See also the note discussing the log tangent and cotangent sheaf for a pair \((X,D)\).
Definition
Let \((X,\mathcal M_X)\) be a log scheme over \((S,\mathcal M_S)\) and let \(E\) be an \(\mathcal O_X\)-module. A log derivation of \(X\) over \(S\) with values in \(E\) is a pair \((\theta, \delta)\) where
- \(\theta:\mathcal O_X\to E\) is an \(f^{-1}\mathcal O_S\)-linear derivation, i.e. \(\theta(ab) = a\theta(b) + \theta(a)b\)
- \(\delta:\mathcal M^{gp}_X\to E\) is a homomorphism of sheaves of abelian groups
such that (i) \(\theta(\alpha_X(m)) = \alpha_X(m)\delta(m)\) for all \(m\in \mathcal M_X\) and (ii) \(\delta\circ f^\flat = 0\) on \(f^{-1}\mathcal M^{gp}_S\).
Read: \(\theta\) is an ordinary derivation and \(\delta\) records how it acts on the log structure. The first condition merely requires the two are compatible, and (ii) says that the derivation must be trivial on the base. Think of \(\delta\) as the specification of \(\theta\) on \(\mathcal M_X\).
The collection of log derivations form a sheaf of \(\mathcal O_X\)-modules under \(a\cdot (\theta,\delta) = (a\theta, a\delta)\). We denote this sheaf \(\Der^{log}_{X/S}(E)\) or \(\Der_{X/S}(\mathcal M_X)(E)\) in Ogus’s notation. Note that we could have defined \(\delta:\mathcal M_X\to E\) to be a monoid homomorphism instead, it is the same thing as the stated definition because \(E\) is already a group.
Remark
If \(X\) is integral and \(\alpha_X(m) \neq 0\) for all \(m\), as in the case of a divisorial log structure, then \(\theta\) determines \(\delta\). If \(\alpha_X\) evaluates to \(0\) somewhere then \(\delta\) is genuinely independent data. See also the special case of monoid algebras.