Author: Eiko

Tags: algebraic geometry, universal extension, connections, de Rham cohomology, homological algebra

Time: 2024-10-08 13:19:48 - 2024-10-08 13:19:48 (UTC)

Construction

Associated De Rham Complex

Given a vector bundle with connection $(\mathcal{V},\mathrm{\nabla })$ over $\pi :X\to S$, i.e. the connections are ${\mathcal{O}}_S$ linear. We can associate a de-Rham complex

$$(\mathcal{V},\mathrm{\nabla })\mapsto {\mathrm{\Omega }}_{X/S}^{\bullet }(\mathcal{V})=\mathcal{V}{\otimes }_{{\mathcal{O}}_X}{\mathrm{\Omega }}_{X/S}^{\bullet }$$

Side Note

I think the category of flat connections on $X/S$ can be viewed as the category of $D_{X/S}$-modules where $D_{X/S}=⟨{\mathcal{O}}_X,{\mathrm{\Theta }}_{X/S}⟩$ is the relative $D$-ring, with ${\mathrm{\Theta }}_{X/S}=\ker ({\pi }_{\ast }:{\mathrm{\Theta }}_X\to {\mathrm{\Theta }}_S)$ the relative tangent sheaf, those tangent vectors that does not project to $S$.

Two Way Pushforward Spectral Sequence

The two projection maps

$$X\stackrel{\pi }{\to }S\stackrel{p_S}{\to }\mathrm{Spec}k$$

Gives pushforward functors

$${\pi }_{\ast }:{}_{D_{X/S}}\mathbf{Mod}\to {}_{{\mathcal{O}}_S}\mathbf{Mod}$$