This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (we can assume that $T$ is affine). Let $\mathbb{T}$ be a smooth lift of $T$ over the ring $W\_{2}(k)$, the ring of witt vectors of length 2 over k. Now, if $H\_{\mathbb{T}}$ is an F-crystal on $\mathbb{T}$ and $F\_{1}$ and $F\_{2}$ are two lifts of Frobenius $F\_{T}$ on T to $\mathbb{T}$, I know that there must exist an isomorphism 
$X(F\_{1}, F\_{2})$ between $F\_{1}^\*H\_{\mathbb{T}}$ and $F\_{2}^\*H\_{\mathbb{T}}$ coming from the nconnection on $H\_{\mathbb{T}}$. What is the explicit formula of this isomorphism? 



Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius 
This is parallel transport. Note that you only get an explicit formula for the isomorphism between $F\_1^\*H(\mathbb{T})$ and $F\_2^\*H(\mathbb{T})$. To get explicit formulas over other thickenings of $\mathbb{T}$, you would need lifts of both $\mathbb{T}$ and also of the Frobenius lifts. For convenience, set $H\_1=H(\mathbb{T})$. We will assume that $\mathbb{T}$ has \'etale co-ordinates $x\_1,\ldots,x\_n$.


Since $H\_{\mathbb{T}}$ is a crystal over $\mathbb{T}$, there is a flat connection $\nabla:H\_1\to H\_1\otimes\Omega^1\_{\mathbb{T}/W\_2(k)}$. Note that, for any $a\in\mathcal{O}\_{\mathbb{T}}$, $\delta(a)=F\_1(a)-F\_2(a)$ is in $p\mathcal{O}\_{\mathbb{T}}$; in particular, $\delta(a)^2=0$. Now, for any $h\in H\_1$, the isomorphism sends $F\_1^\*h$ to $F\_2^\*h+\sum\_{i=1}^n\nabla(\partial\_i)(m)\delta(x\_i)$. This is just a Taylor expansion up to the first degree.


