  We define the notion of projective limit of local shift morphisms of type
$\left( r,s\right) $ and endow the space of such mathematical objects with an
adapted differential structure. The notion of shift Poisson tensor $P$ on a
Hilbert tower corresponds to such morphisms which are antisymmetric and whose
Schouten bracket $\left[ P,P\right] $ vanishes. We illustrate this notion with
the example of the famous KdV equation on the circle $\mathbb{S}^{1}$ for which
one can associate a couple of compatible Poisson tensors of this type on the
Hilbert tower $\left( H^{n}(\mathbb{S}^{1})\right) _{n\in\mathbb{N}% ^{\ast}}$.

Differential Geometry