Categories
Let \(h:Y\to Z\) be a morphism of fine log schemes of finite type.
The category of locally free \(\mathcal{O}_Y\)-modules of finite rank with log connection is denoted by \(\mathrm{MC}(Y/Z)\).
The coherent one is denoted by \(\widetilde{\mathrm{MC}}(Y/Z)\).
The full subcategory consisting of integrable ones are denoted by \(\mathrm{MIC}(Y/Z)\) and \(\widetilde{\mathrm{MIC}}(Y/Z)\) respectively.
Here \(Z\) should be thought as the parameter space, where we only take derivative inside the fibres. So when \(Y=Z\), the above categories collapse to the category of modules with no connection.
Functors
A diagram of arrows
- Pull-back functors
\[ f^* : \mathrm{MC}(Y/Z) \to \mathrm{MC}(Y'/Z')\]
\[ f^* : \widetilde{\mathrm{MC}}(Y/Z) \to \widetilde{\mathrm{MC}}(Y'/Z') \]
Particular Cases
pulling back along the identity morphism \(Y\to Y\) gives the forgetful functor \[ 1_Y^* : \mathrm{MIC}(Y/Z) \to \mathrm{MIC}(Y/Y). \]
pulling back from the identity morphism \(Z\to Z\) gives \[ h^*: \mathrm{MIC}(Z/Z) \to \mathrm{MIC}(Y/Z)\]
push-forward functor \[ h_*: \mathrm{MIC}(Y/Z) \to \mathrm{MIC}(Z/Z) \]
De Rham Complexes
An object \(E=(E,\nabla)\) in \(\widetilde{\mathrm{MIC}}(Y/Z)\) gives a de Rham complex (recall that a connection automatically gives such a complex by differentiating using the multiplicative rule, with Koszul sign rule)
\[ E\mapsto E \otimes_{\mathcal{O}_Y} \Omega_{Y/Z}^\bullet \in D^b_{coh}(Y) \]
So the \(i\)-th relative de Rham cohomology associated with \(E\) on \(Y/Z\) is given by
\[R^ih_{dR*}(E):= H^\bullet_{dR}(Y/Z, E) = H^\bullet(E \otimes_{\mathcal{O}_Y} \Omega_{Y/Z}^\bullet) \in D^b_{coh}(Z)\]
flat connections on \(X/K\) to flat connections on \(S/K\), take \(\mathcal{V}\) to \((\pi_*\mathcal{V})^{\nabla_{X/S}=0} = R^0\pi_*(\Omega_{X/S}(\mathcal{V}))\).