Log-smooth saturated morphism and relative log-canonical line

Let $f:(X,M_X)\to (S,M_S)$ be a morphism of fs log schemes over a field. In this page, “log smooth” means that étale locally it has charts $P_S\to P_X$ for which the induced scheme map $X⟶S{\times }_{\mathrm{Spec}\mathbb{Z}[P_S]}\mathrm{Spec}\mathbb{Z}[P_X]$ is smooth and the kernel of $P_S^{\mathrm{g}\mathrm{p}}\to P_X^{\mathrm{g}\mathrm{p}}$ and torsion subgroup of its cokernel are finite with orders invertible in ${\mathcal{O}}_X$. In characteristic zero this invertibility condition is automatic. This is the Kato chart convention; no underlying-scheme smoothness of $X\to S$ is asserted.

The chart square is

The diagram commutes by the definition of a morphism with charts; the map to the fibre product in the displayed criterion is induced by its top and left arrows. The diagram records the convention, not a proof of the chart criterion.

A homomorphism $u:P\to R$ of integral monoids is integral if every pushout $R{\oplus }_P{P}^{\mathrm{\prime }}$ in the category of monoids is integral for every map from $P$ to an integral monoid $P^{\mathrm{\prime }}$. A map $v:H\to K$ of integral monoids is exact if $H=(v^{\mathrm{g}\mathrm{p}}{)}^{-1}(K)$ inside $H^{\mathrm{g}\mathrm{p}}$. For each prime $p$, form the pushout $R_p=R{\oplus }_{P,p}P$ in integral monoids, using $u:P\to R$ and multiplication by $p:P\to P$. Its canonical map $h_p:R_p\to R$ sends the image of $r\in R$ to $pr$ and the image of $a\in P$ to $u(a)$. The map $u$ is saturated if it is integral and every $h_p$ is exact. For saturated $P,R$, equivalently it is integral and every pushout $R{\oplus }_P{P}^{\mathrm{\prime }}$ is saturated whenever $P^{\mathrm{\prime }}$ is saturated. A morphism of integral log schemes is saturated if its induced characteristic-monoid homomorphism ${\bar{M}}_{S,\bar{s}}\to {\bar{M}}_{X,\bar{x}}$ is saturated at every geometric point $\bar{x}\mapsto \bar{s}$. For fs monoids this agrees with the local chart definition in the cited source.

In this item a log morphism is proper when its underlying scheme morphism is proper, meaning finite type, separated, and universally closed. Its log-relative dimension is $d$ when ${\mathrm{\Omega }}_{X\mathrm{/}S,\log }^1$ is locally free of rank $d$. These clauses fix the use of “proper” and “relative dimension” in the log-BTT interface.

The sheaf of relative log differentials ${\mathrm{\Omega }}_{X\mathrm{/}S,\log }^1$ is characterized by its universal property: its sections represent pairs consisting of an ordinary relative derivation $d:{\mathcal{O}}_X\to \mathcal{E}$ and an additive map $d\log :M_X^{\mathrm{g}\mathrm{p}}\to \mathcal{E}$ with $d(\alpha (m))=\alpha (m)d\log (m)$, vanishing on the image of $M_S^{\mathrm{g}\mathrm{p}}$; $\mathcal{E}$ ranges over ${\mathcal{O}}_X$-modules. When it is locally free of rank $d$, the relative log-canonical line is ${\omega }_{X\mathrm{/}S}:={\bigwedge }^d{\mathrm{\Omega }}_{X\mathrm{/}S,\log }^1$. The phrase “trivial relative log-canonical line” means a specified isomorphism ${\omega }_{X\mathrm{/}S}\cong {\mathcal{O}}_X$.

Fine saturated log schemes and toric monoids

Sheaf of relative Kähler differentials