Let \(K\) be a complete DVR with integer ring \(R\), residue field \(\kappa\) of characteristic \(p\), \(\pi\) an uniformizer.
Recall that the Tate algebra
\[T_n = K\langle t_1,\dots, t_n\rangle = \left\{ \sum a_I t^I : a_I\in K, |a_I|\to 0\right\}\]
is the algebra of the converging power series on the unit polydisk \(B_n = \{ |z_i|\le 1\}\).
An affinoid algebra \(A\) is a finite extension of \(T_n\), or any surjective morphism \(T_m\to A\). The space associated with it is the max-spectrum, equivalently it can be seen as a set of Galois orbits of its points in \(\overline{K}\), or Galois orbits of ‘evaluation mappings’ into \(\overline{K}\).
\[\begin{align*} X &= \mathrm{mSpec}(A) \\ &= \{ \psi: A\to_K \overline{K} \} / \mathrm{Gal}(\overline{K}/K). \end{align*}\]
For example let us denote \(G_K = \mathrm{Gal}(\overline{K}/K)\),
\(\mathrm{mSpec}(T_n) = B_n/G_K\)
\(\begin{align*} \mathrm{mSpec}(T_2/(t_1t_2-1)) &= \{(z_1, z_2)\in B_2 : z_1z_2 = 1\} / G_K\\ &= \{z\in \overline{K} : |z| = 1\}/G_K. \end{align*}\)
The space \(X=\mathrm{mSpec}(A)\) with Grothendieck topology and holomorphic function sheaves together will be called an affinoid space.
In rigid geometry we glue these affinoid spaces (instead of affine spaces in algebraic geometry). For example the open polydisk \(B_n^\circ\) is the union of the affine spaces \(\{|z_i|\le 1-\frac{1}{k}\}\), which corresponds to the power series that are under convergent (convergent on any radius \(1-\varepsilon\) disk).
The algebra \(T_1\) is not very suitable for defining de-Rham cohomology because differentiation slightly increases convergence and integration slightly reduces convergence. This makes the first cohomology of the following complex an infinite dimensional space
\[ 0\to T_1 \xrightarrow{d} T_1 \,\mathrm{d}t \to 0 \]
because there are some forms in \(T_1\,\mathrm{d}t\) that converges slowly enough whose integration does not lie in \(T_1\).
Consider the following ‘overconvergent’ version of Tate algebra
\[\mathcal{T}_n^\dagger=\left\{ \sum a_I t^I : a_I\in R, \exists r>1, |a_I| r^I \to 0 \right\}\]
which aims to solve the above problem by introducing overconvergence. Notice that we are now requiring the coefficients lie in \(R\) because we want to normalize it for reduction.
A weakly complete finitely generated algebra \(A^\dagger\) is the homomorphic image of surjective map \(\mathcal{T}_m^\dagger \to A^\dagger\) for some \(m\), \(\mathcal{T}^\dagger\) is Noetherian and we have presentations
\[ A^\dagger = \mathcal{T}_n^\dagger\]
and define (it is not the ordinary module of differentials as we are passing limit in \(\mathcal{T}^\dagger\))
\[ \Omega_{A^\dagger}^1 = \bigoplus_{i=1}^n A^\dagger \,\mathrm{d}t_i / (\,\mathrm{d}f_i), \quad \Omega_{A^\dagger}^n = \wedge^n \Omega^1_{A^\dagger}. \]
Under reduction, this \(\mathcal{T}^\dagger/\pi\) reduces to a polynomial \(\kappa\)-algebra \(\kappa[x_{1,\dots,m}]\) and \(\overline{A} = A^\dagger/\pi\) is a f.g. \(\kappa\)-algebra.
Note that \(A^\dagger\) and \(\mathcal{T}_n^\dagger\) are not actually finitely generated, they are finitely generated in completion. That’s why they are called weakly complete finitely generated algebras.
Any finitely generated smooth \(\kappa\)-algebra can be lifted to \(A^\dagger\) for which \(\overline{A}=A^\dagger/\pi\) is the \(\kappa\)-algebra we start with. i.e. the reduction
\[ \{\text{WCFG algebras }A^\dagger\} \xrightarrow{/\pi} \{\text{smooth $\kappa$-algebras } \overline{A}\}\]
is surjective. Moreover we have the following magical properties
Any two such lift \(A^\dagger\) and \(B^\dagger\) of the same algebra are isomorphic.
Any map on the reduction \(\overline{f}:\overline{A}\to \overline{B}\) lifts to maps on WCFG \(f:A^\dagger\to B^\dagger\).
Any two maps \(A^\dagger\to B^\dagger\) with the same reduction induce homotopic maps on the differential complex \[ \Omega^\bullet_{A^\dagger}\otimes K \to \Omega^\bullet_{B^\dagger}\otimes K.\]
The Monsky-Washnitzer cohomology of \(\overline{A}\) is the cohomology of the \(K\)-complex obtained above
\[ H^\bullet_{MW}(\overline{A}/K) := H(\Omega^\bullet_{A^\dagger}\otimes K).\]
which can be proved to be a finite dimensional space.
A locally analytic function on \(X\) is a map \(f:X^{geo}\to \overline{K}\) that satisfies
\(f\) is locally a convergent power series on each open residue disk, which is identifiable with a unit polydisk over a finite extension field, with variables \(z_{1,\dots,d}\).
\(f\) is \(K\)-rational, i.e. it is \(\mathrm{Gal}(\overline{K}/K)\)-equivariant.
This enables us to define two spaces, the space of locally analytic funtions \(A_{loc}\) and the space of locally analytic differential forms \(\Omega^\bullet_{loc}\).
Let \(K/\mathbb{Q}_p\) be finite extension (which we call a \(p\)-adic field), then there is a \(K\)-map
\[ \int: Z^1(\Omega^\bullet_{A^\dagger}\otimes K) \to A_{loc}/K \]
such that
\(\int\) is Frobenius equivariant, i.e. \(\phi_a^* \circ \int = \int \circ \phi_a^*\).
\(d\circ \int\) is the injection \(Z^1(\Omega^\bullet_{A^\dagger}\otimes K)\to \Omega^1_{loc}\).
\(\int\circ d\) is the projection \(A^\dagger_K\to A_{loc}/K\).
The construction goes as follows,
take a basis \(\omega_1,\dots,\omega_n\) of the finitely generated \(K\)-vector space \(H^1_{MW}(\overline{A})=H^1(\Omega^\bullet_{A^\dagger}\otimes K)\), we know that any \(\omega\in Z^1(\Omega^1_{A^\dagger}\otimes K)\) falls into some cohomology class so we have
\[ \omega = \sum_{i=1}^n \alpha_i \omega_i + \,\mathrm{d}g, \]
Thus for the integration to be defined on any \(\omega\) it suffices to compute \(F_i = \int \omega_i\), and \(\int_a^b \,\mathrm{d}g = g(b)-g(a)\).
We can consider the decomposition of the pullback of each of \(\omega_i\) and write \(\phi_a^*\omega_i = \sum_{j=1}^n M_{ij}\omega_j + \,\mathrm{d}g_i\), writting
\[ \phi_a^* \underline{\omega} = M\underline{\omega} + \,\mathrm{d}\underline{g}\]
which implies that the integration
\[ \phi_a^*\int \underline{\omega} = \int \phi_a^*\underline{\omega} = M \int \underline{\omega} + \int \,\mathrm{d}\underline{g}\]
which means
\[ \phi_a^*\int^b_a \underline{\omega} - \int^b_a \underline{\omega} = (M-I)\int^b_a \underline{\omega} + \underline{g}(b)-\underline{g}(a).\]
Here we have two facts about the Frobenius map that allows us to compute the integration,
\((\phi^*\int_a^b - \int_a^b)\omega\) is computable locally by series expansions of the following
\[\int^{\phi(b)}_{\phi(a)}\omega -\int_a^b\omega = \int_{\phi(a)}^a\omega + \int_b^{\phi(b)}\omega. \]
\(M-I\) is invertible.
This gives the following algorithm to compute the integration
\[ \int_a^b \omega = (I-M)^{-1}\left( \int_a^b \omega - \int_{\phi(a)}^{\phi(b)} \omega + g(b)-g(a)\right).\]
Besser, Heidelberg Lectures on Coleman Integration