Author: Eiko

Tags: algebraic geometry, connections

Time: 2024-10-04 08:42:29 - 2024-10-04 08:42:29 (UTC)

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

$$f^{\ast }:\mathrm{MC}(Y/Z)\to \mathrm{MC}(Y^{\prime }/Z^{\prime })$$

$$f^{\ast }:\widetilde{\mathrm{MC}}(Y/Z)\to \widetilde{\mathrm{MC}}(Y^{\prime }/Z^{\prime })$$

• Particular Cases

  • pulling back along the identity morphism $Y\to Y$ gives the forgetful functor $$1_Y^{\ast }:\mathrm{MIC}(Y/Z)\to \mathrm{MIC}(Y/Y).$$

  • pulling back from the identity morphism $Z\to Z$ gives $$h^{\ast }:\mathrm{MIC}(Z/Z)\to \mathrm{MIC}(Y/Z)$$

  • push-forward functor $$h_{\ast }:\mathrm{MIC}(Y/Z)\to \mathrm{MIC}(Z/Z)$$

De Rham Complexes

• An object $E=(E,\mathrm{\nabla })$ 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)

  $$E\mapsto E{\otimes }_{{\mathcal{O}}_Y}{\mathrm{\Omega }}_{Y/Z}^{\bullet }\in D_{coh}^b(Y)$$

• So the $i$-th relative de Rham cohomology associated with $E$ on $Y/Z$ is given by

  $$R^i{h}_{dR\ast }(E):=H_{dR}^{\bullet }(Y/Z,E)=H^{\bullet }(E{\otimes }_{{\mathcal{O}}_Y}{\mathrm{\Omega }}_{Y/Z}^{\bullet })\in D_{coh}^b(Z)$$

flat connections on $X/K$ to flat connections on $S/K$, take $\mathcal{V}$ to $({\pi }_{\ast }\mathcal{V}{)}^{{\mathrm{\nabla }}_{X/S}=0}=R^0{\pi }_{\ast }({\mathrm{\Omega }}_{X/S}(\mathcal{V}))$.