The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{ D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincar\'{ e} duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{ D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead us to an analog of "Weil II" and a product formula for $p$-adic epsilon factors.
Categories: math.AG