connections extended

Let $X$ be a separated scheme over a scheme $S$ . Let $\mathcal{V}$ be a quasi coherent sheaf over $X$ with a connection

$$\nabla : \mathcal{V}\to \Omega^1_{X/S} \otimes_{\mathcal{O}_X} \mathcal{V}$$

on it. We extend $\nabla$ to

$$\nabla:\Omega^\bullet_{X/S} \otimes_{\mathcal{O}_X}\mathcal{V} \to \Omega^\bullet_{X/S} \otimes_{\mathcal{O}_X} \mathcal{V}.$$

by defining

$$\nabla(\alpha\otimes v) =(d\alpha )\otimes v + (-1)^k \alpha \wed... ...abla v) \qquad (\forall \alpha \in \Omega^k_{X/S}, \forall v \in \mathcal{V}).$$

In other words,

$$\nabla \vert _{\Omega^k_{X/S}\otimes \mathcal{V}}= d\otimes \operatorname{id}+(-1)^k \operatorname{id}\wedge \nabla.$$

It is easy to verify that $\nabla$ is well-defined and satisfy

$$\nabla(\alpha\wedge x)= (d \alpha) \wedge x + (-1)^k \alpha \wed... ...{X/S}, \forall x \in \Omega^\bullet_{X/S}\otimes_{\mathcal{O}_X}\mathcal{V}).$$