Integrable algebraic connection

Definition

Let $X$ be a smooth scheme over a field $k$ , and let $\mathcal E$ be an $\mathcal O_X$ -module. An algebraic connection on $\mathcal E$ is a $k$ -linear map

\nabla:\mathcal E\longrightarrow \mathcal E\otimes_{\mathcal O_X}\Omega^1_{X/k}

satisfying the Leibniz rule

\nabla(f s)=f\nabla(s)+s\otimes df.

The connection extends to $\mathcal E\otimes\Omega^\bullet_{X/k}$ . It is integrable, or flat, if its curvature vanishes:

\nabla^2: \mathcal E\longrightarrow \mathcal E\otimes\Omega^2_{X/k}, \qquad \nabla^2=0.

Relation to D-modules

In characteristic $0$ , a finite-rank locally free sheaf with an integrable connection is equivalently an $\mathcal O_X$ -coherent $D$ -module.

References

Nicholas M. Katz, “Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin,” Publications Mathématiques de l’IHÉS 39 (1970), 175–232. DOI.