Sheaf Of Log Differentials

Definition

Let \((X, \mathcal M_X)\) be a log scheme over \((S, \mathcal M_S)\). The sheaf of log differentials is

\begin{equation*} \Omega^1_{X/S}(\log) = \left(\Omega^{1}_{\underline X/\underline X} \oplus (\mathcal O_X\otimes_{\mathbb Z}\mathcal M^{gp}_{X})\right)\Big/\mathcal K \end{equation*}

where \(\mathcal K\) is the \(\mathcal O_X\)-submodule generated locally by elements that look like

\begin{align*} (d\alpha_X(m), ~-\alpha_X(m)\otimes m), \quad m\in \mathcal M_X \end{align*}

and

\begin{align*} (0, ~1\otimes f^{\flat}(n)), \quad n\in f^{-1}\mathcal M^{gp}_S. \end{align*}

If we write \(d\log m\) for the class of \((0, 1\otimes m)\) then the relations read

\begin{align*} d\alpha_X(m) = \alpha_X(m)d\log m, \quad d\log f^\flat(n) = 0, \end{align*}

and \(d\log\) is additive: \(d\log(m + m') = d\log m + d\log m'\).