieudonné algebras upgrades to $(\rho , \text {id})$ as a morphism in $\mathrm {DA}^{\log }$ , which we still denote by $\rho $ . By Remark 2.8, we know that, if $A^* \in \mathrm {DA}^{\log }$ is saturated (respectively saturated and p-compatible), then $W(A^*)$ is again saturated (respectively saturated and p-compatible).

Notation. The full subcategory of $\mathrm {DA}_{\mathfrak {sat}}^{\log }$ (respectively $\mathrm {DA}_{\mathfrak {sat}}^{\log ,p}$ ) spanned by strict objects is denoted by $\mathrm {DA}_{\mathfrak {str}}^{\log }$ (respectively $\mathrm {DA}_{\mathfrak {str}}^{\log ,p}$ ).

The condition of being strict and p-compatible imposes some rather strict restrictions on the log structures. For the setup, let $A^* \in \mathrm {DA}_{\mathfrak {str}}^{\log ,p}$ , and recall from Section 2 that we have a canonical isomorphism $\mu : A^0 \;\xrightarrow {\: {}_{\sim } \:} \; W(A^0/ V A^0)$ respecting Frobenii on both sides. The following proposition will be important for the construction of log de Rham–Witt complexes in Section 4.

Proof. It suffices to show that $x - [\overline x] \in \text {Fil}^s (A^0) = V^s (A^0)$ for every $s \in \mathbb {Z}_{\ge 1}$ because $A^0$ is complete with respect to the V-filtration. We proceed by induction. By assumption, $x - [\overline x] \in \text {Fil}^1$ . We claim that if $x - [\overline x] \in \text {Fil}^{r}$ for $r \ge 1$ , then $x- [\overline x] \in \text {Fil}^{2r}.$ Suppose that $x - [\overline x] = V^rb$ for some $b \in A^0$ ; then

$$ \begin{align*}F^r(x) - F^r([\overline x]) = x^{p^r} - [\overline x]^{p^r} = p^r \cdot b.\end{align*} $$

In the first equality we use the assumption on x. On the other hand,

$$ \begin{align*} x^{p^r} & = \Big( [\overline x] + V^rb \Big)^{p^r} \\ & = [\overline x]^{p^r} + p^r [\overline x]^{p^r -1} \cdot V^rb + \sum_{k = 2}^{p^r} {p^r \choose k} [\overline x]^{p^r - k} \cdot \big( V^rb \big)^k. \end{align*} $$

Note that because $x V^r y = V^r ((F^rx) y)$ , we have

$$ \begin{align*}(V^rb)^2 = V^r((F^r V^r b)b) = V^r (p^r b^2) = p^r V^r(b^2).\end{align*} $$

Therefore,

$$ \begin{align*}p^r b = p^r [\overline x]^{p^r -1} \cdot V^rb + p^r V^r(b^2) \cdot \sum_{k = 2}^{p^r} {p^r \choose k} [\overline x]^{p^r - k} \cdot \big( V^rb \big)^{k-2}.\end{align*} $$

Because $A^0$ is p-torsion free, we conclude that $b \in V^r (A^0)$ and this completes the proof.

3.5. Examples

We end this subsection with some toy examples of saturated and strict log Dieudonné algebras in the form of saturation and V-completion.

Example 3.15 Affine spaces with partial log structures

In this example we consider the log Dieudonné algebra $E^*$ where

$$ \begin{align*}E^0 = \mathbb{Z}_p [t_1, ..., t_{r+s}]\end{align*} $$

and $E^i$ is a free $E^0$ module on generators

$$ \begin{align*}\eta_1 \wedge \cdots \wedge \eta_i , \: \: \eta_i \in \left\{ \frac{dt_1}{t_1}, \cdots, \frac{dt_r}{t_r}, d t_{r+1}, ..., d t_{s+r} \right\}.\end{align*} $$

As a log Dieudonné algebra, $E^*$ is equipped with

• • the log structure $\mathbb {N}^r \rightarrow \mathbb {Z}_p [t_1, .., t_{r+s}]$ that sends $1$ in the kth position in $\mathbb {N}^r$ to $t_k$ ,

• • the log Frobenius map $F_L= p$ on $\mathbb {N}^r$ ,

• • the Frobenius map F that sends $t_k \mapsto t_k^p$ and $dt_k \mapsto t_k^{p-1} dt_k $ for each k.

To describe its saturation, we first note that $E^*_{\mathfrak {sat}}$ is a subcomplex of

$$ \begin{align*}E^0[F^{-1}] \xrightarrow{ \: d \: } E^1[F^{-1}][p^{-1}] \xrightarrow{ \: d \: } \cdots \xrightarrow{ \: d \: } E^{r+s}[F^{-1}][p^{-1}]\end{align*} $$

where $E^i[F^{-1}]$ denotes the direct limit of $E^i$ under the Frobenius map F. More concretely,

• • $ E^0 [F^{-1}] \cong \mathbb {Z}_p [t_1^{1/p^\infty }, ..., t_{r+s}^{1/p^\infty }] = \underset {\alpha _1, ..., \alpha _{r+s} \in \mathbb {N}[\frac {1}{p}]}{\bigoplus } \mathbb {Z}_p t_1^{\alpha _1} \cdots t_{r+s}^{\alpha _{r+s}} $

• • $E^1[F^{-1}] [p^{-1}] \cong \Big ( \bigoplus \mathbb {Q}_p t_k^{\alpha _k} \frac {dt_k}{t_k} \Big ) \bigoplus \Big ( \bigoplus \mathbb {Q}_p t_l^{\alpha _l} d t_l \Big )$ ,

where the first direct sum is taken over $ k \in \{1,..., r\}, \alpha _k \in \mathbb {N}[\frac {1}{p}] $ and the second direct sum is taken over $l \in \{r+1, ..., r+s\}, \alpha _l \in \mathbb {N}[\frac {1}{p}] \backslash 0$ . (Also note that $E^1 [F^{-1}] [p^{-1}]$ is isomorphic to $\bigoplus _k \mathbb {Q}_p [t_k^{1/p^\infty }] \frac {dt_k}{t_k} \bigoplus \bigoplus _{l} \Omega ^1_{\mathbb {Q}_p [t_l^{1/p^\infty }]/\mathbb {Q}_p}$ .)

Unwinding definitions, the saturation of $E^*$ is given by the subcomplex of $E^*[F^{-1}]$ consisting of ‘integral forms’, which are forms with integral p-adic coefficients and whose differentials also have integral coefficients. In other words, we have

$$ \begin{align*}E^i_{\mathfrak{sat}} = \{ \omega \in E^i [F^{-1}] \: : \: d \omega \in E^{i+1} [F^{-1}] \}.\end{align*} $$

The log structure on $E^*_{\mathfrak {sat}}$ is clearly given by

$$ \begin{align*}\alpha: 1_k \in \mathbb{N}^r \longmapsto t_k \in E^0_{\mathfrak{sat}}, \qquad \delta: 1_k \in \mathbb{N}^r \longmapsto \frac{d t_k}{t_k} \in E^1_{\mathfrak{sat}}\end{align*} $$

where $1_k \in \mathbb {N}^r$ denotes the element $(0, ..., 1, ..., 0) \in \mathbb {N}^r$ where $1$ is the kth-position. Finally, to obtain a strict log Dieudonné algebra (with trivial log structure) we take the V-adic completion.

Next we discuss what the saturation looks like in the special cases where $r = 0$ or where $s = 0$ in the previous example.

Special case: $\mathbb {A}^1$ with no log structure. Take $r = 0$ and $s = 1$ in the previous example, then we have $E^* = (\mathbb {Z}_p [t] \xrightarrow {\: d \:} \mathbb {Z}_p [t] dt)$ with trivial log structure $0 \rightarrow \mathbb {Z}_p [t]$ and with Frobenius $t \mapsto t^p$ . In this case, the saturation (given by the subcomplex of ‘integral forms’) is

$$ \begin{align*} E^0_{\mathfrak{sat}} & = \left\{x \in \mathbb{Z}_p [t^{1/p^\infty}] : dt \in \mathbb{Z}_p[t^{1/p^\infty}] \frac{dt}{t} \right\} = \{\sum c_k t^{\alpha_k} \in \mathbb{Z}_p [t^{1/p^\infty}] : \text{ord}_p (c_k \alpha_k ) \ge 0 \} \\ E^1_{\mathfrak{sat}} & = \big( \mathbb{Z}_p [t^{1/p^\infty}] \backslash \mathbb{Z}_p \big) \frac{dt}{t} := \bigoplus_{\alpha \in \mathbb{N}[\frac{1}{p}] \backslash 0} \mathbb{Z}_p t^\alpha \frac{dt}{t}. \end{align*} $$

This is essentially the example discussed in [Reference Illusie10, Section I.2].

Special case: logarithmic $\mathbb {A}^1$ . For a more relevant version of the first example, we take $r = 1$ and $s = 0$ . In other words, we consider the log Dieudonné algebra $E^* = (\mathbb {Z}_p [t] \xrightarrow {\: d \:} \mathbb {Z}_p [t] \frac {dt}{t})$ with log structure $\mathbb {N} \xrightarrow {1 \mapsto t} \mathbb {Z}_p [t]$ and Frobenius $F_L = p$ . Similar to the previous example, we have

$$ \begin{align*} E^0_{\mathfrak{sat}} & = \left\{x \in \mathbb{Z}_p [t^{1/p^\infty}] : dt \in \mathbb{Z}_p[t^{1/p^\infty}] \frac{dt}{t} \right\} \\ E^1_{\mathfrak{sat}} & = \mathbb{Z}_p [t^{1/p^\infty}] \frac{dt}{t}. \end{align*} $$

In this example, the log structure of the saturation is given by

$$ \begin{align*}\alpha: \mathbb{N} \xrightarrow{1 \mapsto t} E^0_{\mathfrak{sat}}, \qquad \delta: \mathbb{N} \xrightarrow{1 \mapsto dt/t} E^1_{\mathfrak{sat}}.\end{align*} $$

We further remark that $E^0_{\mathfrak {sat}}$ consists of elements of the form $\cdots , t^{p^2}, t^p, t, p t^{1/p}, p^2 t^{1/p^2}, \cdots $ but not $t^{1/p}$ . The V-filtration on $E^1_{\mathfrak {sat}}$ agrees with the p-adic filtration, whereas on $E^0_{\mathfrak {sat}}$ it is given by

• • $ V (E^0_{\mathfrak {sat}})$ is generated by ${}^{\;} \left \{pt^{i}, p^m t^{\frac {k}{p^m}} \; | \: i, k\in \mathbb {Z}_{\ge 0}, m \in \mathbb {Z}_{\ge 1}, (k, p) = 1 \right \}$ .

• • More generally, $ V^n (E^0_{\mathfrak {sat}})$ is generated by

  $$ \begin{align*}\left\{p^n t^{i}, p^n t^{\frac{j_1}{p}}, \cdots, p^n t^{\frac{j_{n}}{p^{n}}}, p^m t^{\frac{k}{p^m}} \; | \: i, j_*, k\in \mathbb{Z}_{\ge 0}, m \in \mathbb{Z}_{\ge n+1}, (j_*, p) = (k, p) = 1 \right\}.\end{align*} $$

Example 3.16 Normal crossing

Next we examine examples related to a simple normal crossing over $\mathbb {F}_p$ . Let

$$ \begin{align*}A^* = (A^0 \xrightarrow{\: d\:} A^1) = (\mathbb{Z}_p [x, y]/(xy) \xrightarrow{\: d \: } \mathbb{Z}_p [x, y]/(xy) \frac{dx}{x})\end{align*} $$

be a log Dieudonné algebra, with

• • differential $ d: x \mapsto dx, y \mapsto -y \frac {dx}{x},$

• • log structure $\alpha : \mathbb {N}^2 \xrightarrow {(a, b) \mapsto x^{a} y^{b}} A^0$ , and log derivation $\delta : \mathbb {N}^2 \xrightarrow {(a, b) \mapsto (a-b)\frac {dx}{x}} A^1$ .

• • Frobenius given by p on $\mathbb {N}^2$ and by $F: x \mapsto x^p, y \mapsto y^p, \frac {dx}{x} \mapsto \frac {dx}{x}$ on $A^*$ .

The log saturation of $A^*$ is given by

$$ \begin{align*}A_{\mathfrak{sat}}^0 = \left\{ \omega \in A^{0} [F^{-1}] : d \omega \in A^1[F^{-1}] \right\} \xrightarrow{\quad d \quad} A_{\mathfrak{sat}}^1 = A^1 [F^{-1}]\end{align*} $$

where

• • $A^0 [F^{-1}] \cong \mathbb {Z}_p [x^{1/p^\infty }, y^{1/p^\infty }]/ \big ( x^{1/p^m}y^{1/p^m}\big )_{m \ge 0 } = \mathbb {Z}_p[x^{1/p^\infty }] \cup _{\mathbb {Z}_p} \mathbb {Z}_p [y^{1/p^\infty }]$ ,Footnote ⁷

• • $A^1 [F^{-1}] \cong \left (\mathbb {Z}_p [x^{1/p^\infty }, y^{1/p^\infty }]/ ( x^{1/p^m}y^{1/p^m})_{m \ge 0 } \right ) \frac {dx}{x} = \Big (\mathbb {Z}_p[x^{1/p^\infty }]\cup _{\mathbb {Z}_p}$ $\mathbb {Z}_p [y^{1/p^\infty }] \Big ) \frac {dx}{x}$ .

The V-filtration can be described similarly as the example above. Finally, we define the following useful submodule of $A^0_{\mathfrak {sat}}$ consisting of all terms that only involve powers of $x^{1/p^m}$ :

• • Write $A^0_{\mathfrak {sat}} (x) := \{\omega \in \mathbb {Z}_p [x^{1/p^\infty }] \subset A^0 [F^{-1}] \: : \: d \omega \in A^1 [F^{-1}] \}. $

In particular, $A^0_{\mathfrak {sat}} = A^0_{\mathfrak {sat}}(x) \cup A^0_{\mathfrak {sat}} (y)$ and $A^0_{\mathfrak {sat}}(x) \cap A^0_{\mathfrak {sat}} (y) = \mathbb {Z}_p$ .

Example 3.17 Normal crossing over a trivial log point

In this example, we consider the 3-term Dieudonné complex $B^* = B^0 \xrightarrow {d } B^1 \xrightarrow {d} B^2$ given by

$$ \begin{align*}\mathbb{Z}_p [x, y]/(xy) \xrightarrow{\: d\: } \mathbb{Z}_p [x, y]/(xy) \langle {\frac{dx}{x}, \frac{dy}{y} } \rangle \xrightarrow{\: d \: } \mathbb{Z}_p[x, y]/(xy) \frac{dx}{x} \wedge \frac{dy}{y}\end{align*} $$

with log structure given by

$$ \begin{align*} & \alpha: \mathbb{N}^2 \xrightarrow{(a, b) \mapsto x^a y^b \quad } \; B^0 = \mathbb{Z}_p [x, y]/(xy), \\ & \delta: \mathbb{N}^2 \xrightarrow{(a, b) \mapsto a \frac{dx}{x} + b \frac{dy}{y}} B^1 = \mathbb{Z}_p[x, y]/(xy) \frac{dx}{x} \oplus \mathbb{Z}_p [x, y]/(xy) \frac{dy}{y} \end{align*} $$

and Frobenius given by p on $\mathbb {N}^2$ , while sending $x \mapsto x^p, y \mapsto y^p$ as above. Now we describe the saturation of $B^*$ (we use notations from the previous example regarding $A^*$ ).

$$ \begin{align*} B^0_{\mathfrak{sat}} = & \left \{\omega \in B^{0} [F^{-1}] : d \omega \in B^1[F^{-1}] \right \} \cong A^0_{\mathfrak{sat}} \\ B^1_{\mathfrak{sat}} = & \left \{\omega \in B^{1} [F^{-1}] : d \omega \in B^2[F^{-1}] \right \} \\ \cong & \Big(\mathbb{Z}_p[x^{1/p^\infty}]\cup_{\mathbb{Z}_p} A^0_{\mathfrak{sat}}(y) \Big) \frac{dx}{x} \oplus \Big(\mathbb{Z}_p[y^{1/p^\infty}]\cup_{\mathbb{Z}_p} A^0_{\mathfrak{sat}}(x) \Big) \frac{dy}{y} \\ B^2_{\mathfrak{sat}} = & B^2 [F^{-1}] \cong \Big(\mathbb{Z}_p [x^{1/p^\infty}] \cup_{\mathbb{Z}_p} \mathbb{Z}_p [y^{1/p^\infty}] \Big) \frac{dx}{x} \wedge \frac{dy}{y}. \end{align*} $$

4.1. Saturated log de Rham–Witt complexes for log algebras

Let $\text {Alg}_{\underline {k}}^{\log }$ be the category of log algebras over $\underline {k}$ . An element $\underline {R} = (R, M) \in \text {Alg}_{\underline {k}}^{\log }$ can be equipped with a pair of Frobenius $(F, F_M)$ where $F(x) = x^p$ for all $x \in R$ and $F_M (m) = pm$ for all $m \in M$ . From the previous section, we have a functor from p-compatible strict log Dieudonné algebras over $\underline {W}$ to log algebras over $\underline {k}$ :

$$ \begin{align*}\mathrm{DA}_{\mathfrak{str}}^{\log,p} \longrightarrow \text{Alg}_{\underline{k}}^{\log} \qquad \text{given by } \:\: A^* \mapsto \underline{A}^0/ V(A^0),\end{align*} $$

where $\underline {A}^0/ V(A^0) $ consists of the data $ (A^0/V(A^0), \alpha : L \rightarrow A^0 \twoheadrightarrow A^0/V(A^0))$ . Now we are ready to give the definition of saturated log de Rham–Witt complexes for log algebras (Definition 1 in the Introduction).

Definition 4.1. The saturated log de Rham–Witt complex $\mathcal {W} \omega ^*_{-/\underline {W}}$ is a functor

$$ \begin{align*}\mathcal{W} \omega^*_{-/\underline{W}}: \text{Alg}_{\underline{k}}^{\log} \longrightarrow \text{DA}_{\mathfrak{str}_{/\underline{W}}}^{\log, p}\end{align*} $$

that is the left adjoint of the functor $\text {DA}_{\mathfrak {str}_{/\underline {W}}}^{\log , p} \longrightarrow \text {Alg}_{\underline {k}}^{\log }$ sending $A^* \longmapsto (A^0/V\!(A^0), L).$

In other words, the saturated log de Rham–Witt complex of a log algebra $\underline {R}$ over $\underline {k}$ is a strict p-compatible log Dieudonné $W(\underline {k})$ -algebra $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ , equipped with an isomorphism

$$ \begin{align*}e: \underline{R} \;\xrightarrow{\: {}_{\sim} \:} \; \mathcal{W}_1 \omega^0_{\underline{R}/\underline{k}}\end{align*} $$

such that for any $A^* \in \text {DA}^{\log , p}_{\mathfrak {str}}$ , we have a natural identification

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}_{\mathfrak{str}}^{\log}} (\mathcal{W} \omega^*_{\underline{R}/\underline{k}}, A^*) \;\xrightarrow{\: {}_{\sim} \:} \; \operatorname{\mathrm{Hom}}_{\text{Alg}_{\underline{k}}^{\log}} (\underline{R}, \underline{A}^0/ V(A^0)).\end{align*} $$

The main goal of this section is to show the existence of this left adjoint $\mathcal {W} \omega ^*_{-/\underline {W}}$ by construction.

4.2. The construction

4.2.1. Preliminaries I

Notation 4.3. Let $\underline {A} = (A, L_A)$ and $\underline {B} = (B, L_B)$ be two log $\underline {W}$ -algebras equipped with Frobenius maps $(\varphi _A, F_{L_A})$ and $(\varphi _B, F_{L_B})$ , where $\varphi _A$ (respectively $\varphi _B$ ) lifts the p-power Frobenius on $A/p$ (respectively $B/p$ ). We denote by $\operatorname {\mathrm {Hom}}_{F} (\underline {A}, \underline {B})$ the set of Frobenius-compatible $\underline {W}$ -morphisms between $\underline {A}$ and $\underline {B}$ , which are pairs $(f, \psi )$ of morphisms making every square in the cube below commute:

The following setup is of particular interest. Let $\underline {R}, \underline {R'} \in \text {Alg}_{\underline {k}}^{\log }$ be two log algebras over $\underline {k}$ . Let $W(\underline {R}) = (W(R), [\alpha ])$ , where $[\alpha ] = [\;] \circ \alpha : M \rightarrow R \rightarrow W(R)$ is the Teichmuller lift of the log structure on R. $W(\underline {R})$ is equipped with the Frobenius map $(F, p)$ where F is the Witt vector Frobenius. Similarly define $W(\underline {R}')$ . Then $\operatorname {\mathrm {Hom}}_{F} (W(\underline {R}), W(\underline {R'}))$ consists precisely of the homomorphisms $(f, \psi ): W(\underline {R}) \rightarrow W(\underline {R}')$ of log $\underline {W}$ -algebras, such that $f \circ F = F \circ f$ .

Lemma 4.4. Let $\underline {R}, \underline {R'} \in \text {Alg}_{\underline {k}}^{\log }$ be as above. Suppose that both R and $R'$ are reduced; then

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{F} (W(\underline{R}), W(\underline{R'})) \cong \operatorname{\mathrm{Hom}}_{\text{Alg}_{\underline{k}}^{\log}} (\underline{R}, \underline{R}').\end{align*} $$

Proof. Let $(\overline f, \psi ): \underline {R} \rightarrow \underline {R}'$ be a map over $\underline {k}$ . Because $W(R)$ is p-torsion free (as R is reduced) and $F: W(R) \rightarrow W(R)$ lifts the Frobenius on R, the map $W(R) \rightarrow R \xrightarrow {\overline f} R'$ lifts uniquelyFootnote ⁹ to a map $f: W(R) \rightarrow W(R')$ that is compatible with Frobenius on both $W(R)$ and $W(R')$ . It remains to show the compatibility between f and $\psi $ , but this is implied by the compatibility between f and taking Teichmuller lifts: namely, that $f([x]) = [\overline f (x)]$ for every $x \in R$ . To prove this claim, it suffices to show that $f([x]) - [\overline f (x)] \in \text {Fil}^s = V^s W(R')$ for every $s \in \mathbb {N}$ . This can be done by a simple induction on s, and we leave the details to the reader.

4.2.2. A key example of log Dieudonné algebra

The construction of log de Rham–Witt complexes for $\underline {R} \in \text {Alg}_{\underline {k}}^{\log }$ relies on the following proposition.

Proposition 4.5. Let $(A, \alpha :L \rightarrow A)$ be a log algebra over $\underline {W}$ . Suppose that A is p-torsion free and that the pair $(A, L)$ is equipped with morphisms $\varphi : A \rightarrow A$ and $F_L: L \rightarrow L$ such that

• • $\varphi (x) \equiv x^p \mod p$ is a lifting of the absolute Frobenius mod p;

• • $F_L = p$ on L and is compatible with $\varphi $ , that is, $\varphi \circ \alpha = \alpha \circ F_L$ .

Then there exists a unique graded ring homomorphism $ F: \omega ^*_{\underline {A}/\underline {W}} \longrightarrow \omega ^*_{\underline {A}/\underline {W}} $ that extends $\varphi $ on A, such that

$$ \begin{align*}F (d x) = x^{p-1} dx + d (\frac{\varphi(x) - x^p}{p}) \quad \text{for all } x \in A.\end{align*} $$

Moreover, this makes $ \underline {\omega ^*} := (\omega ^*_{\underline {A}/W(\underline {k})}, L, d, \delta , F, F_L)$ a p-compatible log Dieudonné $\underline {W}$ -algebra.

Proof. For the first claim we need to show the existence of $F: \omega ^1_{\underline {A}/W(\underline {k})} \rightarrow \omega ^1_{\underline {A}/W(\underline {k})}$ . We regard the second copy of $ \omega ^1_{\underline {A}/W(\underline {k})}$ as an A-module where A acts by a twist of $\varphi $ , namely, $a \cdot \lambda = \varphi (a) \lambda $ for all $\lambda \in \omega ^1_{\underline {A}/W(\underline {k})}$ . By the universal property of $ \omega ^1_{\underline {A}/W(\underline {k})}$ and Remark A.5, it suffices to supply the structure of a log derivation from $(A, L)$ to $\omega ^1_{\underline {A}/W(\underline {k})}$ with this twisted A-module structure. For this we define

$$ \begin{align*}\widetilde \delta: L \rightarrow \omega^1_{\underline{A}/W(\underline{k})}, \quad \text{and } \:\: \widetilde d: A \rightarrow \omega^1_{\underline{A}/W(\underline{k})}\end{align*} $$

respectively as follows:

• • we set $\widetilde \delta (l) := \delta (l)$ and

• • $\widetilde d (x) := x^{p-1}dx + d \theta (x)$ where $\theta (x) = \displaystyle \frac {\varphi (x) - x^p}{p} $ .

To proceed we need to show that $\widetilde \delta $ and $\widetilde d$ form a log derivation of $\underline {A}$ over $W(\underline {k})$ , which requires

1. (1) $\widetilde d (\gamma x) = \gamma \cdot \widetilde d x := F(\gamma ) \widetilde d x$ for all $ \gamma \in W(k), x \in A$ ;

2. (2) $\widetilde d (x + y) = \widetilde d (x) + \widetilde d(y)$ for all $x, y\in A $ ;

3. (3) $\widetilde d (x y) = x \cdot \widetilde d y + y \cdot \widetilde d x := \varphi (x) \widetilde d y + \varphi (y) \widetilde d x$ for all $x, y \in A$ ;

4. (4) $\alpha (l) \cdot \widetilde \delta (l) : = \varphi (\alpha (l)) \widetilde \delta (l) =\widetilde d (\alpha (l))$ for all $l \in L$ ;

5. (5) $\widetilde \delta (\iota (n)) = 0 $ for all $n \in N$ , where $\iota : N \rightarrow L$ is map of monoids in $W(\underline {k}) \rightarrow \underline {A}.$

(1) Follows from (3) because $ p \widetilde d (\lambda ) = d (F (\lambda )) = 0$ , where $F: W(k) \rightarrow W(k)$ is the Frobenius map on Witt vectors. (2) and (3) follow from the explicit description of $\theta (x)$ . Now we check (4). Because $\theta (\alpha (l)) = 0$ , we have

$$ \begin{align*}\varphi (\alpha(l)) \widetilde \delta (l) = \alpha(l)^p \delta (l) = \alpha(l)^{p-1} d (\alpha(l)) = \widetilde d(\alpha (l)).\end{align*} $$

(5) Is clear because $\delta (\iota (n)) = 0 $ for all $n \in N$ . We thus get the desired map $F: \omega ^1_{\underline {A}/W(\underline {k})} \rightarrow \omega ^1_{\underline {A}/W(\underline {k})} $ that satisfies $F(\delta (l)) = \widetilde \delta (l) = \delta (l)$ and $F d(x) = \widetilde d (x) = x^{p-1} dx + d \theta (x)$ .

To finish the proof of the proposition, we need to check that (i) $ d F = p F d$ on all of $ \omega ^*_{\underline {A}/W(\underline {k})}$ and (ii) $\delta F_L = p F \delta $ on L. For (i), note that $\omega ^*_{\underline {A}/W(\underline {k})}$ is generated over A by $\omega ^1_{\underline {A}/W(\underline {k})}$ , and it is clear that we only need to check the relation on $x, dx $ for all $x \in A$ and on $\delta (l)$ for all $l \in L$ . The fact that $d F (x) = p F d (x)$ follows directly from the construction of F on $\omega ^1_{\underline {A}/W(\underline {k})}$ , because $F d = \widetilde d$ by construction; on $d x$ and $\delta (l),$ both $d F$ and $p F d$ evaluate to $0$ (because $d \delta = 0$ on L). Part (ii) is automatic by construction of F again, because $F \delta =\delta $ .

We record a variant of the proposition above, which allows slightly more flexible Frobenius $F_L$ on the monoid L.

Proposition 4.6. Let $(A, \alpha :L \rightarrow A)$ be a log algebra over $\underline {W}$ . Suppose that A is p-torsion free and p-separated and that $\omega ^1_{\underline {A}/\underline {W}}$ is p-torsion free. Also suppose that the pair $(A, L)$ is equipped with Frobenius morphisms $\varphi : A \rightarrow A$ and $F_L: L \rightarrow L$ such that

• • $\varphi (x) \equiv x^p \mod p$ is a lifting of the absolute Frobenius mod p;

• • $F_L$ is compatible with $\varphi $ .

Then the same conclusions of Proposition 4.5 hold except for p-compatibility.

Proof. First we need the following sublemma.

Sublemma 4.7. Retain the notation and assumption of the proposition. For any $k \in \mathbb {N}$ , if $p^k | \varphi (x)$ , then $p^k | \big (x^{d-1} dx + d \theta (x) \big ).$

Proof of the sublemma. The case $k = 0$ is tautological. We proceed by induction. Suppose that $k \ge 1$ and that the sublemma has been verified up to $k - 1$ . Now let $\varphi (x) = p^{k} y = x^p + p \theta (x)$ ; because $p | \varphi (x)$ , we know that $p | x$ , so $x = p z$ for some $z \in A$ . Because A is p-torsion free, $\varphi (x) = p \varphi (z) = p^{k} y$ implies that $\varphi (z) = p^{k - 1} y$ , so $p^{k - 1} | \big ({z}^{p-1} dz + d \theta (z) \big ) $ by induction hypothesis. Note that ${x}^{p-1} dx + d \theta (x) = (pz)^{p-1} d(pz) + d \theta (pz) = p \big ( {z}^{p-1} dz + d \theta (z) \big )$ , so the lemma follows.

Now we proceed as in the proof of Proposition 4.5 but replace $\widetilde \delta $ by

$$ \begin{align*}\widetilde \delta (l) := \displaystyle {\delta (F_L(l))}/{p}.\end{align*} $$

We need to show that $p | \delta (F_L (l))$ for any $l \in L$ for $\widetilde \delta $ to be well defined. For this we need the assumption that A is p-separated: write $x = \alpha (l)$ ; then there exists some $k \in \mathbb {N}$ such that $p^k | \varphi (x)$ but $p^{k + 1} \nmid \varphi (x)$ . By the sublemma above we know that $p^{k+1} | d (\varphi (x))$ . Now because

$$ \begin{align*}\varphi(x)\; \delta(F_L (l)) = \alpha (F_L (l))\; \delta(F_L(l)) = d \alpha (F_L(l)) = d \varphi(x)\end{align*} $$

and $p^{k+1} \nmid \varphi (x)$ , $p^{k+1} | d \varphi (x)$ , we know that $p | \delta (F_L (l))$ (here we use the assumption that $ \omega ^1_{\underline {A}/W(\underline {k})} $ is p-torsion free). The rest of the proof of part (1) is the same, except that to check $\alpha (l) \cdot \widetilde \delta (l) = \widetilde d(\alpha (l))$ , we need to use

$$ \begin{align*} p \widetilde d (\alpha (l)) & = d \varphi (\alpha (l )) \\ & = \alpha (F_L (l)) \delta(F_L (l)) \\ & = p \varphi (\alpha(l)) \widetilde \delta (l) \quad = p \alpha(l) \cdot \widetilde \delta(l). \end{align*} $$

The rest of the proof is identical to that of Proposition 4.5.

4.2.3. An interlude: Frobenius on finite level

Proposition 4.5 will be applied to $A = W(R)$ where R is a reduced $\mathbb {F}_p$ -algebra. For applications later, we also record the following remark, which the reader may skip in first reading.

Remark 4.8. The graded ring homomorphism F can be constructed on the finite-length Witt vectors; namely, there is a natural graded ring homomorphism

$$ \begin{align*}F: \omega^*_{W_n(\underline{R})/W_n(\underline{k})} \rightarrow \omega^*_{W_{n-1}(\underline{R})/W_{n-1}(\underline{k})}\end{align*} $$

extending the Frobenius $F: W_{n} (R) \rightarrow W_{n-1} (R)$ . We prove the claim using the same strategy as Proposition 4.5, by constructing a log derivation $(\widetilde d, \widetilde \delta )$ of $W_n(\underline {R})/W_n(\underline {k})$ into $F_*\omega ^1_{W_{n-1}(\underline {R})}$ , where $\omega ^1_{W_{n-1}(\underline {R})}$ is regarded as a $W_n(R)$ structure via $F: W_n(R) \rightarrow W_{n-1} (R)$ . We define $\widetilde d$ as follows: write $x \in W_n(R)$ uniquely as $x = [x_0] + V x'$ with $x' \in W_{n-1}(R)$ and $x_0 \in R$ and then define

$$ \begin{align*}\widetilde d x := [x_0]^{p-1} d[x_0] + d x'\end{align*} $$

where $[x_0]$ now lives in $W_{n-1}(R)$ . We then define $\widetilde \delta (m) := \delta (m)$ .

$\bullet $ Let us first check that $\widetilde d (x + y) = \widetilde d(x) + \widetilde d(y)$ , for which it suffices to show that if $[x_0] + [y_0] = [x_0+y_0] + V z'$ , then

$$ \begin{align*}[x_0]^{p-1} d[x_0] + [y_0]^{p-1} d[y_0] = [x_0+y_0]^{p-1} d[x_0+y_0] + dz'.\end{align*} $$

We check this equality in $\omega ^1_{W(\underline {R})/W(\underline {k})}$ . For any element $z = [x] + Vy \in W(R)$ , we claim that

$$ \begin{align*}[x]^{p-1}d[x] + d y = z^{p-1} dz + d \Big(\frac{F(z) - z^p}{p} \Big).\end{align*} $$

This is straightforward by expanding the terms on the right-hand side:

$$ \begin{align*}\big([x] + Vy \big)^{p-1} d\big([x] + Vy \big) + d \left(\frac{ [x]^p + py - \big([x] + Vy \big)^p}{p}\right).\end{align*} $$

Now the desired additivity follows from the proof of Proposition 4.5.

$\bullet $ Next we check that

$$ \begin{align*} \widetilde d (x y) & = \Big([x_0]^{p} + px' \Big)\Big([y_0]^{p-1} d[y_0] + d y' \Big) + \Big([y_0]^{p} + py' \Big)\Big([x_0]^{p-1} d[x_0] + d x' \Big) \\ & = F(x) \widetilde d y + F(y) \widetilde d x = x \cdot \widetilde d y + y \cdot \widetilde d x. \end{align*} $$

Because $(\widetilde d, \widetilde \delta )$ forms a log $W_n(\underline {k})$ -derivation from $W_n(\underline {R})$ into $\omega ^1_{W_{n-1}(\underline {R})/W_{n-1} (\underline {k})}$ , we have the desired map $F: \omega ^1_{W_{n}(\underline {R})/W_{n}(\underline {k})} \rightarrow \omega ^1_{W_{n-1}(\underline {R})/W_{n-1}(\underline {k})}$ . Finally, this extends to a graded ring homomorphism on $\omega ^*_{W_{n}(\underline {R})/W_n(\underline {k})}$ .

4.2.4. Preliminaries II

Now the final ingredient for the construction of the saturated log de Rham–Witt complexes.

Lemma 4.9. Let $\underline {A}$ be a log algebra over $\underline {W} = W(\underline {k})$ satisfying the conditions in either Proposition 4.5 or 4.6 and $\underline {\omega ^*} = \underline {\omega }^*_{\underline {A}/ \underline {W}}$ be the corresponding log Dieudonné $\underline {W}$ -algebra constructed there. Let $B^*$ be a p-torsion-free log Dieudonné $\underline {W}$ -algebra. There is a canonical bijection between

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\text{DA}^{\log}} (\underline{\omega^*}, B^*) \cong \operatorname{\mathrm{Hom}}_{F} (\underline{A}, \underline{B}^0),\end{align*} $$

where $\operatorname {\mathrm {Hom}}_F$ is as defined in Notation 4.3.

Proof. Given a morphism $(f, \psi ) \in \operatorname {\mathrm {Hom}}_{F} (\underline {A}, \underline {B}^0)$ , we need to extend it to a morphism $f: \underline {\omega ^*} \rightarrow \underline {B}^*$ of log Dieudonné $\underline {W}$ -algebras. As in the proof of Proposition 4.5, we first construct a map $f: \omega ^1_{\underline {A}/\underline {W}} \rightarrow B^1$ that fits into the following commutative diagram:

Define $d_f: A \longrightarrow B^1$ by $d_f = d_B \circ f$ and $\delta _f: L \longrightarrow B^1$ by $\delta _f = \delta _B \circ \psi $ . One easily verifies that $d_f$ is a derivation of $A/W(k)$ into $B^1$ . To show that $(d_f, \delta _f) $ is a log derivation of $\underline {A}/\underline {W}$ , we check that $\delta _f (\iota (n)) = 0$ and that

$$ \begin{align*} \alpha (l) \cdot \delta_f (l) & = \alpha_B (\psi (l)) \: \delta_B (\psi (l)) \\ & = d_B \: \alpha_B (\psi (l)) = d_B \: f (\alpha (l)) = \delta_f (\alpha (l)). \end{align*} $$

Therefore, we have the dotted map $f: \omega ^1_{\underline {A}/{W(\underline {k})}} \rightarrow B^1$ in the diagram above.

Next we extend f to $\omega ^*_{\underline {A}/\underline {W}} $ as a differential graded morphism (because $\omega ^*_{\underline {A}/\underline {W}} $ is generated over A by $\omega ^1_{\underline {A}/\underline {W}}$ ). Note that f is compatible with $\delta $ by construction, so it remains to show its compatibility with F. For this we proceed as the proof of Proposition 4.5; namely, we check it on $x, dx $ for $x \in A$ and $\delta (l)$ for $l \in L$ because both $F $ and f are algebra morphisms. We know that $F f (x) = f F (x)$ by assumption. For $d x$ (where $x \in A^0$ ), we have

$$ \begin{align*} p f F (dx) & = f d F(x) = d( f F(x)) \\ & = d F (f (x)) = p F (d f(x)) = p F f (d x), \end{align*} $$

and because $B^*$ is p-torsion free, we get $f F (dx) = F f (dx)$ . The proof for $\delta (l)$ is similar.

4.2.5. The construction

Now we prove our first main result (Theorem 2 in the Introduction). We continue to assume that $\underline {k}$ has the form $(k, N)$ where $N \backslash \{0\} \mapsto 0 \in k$ .

Theorem 4.10. Let $\underline {R} = (R, M) \in \text {Alg}_{\underline {k}}^{\log }$ be a log algebra over $\underline {k}$ . The saturated log de Rham–Witt complex $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ of $\underline {R}/\underline {k}$ exists. Moreover, the association of $\underline {R} \mapsto \mathcal {W} \omega ^*_{\underline {R}/\underline {k}} $ is functorial.

Proof. Without loss of generality, assume that R is reduced by replacing it with its reduced quotient if necessary (for an object $B^* \in \mathrm {DA}_{\mathfrak {str}}^{\log ,p}$ , $B^0/V(B^0)$ is reduced by Remark 2.8). Let $W(\underline {R}) = (W(R), L =M \xrightarrow {[\alpha ]} W(R))$ be the Witt vector of $\underline {R}$ . Because R is reduced, $W(R)$ is p-torsion free. By Proposition 4.5, there exits a unique log Dieudonné $\underline {W}$ -algebra on the relative log de Rham complexes:

$$ \begin{align*}\underline{\omega^*} = (\omega^*_{W(\underline{R})/W(\underline{k})}, L, d, \delta, F, F_L)\end{align*} $$

where $L = M, F_L = p$ and $F (dx) = x^{p-1} dx + d(\frac {\varphi (x) - x^p}{p})$ . Define

$$ \begin{align*}\mathcal{W} \omega^*_{\underline{R}/\underline{k}} = W (\underline{\omega^*})_{\mathfrak{sat}}.\end{align*} $$

The construction is clearly functorial in $\underline {R}$ . We claim that $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ is the log de Rham–Witt complex of $\underline {R}/\underline {k}$ . To prove the claim, consider any strict p-compatible log Dieudonné $W(\underline {k})$ -algebra $B^* \in \text {DA}^{\log , p}_{\mathfrak {str}}$ ; then we have

$$\begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}_{\mathfrak{str}}^{\log,p}} (\mathcal{W} \omega^*_{\underline{R}/\underline{k}}, B^*) & = \operatorname{\mathrm{Hom}}_{\mathrm{DA}^{\log,p}} (\underline{\omega^*}, B^*) &\text{by Lemma 3.12} \\& = \operatorname{\mathrm{Hom}}_F(W(\underline{R}), \underline{B}^0) & \text{by Lemma 4.9} \\& = \operatorname{\mathrm{Hom}}_F(W(\underline{R}), W(\underline{B}^0/V \underline{B}^0)) & \text{by Corollary 3.14} \\& = \operatorname{\mathrm{Hom}}_{\text{Alg}^{\log}_{\underline{k}}} (\underline{R}, \underline{B}^0/V \underline{B}^0) & \text{by Lemma 4.4}. \end{align*}$$

Notation. We denote $ \mathcal {W}_n \omega ^*_{\underline {R}/\underline {k}} := W_n \big ( (\underline {\omega ^*})_{\mathfrak {sat}} \big )$ .

Remark 4.11. For each $n \ge 1$ , write $\omega ^*_n := \omega ^*_{W_n(\underline {R})/W_n (\underline {k})}$ ; then in the construction above we may use $\varprojlim \omega ^*_n$ instead of $\omega ^*_{W(\underline {R})/ W(\underline {k})}$ . In other words, consider $\varprojlim \omega ^*_n$ as a p-compatible log Dieudonné algebra (see, for example, Remark 4.8); then we may define $\mathcal {W} \omega ^*_{\underline {R} /\underline {k}}$ as $W (\varprojlim \omega ^*_n)_{\mathfrak {sat}}$ . It suffices to show that the canonical map $\omega ^*_{W(\underline {R})/W(\underline {k})} \rightarrow \varprojlim \omega ^*n$ induces a bijectionFootnote ¹⁰

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}^{\log,p}} (\underline{\omega^*}, B^*) \;\xrightarrow{\: {}_{\sim} \:} \; \operatorname{\mathrm{Hom}}_{\mathrm{DA}^{\log,p}} (\varprojlim \omega^*_n, B^*).\end{align*} $$

For this, we observe that any map $\omega ^* \rightarrow B^*$ of log Dieudonné algebras induces a map $\omega ^*_n \rightarrow W_n(B^*)$ , which is compatible with transition maps as well as Frobenius on both sides, hence giving rise to $\varprojlim \omega ^*_n \rightarrow B^*$ on the inverse limit. Note that we have $\omega ^0_n = W_n(R) \rightarrow W_n B^0 = B^0 /V^n (B^0)$ . Similar to the proof of Lemma 4.9, we claim that the map $\omega ^1_{W(\underline {R})/W(\underline {k})} \rightarrow B^1$ factors through $\omega _n^1 \rightarrow W_n B^1$ . This follows from the fact that kernel of $\omega ^1_{W(\underline {R})/W(\underline {k})} \rightarrow \omega ^1_n$ is generated by $\operatorname {\mathrm {im}} (V^n)$ and $\operatorname {\mathrm {im}} (d V^n)$ .

4.3. Choice of charts of the log structure

We may have different log structures on a k-algebra R that give rise to the same saturated affine log scheme $(\operatorname {\mathrm {Spec}} R, \mathcal {M}^a)$ , which amounts to different choices of charts for $\mathcal {M}^a$ . This choice is almost irrelevant in forming the saturated log de Rham–Witt complexes, because the underlying Dieudonné algebras will be the same. More precisely, let $(R, \alpha : M \rightarrow R)$ be a log algebra over $\underline {k}$ and $\mathcal {M}^a$ be the log structure associated to the constant pre-log structure M on $\operatorname {\mathrm {Spec}} R$ (see Conventions in the Introduction 1.6). Let $M^{\text {sh}} = \Gamma (\operatorname {\mathrm {Spec}} R, \mathcal {M}^a)$ and denote the log algebra $(R, M^{\text {sh}})$ by $\underline {R}^{\text {sh}}$ . We then have the following lemma.

Lemma 4.12. The induced map

$$ \begin{align*}\mathcal{W} \omega^*_{\underline{R}/ \underline{k}} \rightarrow \mathcal{W} \omega^*_{\underline{R}^{\text{sh}}/ \underline{k}}\end{align*} $$

of the saturated log de Rham–Witt complexes from $\underline {R} \rightarrow \underline {R}^{\text {sh}}$ is an isomorphism on the underlying Dieudonné algebras. In particular, if $(R, M)$ and $(R, M')$ are two charts for the affine log scheme $(\operatorname {\mathrm {Spec}} R, \mathcal {M}^a)$ , then the underlying Dieudonné algebra of their saturated log de Rham–Witt complexes is canonically isomorphic.

Proof. It suffices to show that for each $n \ge 1$ , the pre-log structures $\beta : M \rightarrow W_n(R)$ and $\beta ^{\text {sh}}: M^{\text {sh}} \rightarrow W_n(R)$ induce isomorphic log structures over $\operatorname {\mathrm {Spec}} W_n(R)$ ; thus, we have

$$ \begin{align*}\omega^*_{\big(W_n(R), M \big)/W_n(\underline{k})} \;\xrightarrow{\: {}_{\sim} \:} \; \omega^*_{\big(W_n(R), M^{\text{sh}} \big)/W_n(\underline{k})}.\end{align*} $$

The proposition then follows from Remark 4.11. Now write $X = \operatorname {\mathrm {Spec}} R$ and $Y = \operatorname {\mathrm {Spec}} W_n (R)$ . Let $\mathcal {O}_X$ (respectively $\mathcal {O}_Y$ ) be the structure sheaf of $X_{\operatorname{\mathrm{\acute{\rm e}{t}}}}$ (respectively $Y_{\operatorname{\mathrm{\acute{\rm e}{t}}}}$ ) and $W_n (\mathcal {O}_X)$ be the sheaf sending $U \mapsto W_n (\Gamma (U, \mathcal {O}_X))$ . We need to show that the following two pushouts

of $\operatorname{\mathrm{{\acute{\rm e}tale}}}$ sheaves of monoids over $Y_{\operatorname{\mathrm{\acute{\rm e}{t}}}}$ are isomorphic. By topological invariance of $\operatorname{\mathrm{{\acute{\rm e}tale}}}$ sites, we may compute the pushout in $X_{\operatorname{\mathrm{\acute{\rm e}{t}}}}^\sim $ . Over X, the morphism $\beta $ corresponds to $M \xrightarrow {\alpha } \mathcal {O}_X \xrightarrow {\gamma } W_n(\mathcal {O}_X)$ , where $\gamma $ is given by Teichmuller liftings. Note that $\gamma ^{-1} (W_n (\mathcal {O}_X)^\times ) = \mathcal {O}_X^\times $ because for an algebra R we have $W_n(R)^\times \cap [R] = [R^\times ]$ , so we have $\beta ^{-1} (W_n (\mathcal {O}_X)^\times ) = \alpha ^{-1} (\mathcal {O}_X^\times )$ . Thus, the first pushout can be computed via two steps: first take the pushout of $\alpha ^{-1} (\mathcal {O}_X^\times ) \rightarrow M$ along $\alpha ^{-1} (\mathcal {O}_X^\times ) \rightarrow \mathcal {O}_X^\times $ and then along $ \mathcal {O}_X^\times \rightarrow W_n (\mathcal {O}_X)^\times $ . The first step precisely computes $\mathcal {M}^a$ , which is isomorphic to $(\mathcal {M}^{\text {sh}})^a$ by Lemma A.1; therefore, M and $M^{\text {sh}}$ indeed induce the same log structures on $Y = \operatorname {\mathrm {Spec}} W_n (R)$ .

5.1. Log Frobenius liftings of log algebras

We record a (variant of a) definition from [Reference Langer and Zink14].

The requirements on $\varphi _n$ in the definition of log Frobenius liftings can be summarised in the following commutative diagrams:

Also note that, though we stated the definitions on log algebras following [Reference Langer and Zink14], it is clear that one can globalise and extend the definitions to pre-log schemes in general.

5.2. Saturated log de Rham–Witt complex via log Frobenius liftings

For the rest of this subsection we make the following assumption.

Assumption.

(*) $$ \begin{align} \underline{R}/\underline{k}\ \text{is integral and admits a}\ p-\text{compatible log Frobenius lifting}\ (\underline{A}_n, \delta_n, \varphi_n). \end{align} $$

Examples include log algebras $\underline {R}$ that are log-smooth and integral over $\underline {k}$ (Lemma 5.2).Footnote ¹¹

5.2.1. A p-completed log de Rham complex

Write $\underline {\widehat {A}} = (\widehat {A}, M)$ where $\widehat {A} = \varprojlim A_n$ , which is equipped with the log structure from the filtered inverse limit. We then form the p-adically completed log de Rham complex

$$ \begin{align*}\widehat{\omega}^*_{\underline{\widehat{A}} /W(\underline{k})} (= {\widehat{\omega}}^*) := \varprojlim_n \omega^*_{\underline{A}_n/W_n (\underline{k})} \cong \varprojlim_n \big( {\omega}^*_{\underline{\widehat{A}}/W(\underline{k})}/p^n \big).\end{align*} $$

From Subsection A.3.1, f being integral means that the map $\mathbb {Z}[N] \rightarrow \mathbb {Z}[M]$ induced from the map of monoids is flat; therefore, for each n, $\operatorname {\mathrm {Spec}} \underline {A}_n \rightarrow \operatorname {\mathrm {Spec}} W(\underline {k})$ is log-smooth and integral and hence flat. Now, because the restriction map $R: A_{n+1} \rightarrow A_n$ is surjective, the inverse limit $\widehat {A}$ is also flat over $W(k)$ and hence p-torsion free. This allows us to apply Proposition 4.5 and get a log Dieudonné algebra $\omega ^*_{\underline {\widehat {A}}/W(\underline {k})}$ . The completed log de Rham complex $\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ has a unique cdga structure such that the canonical map $\omega ^*_{\underline {\widehat {A}}/W(\underline {k})} \rightarrow \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ is a map of cdgas. Moreover, $\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ inherits the Frobenius structure from $\omega ^*_{\underline {\widehat {A}}/W(\underline {k})}$ , which makes it a p-compatible log Dieudonné algebra.

5.2.2. A key lemma

The following lemma is a variant of Lemma 4.9. In this lemma we do not need to assume that $\underline {A}$ comes from log Frobenius liftings.

Lemma 5.3. Let $\underline {A}$ be a log algebra over $W(\underline {k})$ satisfying the conditions in either Proposition 4.5 or Proposition 4.6, and let ${\omega ^*} = \underline {\omega }^*_{\underline {A}/W(\underline {k})}$ be the log Deiudonne algebra constructed there. Let ${\widehat {\omega }}^*$ be the term-wise p-completion of $\omega ^*$ . Then for any p-torsion-free and p-complete log Dieudonné $W(\underline {k})$ -algebra $B^*$ , the canonical map

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\text{DA}^{\log}} (\widehat{\omega}^*, B^*) \cong \operatorname{\mathrm{Hom}}_{F} ({\underline{A}}, \underline{B}^0)\end{align*} $$

is a bijection. Suppose that $B^*$ is in addition strict and p-compatible; then, in fact,

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}^{\log}} (\widehat{\omega}^*, B^*) \cong \operatorname{\mathrm{Hom}} (\underline{A}/p, \underline{B}^0/V \underline{B}^0),\end{align*} $$

where the second set denotes homomorphisms of log algebras over $\underline {k}$ .

Proof. The first assertion follows directly from Lemma 4.9. For the second assertion, we appeal to the Cartier–Dieudonné–Dwork lemma, which in our setup says that, because A is equipped with a lift of Frobenius $\varphi $ satisfying $\varphi (a) \equiv a^p$ for all $a \in A,$ a homomorphism $\overline h: A \rightarrow B^0/V B^0$ has a unique lift to a homomorphism $h: A \rightarrow W(B^0/V B^0) \xrightarrow {\mu ^{-1}} B^0$ such that $F \circ h = h \circ \varphi $ . We still need to check that, given a morphism of log algebras $(\overline h, \psi )$ , its lift $(h, \psi ): \underline {A} \rightarrow \underline {B}^0$ is a morphism of log algebras; namely, the top square in the left diagram commutes:

This is indeed the case, because the log structure on $\overline B^0$ factors through the Teichmuller lifts by Corollary 3.14, as the diagram on the right indicates. Moreover, this also shows that any map $(h, \psi ): \underline {A} \rightarrow \underline {B}^0$ of log algebras comes from the lifting of a pair $(\overline h, \psi )$ . Finally, because p is $0$ in $B^0 / V B^0$ , we get $\operatorname {\mathrm {Hom}}_{F} ({\underline {A}}, \underline {B}^0) = \operatorname {\mathrm {Hom}} ({\underline {A}}/p, \underline {B}^0/ V \underline {B}^0)$ as desired.

5.2.3. Comparison with log de Rham–Witt complexes

Now let us apply the lemma above to the setup in Subsection 5.2.1, where $\underline {\widehat {A}}$ comes from a log Frobenius lifting of $\underline {R}$ . By Lemma 5.3 there is a canonical bijection

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\text{DA}^{\log, p}} (\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})}, \mathcal{W} \omega^*_{\underline{R}/\underline{k}}) \cong \operatorname{\mathrm{Hom}} ({\underline{R}}, \mathcal{W}_1 \omega^0_{\underline{R}/\underline{k}}).\end{align*} $$

Let $e: R \rightarrow \mathcal {W}_1 \omega _{\underline {R}/\underline {k}}^0$ be the tautological map from the definition of the log de Rham–Witt complex $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ . Then $(e, \text {id}): {\underline {R}} \rightarrow \mathcal {W}_1 \omega ^0_{\underline {R}/\underline {k}}$ gives rise to a map $ \upsilon : \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} \rightarrow \mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ in the category $\mathrm {DA}^{\log ,p}$ of p-compatible log Dieudonné $\underline {W}$ -algebras.

Proposition 5.4. Keep the assumption as in the beginning of Subsection 5.2. The map $\upsilon $ induces a canonical isomorphism (again denoted by $\upsilon $ )

$$ \begin{align*}\upsilon: W \big(\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})} \big)_{\mathfrak{sat}} \;\xrightarrow{\: {}_{\sim} \:} \; \mathcal{W} \omega^*_{\underline{R}/\underline{k}}.\end{align*} $$

Proof. By construction, $W \big (\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} \big )_{\mathfrak {sat}} \in \mathrm {DA}_{\mathfrak {str}}^{\log ,p}$ , so it suffices to show that for any $B^* \in \mathrm {DA}_{\mathfrak {str}}^{\log ,p}$ , the map $\upsilon : W \big (\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} \big )_{\mathfrak {sat}} \rightarrow \mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ induces a bijection between

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}_{\mathfrak{str}}^{\log,p}} (\mathcal{W} \omega^*_{\underline{R}/\underline{k}}, B^*) \stackrel{?}{=} \operatorname{\mathrm{Hom}}_{\mathrm{DA}_{\mathfrak{str}}^{\log,p}}(W \big(\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})} \big)_{\mathfrak{sat}} , B^*).\end{align*} $$

This follows from the following commutative diagram

Therefore, if $\underline {R}/ \underline {k}$ is integral and admits a p-compatible log Frobenius lifting, the saturated log de Rham–Witt complex of $\underline {R}/ \underline {k}$ can be computed by taking $\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ and then taking its associated V-complete saturation (cf. Subsection 3.4).

5.3. Aside: comparison with more general log Frobenius liftings

As an aside, we prove a more general form of Theorem 7 (3). In general, we cannot construct $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ from a non-p-compatible Frobenius lifting of $\underline {R}/\underline {k}$ as in Subsection 5.2. However, in some cases we can still compare the completed log de Rham complexes of non-p-compatible Frobenius liftings with $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ in the derived category. This crucially relies on the notion of log-Cartier type (cf. Subsection A.3.2). In contrast, the Cartier criterion is not needed in Subsection 5.2.

Each $ \omega ^1_{{\underline {A}}_n/W_n(\underline {k})}$ is a finitely generated locally free module over $A_n = \widehat {A}/p^n$ , by log smoothness of $\underline {A}_n / W_n (\underline {k})$ , so $\widehat {\omega }^1_{\underline {\widehat {A}}/W(\underline {k})} = \varprojlim \omega ^1_{{\underline {A}}_n/W_n(\underline {k})}$ is locally free over $\widehat {A}$ and hence p-torsion free. Then $\underline {\widehat {A}}$ satisfies the condition in Proposition 4.6 with $\widehat {\omega }^1_{\underline {\widehat {A}}/W(\underline {k})}$ replacing $\omega ^1_{\underline {\widehat {A}}/W(\underline {k})}$ . The conclusion of Proposition 4.6 holds by the same proof. To summarise, $\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ is equipped with a Frobenius extending $\varphi = \varprojlim \varphi _n$ , which makes it a log Dieudonné $W(\underline {k})$ -algebra,Footnote ¹² and there is a bijection

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\mathrm{DA}^{\log}} (\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})}, \mathcal{W} \omega^*_{\underline{R}/\underline{k}}) \cong \operatorname{\mathrm{Hom}} (\underline{R}, \mathcal{W}_1 \omega_{\underline{R}/\underline{k}}^0)\end{align*} $$

by Lemma 5.3. The tautological map $\underline {R} \rightarrow \mathcal {W}_1 \omega _{\underline {R}/\underline {k}}^0$ again induces a map $\upsilon : \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} \rightarrow \mathcal {W} \omega ^*_{\underline {R}/\underline {k}}.$

Theorem 5.6. Let $\underline {R}$ and $\widehat {A}$ be as in the setup above; then the map

$$ \begin{align*}\upsilon: \widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})} \longrightarrow \mathcal{W} \omega^*_{\underline{R}/\underline{k}}\end{align*} $$

induces a quasi-isomorphism on the underlying cochain complexes.

Proof. (1). First suppose that the log Frobenius lift $(\underline {A}_n, \varphi _n, \delta _n)$ in the Assumption above is p-compatible; then by Proposition 5.4, we have

$$ \begin{align*}\upsilon: \widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})} \rightarrow W(\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})})_{\mathfrak{sat}} \;\xrightarrow{\: {}_{\sim} \:} \; \mathcal{W} \omega^*_{\underline{R}/\underline{k}}.\end{align*} $$

Because both $\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}$ and $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ are p-torsion free and p-complete, it suffices to show that $\upsilon $ is a quasi-isomorphism after mod p. In other words, we need to show that

$$ \begin{align*}\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})}/p \rightarrow (\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})})_{\mathfrak{sat}} /p \rightarrow W(\widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})})_{\mathfrak{sat}} /p \;\xrightarrow{\: {}_{\sim} \:} \; \mathcal{W} \omega^*_{\underline{R}/\underline{k}}/p\end{align*} $$

is a quasi-isomorphism. The rest of the proof is similar to the proof of Corollary 2.17. The key point is that the underlying Dieudonné algebra $ \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} $ satisfies the Cartier criterion (cf. Definition 2.14) by Proposition A.10. In other words, we have

$$ \begin{align*}F: \widehat{\omega}^*_{\underline{\widehat{A}}/W(\underline{k})}/p \cong \omega^*_{\underline{R}/\underline{k}} \;\xrightarrow{\: {}_{\sim} \:} \; H^*(\omega^{\bullet}_{\underline{R}/\underline{k}}) \cong H^*(\widehat{\omega}^{\bullet}_{\underline{\widehat{A}}/W(\underline{k})}/p).\end{align*} $$

To finish the proof, note that both of the maps of $(\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})})_{\mathfrak {sat}} /p \rightarrow W(\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})})_{\mathfrak {sat}} /p$ and $ \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}/p \rightarrow (\widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})})_{\mathfrak {sat}} /p $ are quasi-isomorphisms by Corollary 2.16.

(2). For the general case, again we need to show that $\overline \upsilon : \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})}/p \cong \omega ^*_{\underline {R}/\underline {k}} \xrightarrow { \upsilon \!\!\mod p } \mathcal {W} \omega ^*_{\underline {R}/\underline {k}} /p $ is a quasi-isomorphism. The map $\overline \upsilon $ depends on the lifting $\widehat {A}$ of R and $\varphi $ of Frobenius, but the induced map in the derived category does not. More precisely, let $\underline {\widehat {A}'} / \underline {W}$ be a p-compatible formal lift of $\underline {R}/\underline {k}$ (which exists thanks to Lemma 5.2), and consider the following diagram:

where $\text {pr}: \mathcal {W} \omega ^*_{\underline {R}/\underline {k}} /p \twoheadrightarrow \mathcal {W} \omega ^*_{\underline {R}/\underline {k}} / \text {Fil}^1 = \mathcal {W}_1 \omega ^*_{\underline {R}/\underline {k}} $ is the canonical projection. Unwinding definitions, we see that $\text {pr} \circ \overline \upsilon = \text {pr} \circ \overline \upsilon '$ (the square commutes). From step (1), we know that $\overline \upsilon '$ is a quasi-isomorphism. By Theorem 5.8, $ \text {pr} \circ \overline \upsilon '$ is an isomorphism; hence, $\text {pr}$ is a quasi-isomorphism. This in turn (using Theorem 5.8 again) implies that $\overline \upsilon $ is a quasi-isomorphism.

5.4. Comparison with de Rham complexes in characteristic p

Let $\underline {R}/\underline {k}$ be a log algebra and let $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ be its saturated log de Rham–Witt complex. We have a map $(e, \text {id}): \underline {R} \rightarrow \mathcal {W}_1 \omega ^0_{\underline {R}/\underline {k}}$ , which extends uniquely to a map $\nu : \omega ^*_{\underline {R}/\underline {k}} \rightarrow \mathcal {W}_1 \omega ^*_{\underline {R}/\underline {k}}$ of cdgas that is compatible with log structures and log differentials (in fact, $\nu = \text {pr} \circ \overline \upsilon = \text {pr} \circ \overline \upsilon '$ in the proof of Theorem 5.6). Our first goal of this subsection is to prove the following proposition.

Theorem 5.8. Suppose that $\underline {R} /\underline {k}$ is log-smooth of log-Cartier type; then

$$ \begin{align*}\nu: \omega^*_{\underline{R}/\underline{k}} \;\xrightarrow{\: {}_{\sim} \:} \; \mathcal{W}_1 \omega^*_{\underline{R}/\underline{k}}\end{align*} $$

is an isomorphism.

Proof. The proof is similar to that of [Reference Bhatt, Lurie and Mathew2, Proposition 4.3.2]. Let $(\widehat {A}, M)$ be a p-compatible formal lift of $\underline {R}$ over $\underline {W}$ , which exists by Lemma 5.2. Then we have a commutative diagram of cochain complexes

where both complexes on the right are equipped with the Bockstein differentials (see Subsection 2.2). The vertical maps are both induced by $\upsilon : \widehat {\omega }^*_{\underline {\widehat {A}}/W(\underline {k})} \rightarrow \mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ , and the bottom triangle is the triangle above Lemma 2.11 applied to the saturated log de Rham–Witt complex. From the proof of part (1) of Theorem 5.6, $\overline \upsilon $ is a quasi-isomorphism, so $H(\overline \upsilon )$ is an isomorphism. The map $F_1$ is an isomorphism by part (2) of Lemma 2.11. Finally, by the assumption that $\underline {R}/\underline {k}$ is log-smooth of log-Cartier type, $C^{-1}$ is an isomorphism. Therefore, $\nu $ is an isomorphism.

Corollary 5.9. We continue to assume that $\underline {R} /\underline {k}$ is log-smooth of log-Cartier type. The projection $\mathcal {W} \omega ^*_{\underline {R} /\underline {k}} \rightarrow \mathcal {W}_1 \omega ^*_{\underline {R} /\underline {k}} \xrightarrow {\nu ^{-1}} \omega ^*_{\underline {R}/\underline {k}}$ induces a quasi-isomorphism of cochain complexes

$$ \begin{align*}\text{pr}: \mathcal{W} \omega^*_{\underline{R} /\underline{k}} /p \mathcal{W} \omega^*_{\underline{R} /\underline{k}} \;\xrightarrow{\: {}_{\sim} \:} \; \omega^*_{\underline{R} /\underline{k}}.\end{align*} $$

In other words, $\mathcal {W} \omega ^*_{\underline {R}/\underline {k}}$ is a deformation of $\omega ^*_{\underline {R}/\underline {k}}$ in the derived category.

Our next remark is helpful for the discussion on the monodromy operator and for [Reference Yao20].