# Cohomology of endomorphisms

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Let $E \rightarrow M$ be a vector bundle over the manifold $M$, and $\nabla$ is a connection. I define a differential $d$ for the exterior forms with values in the endomorphisms of $E$:

$$d (A)(X)=[\nabla_X,A]$$

$$d(A_Z)(X,Y)=[\nabla_X, A_Y]-[\nabla_Y,A_X]-A_{[X,Y]}$$

and for the other degrees, I define $d$ by the Leibniz rule:

$$d(\alpha \wedge \beta)=d(\alpha)\wedge \beta +(-1)^{deg(\alpha)} \alpha \wedge d(\beta)$$

Can we have $d^2=0$, and can we define a cohomology?

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