Rational Analytic Prismatization

The paper develops a $Q_p$-linear analogue of prismatization using rational analytic spaces with nontrivial overconvergent structure. The rational analytic prismatization functor places Gelfand stacks between rigid analytic and perfectoid geometry, encoding $(\varphi,\Gamma)$-modules and geometric Sen theory. The central result is a categorical equivalence of perfect complexes on the $p$-adic twistor stack with those on the relative Fargues–Fontaine curve for smooth rigid spaces. Analytic setting and central construction “Rational analytic prismatization” develops a QpQ_p-linear analogue of prismatization for rigid and Berkovich geometry. Its primary objective is to construct a geometric object whose coherent and perfect-coherent sheaf theories encode relative (φ,Γ)(\varphi,\Gamma)-modules, pro-étale QpQ_p-cohomology, and geometric Sen theory. The paper works throughout with rational coefficients and postpones the integral ZpZ_p-theory, refined filtered constructions, and syntomification. The geometric framework is that of Gelfand rings and Gelfand stacks. This choice is essential: Gelfand geometry contains Banach QpQ_p-algebras, admits a descendable topology with nilperfectoid rings as a basis, and retains enough nonreduced analytic structure to model overconvergent thickenings of perfectoid spaces. The perfectoidization functor (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p} forgets precisely the overconvergent thickening data, while the analytic de Rham functor is its right adjoint. The resulting formalism places Gelfand stacks between rigid analytic geometry and arc-stack/perfectoid geometry. The basic analytic analogue of a δ\delta-ring is a rational δ\delta-space (S,X)(S,\mathcal X). Here SS is a perfectoid space in characteristic (φ,Γ)(\varphi,\Gamma)0, (φ,Γ)(\varphi,\Gamma)1 is a Berkovich space with Frobenius, and (φ,Γ)(\varphi,\Gamma)2 induces an isomorphism after perfectoidization. Thus (φ,Γ)(\varphi,\Gamma)3 is an overconvergent or derived thickening of the punctured relative Fargues–Fontaine disc (φ,Γ)(\varphi,\Gamma)4. A rational analytic prism is a (φ,Γ)(\varphi,\Gamma)5-space equipped with a degree-(φ,Γ)(\varphi,\Gamma)6 Cartier divisor (φ,Γ)(\varphi,\Gamma)7. The degree condition means that the divisor restricts on (φ,Γ)(\varphi,\Gamma)8 to the divisor associated with an untilt of (φ,Γ)(\varphi,\Gamma)9. For a Gelfand stack QpQ_p0, the analytic prismatic site consists of quadruples QpQ_p1 with …