Proof. This is a special case of [Reference Hassett and Kovács27, Proposition 3.5]. Note that our (CC) is a stronger assumption than the condition on the codimension in [Reference Hassett and Kovács27, Proposition 3.5].

Proof. Let $\tilde U \subset U$ also satisfy $(CC)$ . We have by Lemma 2.3 that $j_*\Omega ^{\bullet }_{\tilde U/S} = j_*\Omega ^{\bullet }_{U/S}$ because $\Omega ^m_{U/S}$ is finite locally free. Thus, $W^{\bullet }_{X/S}$ depends only on the equivalence class of f. It is clear that it is quasi-coherent. For every sheaf $\mathcal {G}$ on U, $j_*\mathcal {G}$ is Z-closed, so in particular $W^m_{X/S}$ is Z-closed. Set $\mathcal {F}^\vee :=\operatorname {Hom}_{\mathcal {O}_X}(\mathcal {F},\mathcal {O}_X)$ . By Lemma 2.3, $\mathcal {F}^\vee $ is Z-closed for all $\mathcal {F}$ , so in particular $(W^m_{X/S})^{\vee \vee }$ is a Z-closed sheaf and it coincides with $W^m_{X/S}$ on U; hence, $(W^m_{X/S})^{\vee \vee }=W^m_{X/S}$ and $W^m_{X/S}$ is reflexive. By the extension theorem [Reference Grothendieck22, Corollaire 9.4.8], there is a coherent $\mathcal {G}$ that restricts to $W^m_{X/S}$ on U. Now $\mathcal {G}^{\vee \vee }=W^m_{X/S}$ because both are Z-closed and agree on U; hence, $W^m_{X/S}$ is also coherent. The argument for $\Theta ^m_{X/S}$ is similar.

Proof. The statement is clear on U where all sheaves are locally free and then it follows because all sheaves are Z-closed.

Proof. On U, this is [Reference Tsuji49, Theorem 2.21, (ii)], and because both sheaves are Z-closed, the statement follows.

Proof. Because P is free as a Q-set (generated by E), $\operatorname {Spec}\mathbb {Z}[P]$ is a flat $\operatorname {Spec}\mathbb {Z}[Q]$ -module. By [Reference Grothendieck24, Corollary 6.3.5] the total space of a faithfully flat morphism of Noetherian schemes is Cohen–Macaulay if and only if the base and all fibres are. By Hoechster’s theorem, the fibres of $\operatorname {Spec}\mathbb {Z}[P]\to \operatorname {Spec}\mathbb {Z}$ are Cohen–Macaulay; hence, $\operatorname {Spec}\mathbb {Z}[P]$ and $\operatorname {Spec}\mathbb {Z}[Q]$ are Cohen–Macaulay. Now flatness of $\underline f$ implies that it is Cohen–Macaulay.

Proof. By the decomposition (3.1), any element in F has the form $\bar f+q$ with $\bar f \in E, q\in Q$ . Because F is a face, $\bar f,q$ are both in F; hence, $F\subset \bar F+Q'$ . The reverse inclusion is clear.

Proof. Because $\bar F+Q'$ is free as a $Q'$ -set, $\mathbb {Z}[\bar F+Q']$ is a free $\mathbb {Z}[Q']$ -module, so the flatness statement follows. The origin $0$ given by the prime ideal $(z^q|q\in Q^+)$ is contained in $V_{Q'}$ and let $X_0$ be the fibre over it. It suffices to check the codimension condition for this particular fibre. But note that $X_0\cap V_{\bar F+Q'}= \bigcup _{F\subset \bar F}V_F$ where the union runs over faces F of P contained in $\bar F$ and we have $\dim V_F\le d-2$ by the assumption on $\bar F$ .

Proof. Because $U_P$ is a union of torus orbits, it suffices to check that any torus orbit contained in $U_P$ is contained in some $U_F$ for F essential of rank $d-1$ . Every torus orbit is given by $O_G:=\operatorname {Spec}\mathbb {Z}[G^{\operatorname {gp}}]$ for G a face of P. Assume $O_G\subset U$ . We use Lemma 3.5 to write $G=\bar G+Q'$ . If $\operatorname {rk}\bar G\le d-2$ , then $G\in \mathcal {B}$ , so $O_G\not \subset U$ . Hence, $\dim \bar G\ge d-1$ and $\bar G$ contains some essential face F of rank $d-1$ . Then F is also contained in G and thus $O_G$ is contained in $U_F$ . Conversely, because $O_F$ is not in any $V_B$ , the assertion follows.

Proof. If $\mathcal {F}=\mathcal {F}_{{\operatorname {max}}}$ , then f is saturated because $\theta $ is saturated. More generally, because $A_{P,\mathcal {F}_{{\operatorname {max}}}}\to A_{P,\mathcal {F}}$ is locally given by embedding a face, it is exact. Now by [Reference Ogus39, I.4.8.5(2)], f is saturated.

The assertion is clear if $d=0 \iff P=Q$ , so assume $d>0$ . Given Lemma 3.4, we still need to verify that U satisfies (CC) and that f is log smooth on $U_P$ . Note that Lemma 3.6 implies that $U_P$ satisfies (CC) because the complement of $U_P$ is the union of closed sets each of which has codimension at least 2 in each fibre.

To see that f is log smooth on $U_P$ , by Lemma 3.7, it suffices to check that f is log smooth on $U_F$ for F essential of rank $d-1$ . Let F be such a face. Set $\bar P_F:=P_F/F^{\operatorname {gp}}$ and note that the projection of Q to $\bar P_F$ is injective because $F^{\operatorname {gp}}\cap Q=\{0\}$ . There is an isomorphism $P_F\cong F^{\operatorname {gp}}\times \bar P_F$ commuting with the injection of Q that is $\{0\}\times Q$ on the right.

The log structure on $U_F$ is a divisorial log structure given by a set of divisors each of which pulls back from $\operatorname {Spec}\mathbb {Z}[\bar P_F]$ , so we may consider the corresponding divisorial log structure on $\operatorname {Spec}\mathbb {Z}[\bar P_F]$ to upgrade this to a log scheme $\bar U_F$ . We have a factorisation $U_F\to \bar U_F\to A_Q$ with the first map a smooth projection from a product that is therefore strict and hence log smooth. It thus suffices to show that $\bar U_F\to A_Q$ is log smooth. Note that $\bar U_F\to A_Q$ is the log morphism of an ETD with $d=1$ . The following lemma finishes the proof.

Proof. We are done by Lemma 3.8 if $\mathcal {F}=\mathcal {F}_{{\operatorname {max}}}$ , and this always holds if Q meets the interior of P. So assume that Q is contained in a proper face of P; then by Lemma 3.5 it is in fact a facet of P and then $\bar P=\mathbb {N}$ and consequently $P=\mathbb {N}\times Q$ . A facet of P that is not Q is in $\mathcal {F}_{\min }= \{\mathbb {N}\times F\,|\, F\mbox {is a facet of Q}\}$ . Hence, $\mathcal {F}\subsetneq \mathcal {F}_{{\operatorname {max}}}$ implies $\mathcal {F}=\mathcal {F}_{\min }$ and thus f is strict. Because $\underline f$ is smooth, we find that f is log smooth.

Proof. Let $\mathcal {E}_1$ be the set of essential faces of rank $d - 1$ such that when applying the proof of Lemma 3.10 to $\bar U_F \to A_Q$ from the proof of the proposition, we are in the case $\mathcal {F} = \mathcal {F}_{max}$ , and let $\mathcal {E}_2$ be the set of faces where we are in case $\mathcal {F} = \mathcal {F}_{min}$ . Then for $F \in \mathcal {E}_1$ we have $A_P|_{U_F} = A_{P,\mathcal {F}}|_{U_F}$ , and for $F \in \mathcal {E}_2$ the morphism $U_F \to A_Q$ is strict and smooth. Now we define $U_1 = \bigcup _{F \in \mathcal {E}_1} U_F$ and $U_2 = \bigcup _{F \in \mathcal {E}_2} U_F$ .

Proof. At a geometric point $\bar x\in X$ with

for $\mathcal {M}$ defined on some étale $U\to X$ that contains $\bar x$ , let $\mu _M\colon \mathcal {O}_{X,\bar x}\to \mathcal{T}\,^1_{X,\bar x}$ denote the connecting homomorphism at $\bar x$ in the long exact sequence obtained from applying $\mathcal {H} om(-,\mathcal {O}_X)$ to (5.1). By a general fact for extensions, we have $\mu _M(1)=\eta (M)$ . For $\lambda \in \mathcal {O}_{X,\bar x}^{\times }$ , let $M_{\lambda }\in \mathcal {L}\mathcal {S}_{X,\bar x}$ denote the element

. The statement of the lemma comes down to the following claim.

To prove the claim, let $U_1,U_\lambda $ denote the log smooth schemes over S respectively obtained from the log scheme U and the map to S given by and respectively. Let $(U_1)_\varepsilon $ and $(U_\lambda )_\varepsilon $ be the unique deformations of $U_1,U_\lambda $ over $S_\varepsilon $ , respectively. Let $\chi \colon U_\lambda \to U_1$ be the canonical isomorphism over $\underline S=\operatorname {Spec}\big (0\to \Bbbk )$ .

We are now going to use facts about idealised log schemes; see [Reference Ogus39, Chapter III.1.3 & IV-Variant 3.1.21] for an introduction. We give $S_\varepsilon $ the ideal $\langle 2\rangle $ generated by $2\in \mathbb {N}$ and $(U_1)_\varepsilon $ and $(U_\lambda )_\varepsilon $ the pullback ideals $K_1,K_\lambda $ respectively so that $((U_1)_\varepsilon ,K_1)$ and $((U_\lambda )_\varepsilon ,K_\lambda )$ are ideally log smooth over $(S_\varepsilon ,\langle 2\rangle )$ . The map $(S_\varepsilon ,\langle 2\rangle )\to (A_{\mathbb {N}},\emptyset )$ is an étale map of idealised log schemes and $A_{\mathbb {N}}\to \underline S$ is log smooth; hence, the composition $\pi \colon ((U_1)_\varepsilon ,K_1)\to (S_\varepsilon ,\langle 2\rangle )\to \underline S$ is log smooth. We apply the infinitesimal lifting property to the diagram

where $(U,K)$ is the idealised log scheme $U=U_1\stackrel {\chi }{=}U_\lambda $ with ideal given by

or, equivalently,

. The left vertical map $i_\lambda $ is strict for the log structure and ideal and given by a square-zero ideal. We obtain a morphism $\tilde \chi \colon (U_\lambda )_\varepsilon \to (U_1)_\varepsilon $ of log schemes that preserves the ideals and is an isomorphism on ghost sheaves. Consequently, with $\rho _\lambda \in \mathcal {M}_{(U_\lambda )_\varepsilon ,\bar x}$ and $\rho _1\in \mathcal {M}_{(U_1)_\varepsilon ,\bar x}$ , the images of the generator $1\in \mathcal {M}_{S_\varepsilon }$ respectively, we have $\tilde \chi ^*\rho _1=\tilde \lambda \cdot \rho _\lambda $ for some $\tilde \lambda \in \mathcal {O}_{(U_\lambda )^\times _\varepsilon ,\bar x}$ that restricts to $\lambda \in \mathcal {O}^\times _{U,\bar x}$ . This implies that $\tilde \chi $ becomes an isomorphism after shrinking $(U_\lambda )_\varepsilon $ , $(U_1)_\varepsilon $ if needed. Using $i_1 \circ \chi = \tilde \chi \circ i_\lambda $ , we obtain the commutative diagram

and we conclude $\eta (M_\lambda ) = \mu _M(\lambda )$ via standard homological algebra.

Proof. Set $P:=\mathcal {P}_{\bar x}$ . For (1), let $U\to X$ be an étale affine neighbourhood of $\bar x$ where

is defined and

the strict S-morphism whose underlying map is smooth. Possibly after shrinking U, via

, we obtain a strict map of extensions over $S_\varepsilon $ ,

whose underlying morphism is also smooth and hence $\Omega _{\underline {U_\varepsilon }}$ is locally free. This implies that the corresponding term $\mathcal {E} xt^1(\Omega _{\underline {U_\varepsilon }},\mathcal {O}_X)_{\bar x}$ in the long exact sequence for (5.1) vanishes and thus $\mu _{M,\bar x}$ is surjective.

To show (2), note that it suffices to show that any two elements in $\mathcal {L}\mathcal {S}_{X,\bar x}$ are isomorphic over $\underline {S}$ . Equivalently, by [Reference Gross and Siebert19, Definition 3.19 & Corollary 3.12], the composition

needs to be the constant map. By assumption, P is free and then (2) follows from the description of $\mathcal {E} xt^1(\mathcal {P}^{\operatorname {gp}}_{\bar x},\mathcal {O}^\times _{X,\bar x})$ in [Reference Gross and Siebert19, Proposition 3.14].

For (3), if $\bar x\not \in X_{\mathrm {sing}}$ , both stalks are trivial and there is nothing to show, so assume that $\bar x\in X_{\mathrm {sing}}$ . By [Reference Friedman17, Proposition 1.10], we have $\mathcal{T}\,^1_{X,\bar x}\cong \mathcal {O}_{X_{\mathrm {sing}},\bar x}$ , so the kernel of the action of $\mathcal {O}^\times _{X,\bar x}$ on $\mathcal{T}\,^1_{X,\bar x}$ is $K:=\operatorname {ker}\big (\mathcal {O}^\times _{X,\bar x}\to \mathcal {O}^\times _{X_{\mathrm {sing}},\bar x}\big )$ . If we show that K is contained in the kernel of the action of $\mathcal {O}^\times _{X,\bar x}$ on $\mathcal {L}\mathcal {S}_{X,\bar x}$ , then (3) follows from (2) and Proposition 5.2. By assumption, X is normal crossings at $\bar x$ . Let $X_1,\ldots ,X_r$ be the local components of X at $\bar x$ , $r\ge 2$ . Let $\lambda \in K$ be given and write $\lambda =1+\sum _{i=1}^r f_i$ where $f_i|_{X_j}=0$ for $i\neq j$ . We observe that $\lambda =\prod _i (1+f_i)$ because $f_if_j=0$ for $i\neq j$ . If $\mathbb {N}^r\to \mathcal {O}_{X,\bar x},\,e_i\mapsto h_i$ is a chart of X at $\bar x$ representing an element of $\mathcal {L}\mathcal {S}_{X,\bar x}$ with and $V(h_i)=X_i$ , then $e_i\mapsto (1+f_i) e_i$ defines an automorphism of $\mathcal {M}_{X,\bar x}$ compatible with the map to $\mathcal {O}_{X,\bar x}$ because $(1+f_i)h_i=h_i$ . It takes to , so $\lambda $ acts trivially on $\mathcal {L}\mathcal {S}_{X,\bar x}$ .

Proof. The second statement is Lemma 5.3. The first statement also follows from the lemma combined with the fact that for every open $U\subset X$ , the restriction map $\mathcal {L}\mathcal {S}_X(U)\to \mathcal {L}\mathcal {S}_X(U\cap V)$ is injective, which is a consequence of Corollary 6.2. Indeed, in view of the diagram on the right, that the composition of the left vertical and bottom horizontal arrow is injective implies the injectivity of the top horizontal arrow.

Proof. We borrow the notation $P_{\kappa }$ from Remark 5.4. From a careful analysis of the proof of [Reference Gross and Siebert19, Theorem 3.22] one finds that the action becomes $f_\omega \mapsto \lambda ^{\kappa _\omega }f_\omega $ where $\kappa _\omega $ is such that $\mathcal {P}_{\bar x}\cong P_{\kappa _\omega }$ at the generic point $\bar x$ of $X_\omega $ . Indeed, if a local model at $\bar x$ is given by , this is equivalent to , which explains the action. The hypothesis of Theorem 5.5 says that $\kappa _\omega =1$ for all $\omega $ , so indeed the action of $\mathcal {O}^\times _X$ on $\mathcal {L}\mathcal {S}_X$ is compatible with the ordinary action on the coherent sheaf $\bigoplus _{X_{\omega }} q_{\omega ,*}\mathcal {N}_{{\omega }}$ .

Proof. Given Lemma 6.4 and Theorem 5.5 and noting that $V=X$ for a normal crossing space and that the annihilator of $\mathcal{T}\,^1_X$ is contained in the annihilator of $\bigoplus _{X_{\omega }} q_{\omega ,*}\mathcal {N}_{{\omega }}$ , the statement becomes an elementary lemma about a cyclic module whose proof we omit.

Proof. Using assumptions in Definition 6.7, the proof is the same as the one of [Reference Gross and Siebert20, Theorem 2.6]. See also Example 4.6.

Proof. Applying Bertini’s theorem to the line bundle $\mathcal{T}\,_X^1$ on $X_{\mathrm {sing}}$ , we obtain a section $\hat s\in \Gamma (X_{\mathrm {sing}},\mathcal{T}\,_X^1)$ that gives a simple section $s\in \Gamma (X\setminus V(\hat s),\mathcal {L}\mathcal {S}_X)$ by Theorem 6.5.

Proof. We are given E that is transverse to the strata of X. We apply a slight variant of Proposition 6.9 by making sure that the zero locus Z of the section $\hat s$ generated by Bertini is transverse also to E. Theorem 1.7 gives an orbifold smoothing, but we know that it is an actual smoothing from the fact that each $\Delta _1$ is standard.

Proof. On U the result is straightforward and along Z we use the product description to make E the horizontal divisor in the ETDs by adding a summand $\mathbb {N}$ to P and the unique new facet gets included in $\mathcal {F}$ . That these give local models follows the same proof as [Reference Gross and Siebert20, Theorem 2.6], noting that we may treat the local equation for E as one of the $f_i$ in the notation of [Reference Gross and Siebert20].

Proof. This follows via an application of Bertini’s theorem.

Proof. The proof works precisely as in [Reference Gross and Siebert20]. We remark that in Lemma 2.14, the exact sequence in (2) becomes $0\to \Theta _{Y/S}\to \Theta _{\underline X/\Bbbk }(\log E)\to \mathcal {B}\to 0$ where $\Theta _{\underline X/\Bbbk }(\log E)$ denotes ordinary derivations that preserve the ideal of E. In other words, for the ordinary deformations, we consider the ones of the pair $(\underline X, E)$ rather than just $\underline X$ .

Proof. We can compose f with the projection to $\operatorname {Spec} R$ to relate the current situation to that of Proposition 7.2. The open set $U^{\operatorname {abs}}$ in the absolute case is the complement of $Z^{\operatorname {abs}}$ , the union of all codimension 2 strata. Hence, $U^{\operatorname {abs}}$ is covered by $U_F$ where F runs over the facets of P. On the other hand, the open set U for f as given in (3.3) has a cover $U_F$ where F runs over the essential faces of rank $d-1$ by Lemma 3.7. Obviously, $U^{\operatorname {abs}}\subset U$ . Note that because $W^m_f$ is locally free on U and $\mathcal {O}_U$ is $Z^{\operatorname {abs}}$ -closed, we find that $W^m_f$ is not only Z-closed but also $Z^{\operatorname {abs}}$ -closed. Consider the commutative diagram of solid arrows

(7.1)

where the top row is obtained by pushing it forward from $U^{\operatorname {abs}}$ . The bottom sequence is obtained from tensoring the sequence $0\to Q^{\operatorname {gp}}\to P^{\operatorname {gp}}\to P^{\operatorname {gp}}/Q^{\operatorname {gp}}\to 0$ with $\mathcal {O}_{A_P}$ ; in particular, it is exact and splits. Hence, the dotted diagonal arrow exists and commutes with the other maps. Therefore, $\operatorname {coker}(\iota )$ is a direct summand of $W^1_{A_{P,\mathcal {F}}/\operatorname {Spec} R}$ ; in particular, $Z^{\operatorname {abs}}$ -closed. Moreover, $\operatorname {coker}(\iota )\to W^1_{f}$ is an isomorphism on $U^{\operatorname {abs}}$ and because both sheaves are $Z^{\operatorname {abs}}$ -closed, we have $\operatorname {coker}(\iota )=W^1_{f}$ and thus the top row is exact and splits.

Let $\langle f^*\Omega _{A_Q/\operatorname {Spec} R}\rangle $ denote the homogeneous ideal in the sheaf of exterior algebras $W^{\bullet }_{A_{P,\mathcal {F}}/\operatorname {Spec} R}$ generated by $f^*\Omega _{A_Q/\operatorname {Spec} R}$ . The split exactness above gives the split exactness of the following sequence:

$$ \begin{align*}0\to \langle f^*\Omega_{A_Q/\operatorname{Spec} R}\rangle_m \to W^m_{A_{P,\mathcal{F}}/\operatorname{Spec} R} \to W^m_{f}\to 0.\end{align*} $$

Because $A_P$ is affine and $\langle f^*\Omega _{A_Q/\operatorname {Spec} R}\rangle $ coherent, applying $\Gamma (A_P,\cdot )$ to this sequence yields another exact sequence that already gives that $\Gamma (A_P,W^m_f)$ is P-graded. We have $\Gamma (A_P,f^*\Omega _{A_Q/\operatorname {Spec} R})= Q^{\operatorname {gp}}\otimes _{\mathbb {Z}} R[P]$ . Set ${\textbf {F}}_p:=\left (\bigcap _{{F \in \mathcal {F}_{max} \setminus \mathcal {F}}\atop {p \in F}} F^{gp} \otimes _{\mathbb {Z}} R\right )$ and let $\langle Q^{\operatorname {gp}}\otimes R\rangle \subset \bigwedge ^{\bullet }_R {\textbf {F}}_p $ be the homogeneous ideal generated by $Q^{\operatorname {gp}}\otimes R$ . One computes $\Gamma (A_P,\langle f^*\Omega _{A_Q/\operatorname {Spec} R}\rangle _m)_p=\langle Q^{\operatorname {gp}}\otimes R\rangle _m$ . Using Proposition 7.2, in degree $p\in P$ , we obtain the exact sequence

$$ \begin{align*}0 \to \langle Q^{\operatorname{gp}}\otimes R\rangle_m \to \bigwedge^m_R {\textbf{F}}_p \to (W^m_f)_p \to 0.\end{align*} $$

Using a splitting of the injection $(Q^{\operatorname {gp}}\otimes R)\subset {\textbf {F}}_p$ and comparing leads to the assertion.

Proof. Inspecting the result in Proposition 7.3, we find that $\Gamma (A_P,W^m_f)$ is a free $R[Q]$ -module.

Proof. This follows directly from the local statement Corollary 7.9.

Proof. Again, this follows directly from the local statement Corollary 7.10 combined with the fact that we can use a finite cover by local models.

Proof. Because $W^m_{X/S}$ is flat over S – this is Corollary 7.4 – the proof becomes identical to that in [Reference Illusie29, Proposition 6.6].

Proof. By [Reference Grothendieck25, Theorem 8.8.2 (ii)], [Reference Grothendieck25, Theorem 8.10.5] and [Reference Grothendieck25, Theorem 11.2.6 (ii)] we can find a $\lambda \in \Lambda $ and a morphism $f_\lambda \colon X_\lambda \to S_\lambda $ that is finitely presented and flat and an isomorphism $S \times _{S_\lambda } X_\lambda \cong X$ over S. If $f\colon X \to S$ is separated respective proper, we can choose $f_\lambda $ moreover separated respective proper. Using [Reference Grothendieck25, Corollaire 12.1.7(iii)] and [Reference Grothendieck25, Theorem 8.10.5], we can choose $\lambda $ such that $f_\lambda $ is a Cohen–Macaulay morphism. Because these decompose disjointly over the relative codimension, again by increasing $\lambda $ if needed, we may assume that $f_\lambda $ has relative dimension d.

We next spread out U such that $U_\lambda \subset X_\lambda $ satisfies (CC). We do this by spreading out its complement Z. Indeed, by [47, 05M5, Lemma 31.16.1], we can increase $\lambda $ so that every fibre of $Z_\lambda \to S_\lambda $ has dimension $\leq d - 2$ and then define $U_\lambda :=X_\lambda \setminus Z_\lambda $ .

Now a straightforward generalisation of the method of [Reference Tsuji48, Lemma 4.11.1] yields that, for appropriate $\lambda $ , we can find a log structure on $U_\lambda $ and upgrade $f_\lambda $ to a log morphism such that $U_\lambda $ is fs and $f_\lambda $ is log smooth and saturated. More precisely, we choose an affine étale cover $\{U_i\}_i$ of U such that $U_i \to S$ admits a local model by a saturated morphism $\theta _i\colon Q \to P_i$ of monoids; the local model remains a local model for an appropriate spread-out $U_{i,\lambda } \to S_\lambda $ (for appropriate $\lambda $ ), which thus carries the structure of a saturated log smooth morphism to $S_{\lambda }$ . At this point, the $U_{i,\lambda }$ might not cover $U_\lambda $ , the log structures on $U_{i,\lambda }$ might not coincide on overlaps, and even if this was the case, we might not have a log morphism to $S_\lambda $ . We achieve all of this by increasing $\lambda $ .

Finally – again by possibly increasing $\lambda $ – we show that the family $f_\lambda \colon X_\lambda \to S_\lambda $ is log toroidal. We fix a finite covering $\{V_i \to X\}$ with local models $(Q\subset P_i, \mathcal {F}_i)$ as in Definition 4.1, and for each of them we construct a diagram as in Figure 9.1. Namely, we first spread out $V_i \to S$ to $V_{i,\lambda } \to S_\lambda $ . Then $L_{i,\lambda }$ is defined by base change, and we construct the étale morphisms of schemes $g_\lambda \colon V_{i,\lambda } \to X_\lambda $ and $h_\lambda \colon V_{i,\lambda } \to L_{i,\lambda }$ also by spreading out. We can assume that $X_\lambda $ is covered by $\{V_{i,\lambda } \to X_\lambda \}$ and that $\tilde U_i \subset V_i$ spreads out to an open $\tilde U_{i,\lambda } \subset V_{i,\lambda }$ satisfying (CC). We get two log structures $(g_\lambda )^*_{log}\mathcal {M}_{X_\lambda }$ and $(h_\lambda )^*_{log}\mathcal {M}_{L_{i,\lambda }}$ on $\tilde U_{i,\lambda }$ , which we identify by [Reference Tsuji48, Sublemma 4.11.3]. By the same sublemma, the two morphisms $(g \circ f)^*_{log}\mathcal {M}_{S_\lambda } \to \mathcal {M}_{\tilde U_{i,\lambda }}$ coming from $f_\lambda \circ g_\lambda $ respectively $r_\lambda \circ h_\lambda $ coincide. Because $\{V_i \to X\}$ is a finite covering, we can find $\lambda $ that admits the above construction for all $V_i$ simultaneously.

Figure 9.1 The diagram constructed in the text.

Proof. This is [Reference Kato32, Theorem 4.12(1)] once we identify $U''=U'$ : Kato considers the factorisation $U\stackrel {g}\to U''\stackrel {h}\to (U')^{\operatorname {int}}\stackrel {i}\to U'$ of $F|_U$ where i is the integralisation of $U'$ and $g\circ h$ is the unique factorisation of this weakly purely inseparable morphism where h is étale and g purely inseparable, using [Reference Kato32, Proposition 4.10(2)]. Now i is an isomorphism because f is integral. By [Reference Ogus39, Corollary III.2.5.4], because $f\colon U \to S$ is saturated, $F\colon U\to U'$ is exact. The uniqueness of the factorisation $g\circ h$ now implies that h is an isomorphism.

Proof Proof of Lemma 10.4

Set $V := c^{-1}(U_{P})$ and let $Y', V'$ be the base changes by the absolute Frobenius $F_T$ . Let $F\colon Y \to Y'$ be the relative Frobenius. Inspired by the Frobenius decomposition [Reference Deligne and Illusie13, Theorem 2.1], we construct a homomorphism $\phi ^{\bullet } : \bigoplus _{m} W^m_{Y'/T}[-m] \to F_*W^{\bullet }_{Y/T}$ of complexes of $\mathcal {O}_{Y'}$ -modules that induces an isomorphism in cohomology. Because the left-hand side has zero differentials, the assertion then follows from the reflexivity of $W^m_{Y'/T}$ given by Lemma 2.4.

Similar to Subsection 7.1, we find explicitly that $R' := \Gamma (Y',\mathcal {O}_{Y'}) = \bigoplus _{e \in E} z^e \cdot \mathcal{T}\,$ with

$$ \begin{align*}z^{e_1} \cdot z^{e_2} = z^e \cdot \sigma(q)^p \quad \mathrm{whenever} \quad e_1 + e_2 = e + q\end{align*} $$

with $e \in E, q \in Q$ . We have $s^*(z^e \cdot t) = z^e \cdot t^p$ and $F^*(z^e \cdot t) = z^{p \cdot e} \cdot t$ . After writing $W^m_e := (W^m_f)_e \otimes _{\mathbb {F}_p} \mathcal{T}\,$ , the module $\Gamma (Y', W^m_{Y'/T})$ is given by the $\mathcal{T}\,$ -module $\bigoplus _{e \in E} z^e \cdot W^m_e$ on which $R'$ acts as

$$ \begin{align*}(z^{e_1} \cdot t_1) \cdot [z^{e_2} \cdot (w \otimes t_2)] = z^{e} \cdot (w \otimes \sigma( q)^pt_1t_2) \quad \mathrm{whenever} \quad e_1 + e_2 = e + q\end{align*} $$

with $e \in E, q \in Q$ . Similarly, $\Gamma (Y',F_*W^m_{Y/T})$ is given by the same $\mathcal{T}\,$ -module, however now with $R'$ acting via $F^*$ as

$$ \begin{align*}(z^{e_1} \cdot t_1) \cdot [z^{e_2} \cdot (w \otimes t_2)] = z^{e} \cdot (w \otimes \sigma(q)t_1t_2) \quad \mathrm{whenever} \quad p \cdot e_1 + e_2 = e + q.\end{align*} $$

Note the subtle difference. The differential on $F_*W^{\bullet }_{Y/T}$ is given by $d(z^e \cdot (w \otimes t)) = z^e \cdot ([e] \wedge w \otimes t)$ . We define

$$ \begin{align*}\phi^{\bullet} \colon \bigoplus_{m} W^m_{Y'/T}[-m] \to F_*W^{\bullet}_{Y/T}, \quad z^e \cdot (w \otimes t) \mapsto z^{p \cdot e} \cdot (w \otimes t),\end{align*} $$

and claim that $\mathcal {H}^m(\phi ^{\bullet })$ is an isomorphism. Indeed, first note that $\phi ^{\bullet }$ itself is injective. Then set $E_p = \{p \cdot e | e \in E\}$ . We have $\mathrm {im}(\phi ^m) = \bigoplus _{e \in E_p} z^e \cdot W^m_e$ because $W^m_e=W^{m}_{e/p}$ for $e\in E_p$ by Proposition 7.3. Denoting the coboundaries of $F_*W^m_{Y/T}$ by $B^m$ , we have $\mathrm {im}(\phi ^m) \cap B^m = 0$ because $0 = [e] \in W^1_e$ for $e \in E_p$ because $e=pe'$ and p is zero in $\mathcal{T}\,$ . This readily gives that $\mathcal {H}^m(\phi ^{\bullet })$ is injective. For surjectivity, if $e \notin E_p$ , observe that $[e] \not = 0$ , so if $w \in W^m_e$ , then $[e] \wedge w = 0$ if and only if there is some $w' \in W^{m - 1}_e$ with $[e] \wedge w' = w$ .