Zoran Skoda MR4332074

Akhil Mathew, Faithfully flat descent of almost perfect complexes in rigid geometry. J. Pure Appl. Algebra 226 (2022), no. 5

Descent theory is a systematic approach due Grothendieck to describe global geometrical objects in terms of local, where locality is usually understood in terms of some Grothendieck topology. In algebraic geometry, the most standard case is the descent of quasicoherent sheaves along faithfully flat morphisms. It boils down to the basic case of descent of modules along faithfully flat morphisms of commutative rings (which holds more generally along pure morphisms, and for noncommutative rings). In a basic traditional flavour of rigid analytic geometry, rings are replaced by affinoid algebras. A result due Bosch, Görtz and Gabber establishes an effective descent theorem for finitely generated modules along a faithfully flat map of $K$-affinoid algebras $A\to A'$ where $K$ is a complete nonarchimedean field. The finitely generated $A$-modules are described as descent data which live over a standard cosimplicial object whose lowest degree objects are $A'$, $A'\hat{\otimes}_A A'$, $A'\hat{\otimes}_A A'\hat{\otimes}_A A'$, the tensor product $\hat{\otimes}$ is completed and the 2-truncation suffices.

The work under review generalizes the result in two directions. It is first observed that a relatively straightforward argument may extend the descent for modules to a flat hyperdescent theorem for $\infty$-categories of almost perfect $R$-complexes, keeping the same assumption on the base ring $R$ (a $K$-affinoid algebra). Descent for $\infty$-categories involves $\infty$-categorical/homotopy limits and hypercovers which are therefore not determined by a 2-truncation (if presented in simplicial language). Then a number of advanced techniques are used to relax the conditions on the base ring. This is partly motivated by a descent result of V. Drinfeld for finitely generated projectives over $\pi$-torsion free $\pi$-adically complete $\mathcal{O}_K$-algebras, where $\mathcal{O}_K\subset K$ is the ring of integers and $0\neq\pi\in\mathcal{O}_K$ a nonunit. Drinfeld’s theorem states that morphisms $R\to R'$ such that $R/\pi\to R'/\pi$ is faithfully flat are of effective descent.

The setup for author’s generalization involves a connective $E_\infty$-ring $R$, a finitely generated ideal $I\subset\pi_0(R)$, and a subcategory $\mathcal{C}$ of the $\infty$-category $\mathrm{Mod}(R)$ of modules (for example the $\infty$-subcategory of perfect complexes). Some homotopical/spectral algebra is developed in this generality, involving consideration of properties of objects in $\mathcal{C}$ holding up to isogeny, $t$-structures, and the interplay between $I$-completion and $I$-torsion of $R$-modules. The approach to $\infty$-descent is centered around Lurie’s version of $\infty$-categorical monadicity theorem and the study of universal descent functors in the setting of idempotent-complete stable $\infty$-categories. The author constructs a stable $\infty$-category $\mathcal{M}(R)$ interpreted as the generic fiber of the formal spectrum $\mathrm{Spf}(R)$ and proves that the category $\mathrm{APerf}(\mathrm{Spec}(R)\backslash V(I))$ of almost perfect modules on $\mathrm{Spec}(R)\backslash V(I)$ forms a full subcategory of $\mathcal{M}(R)$. It is proved that $R'\to\mathcal{M}(R')$ and $R'\to \mathrm{APerf}(\mathrm{Spec}(\widehat{R'}_I)\backslash V(I))$ are sheaves for the universal descent topology on the $\infty$-category of $E_\infty$-$R$-algebras. Finally, descent for $I$-complete faithfully flat maps $S\to S'$ of $I$-complete connective $E_\infty$-$R$-algebras is proved: $S\mapsto\mathrm{APerf}(\mathrm{Spec}(\hat{S}_I)\backslash V(I))$ is a hypercomplete sheaf for the $I$-completely flat topology.

These are strong results. A part of the difficulty (in comparison to algebraic geometry) lays in the lack of well-behaved category of quasicoherent sheaves in traditional rigid analytic geometry. One of the goals of the program of condensed mathematics of P. Scholze and D. Clausen is to rectify this problem. This may lead to a different approach to faithfully flat descent in rigid geometry, as the author points out.