Finiteness and duality of cohomology of $(\varphi,Γ)$-modules and the 6-functor formalism of locally analytic representations (2504.01780v1)

Abstract: Finiteness and duality of cohomology of families of $(\varphi,\Gamma)$-modules were proved by Kedlaya-Pottharst-Xiao. In this paper, we study solid locally analytic representations introduced by Rodrigues Jacinto-Rodr\'iguez Camargo in terms of analytic stacks and 6-functor formalisms, which are developed by Clausen-Scholze, Heyer-Mann, respectively. By using this, we will provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.

Finiteness and Duality of Cohomology of $(\varphi,\Gamma)$-Modules

The paper addresses the cohomology of $(\varphi,\Gamma)$-modules and explores its finiteness and duality properties. It advances the understanding of these modules by extending known results using analytic stacks and 6-functor formalisms. Here is an academic overview of the paper:

Introduction and Background

The paper of $(\varphi,\Gamma_K)$-modules is a pivotal area in $p$-adic representations, with foundational work by Fontaine, Berger, and others establishing these modules' equivalence to $p$-adic representations over local fields like $\mathbb{Q}_p$. Kedlaya, Pottharst, and Xiao previously generalized results on the cohomology of these structures, creating a linkage between $(\varphi,\Gamma_K)$-modules over the Robba ring and Galois representations.

Analytical Framework and Results

The paper uses advanced tools from recent developments in analytic $D$-stacks and formalisms like Clausen-Scholze's solid $D$-stacks to generalize results and offer new proofs for previously explored cases. A solid $D$-stack is an enhanced algebraic structure that helps handle complex cohomological behavior, enabling work beyond rigid-analytic varieties.

1. Framework Extensions:
   • The paper extends the 6-functor formalism, offering new insights and proofs for finite cohomologies related to $(\varphi,\Gamma_K)$-modules. This is seen as a leap towards encompassing analytic stacks and broader coefficient rings.
2. Key Theorems and Implications:
   • A highlight is Theorem \ref{thm:Local Tate duality} confirming local Tate duality in a previously speculative domain, which enriches the theoretical framework for mathematicians to exploit in understanding these algebraic structures better.
3. Impact on Moduli Spaces:
   • It touches on moduli spaces via the Emerton-Gee stack, proposing that deeper insights are possible by considering these analytic environments. The paper suggests that adapting $D$-stack formalisms provides a more refined understanding of these cohomologies.

Theoretical and Practical Implications

The practical aspects of these theoretical advances envisage outcomes like better tools for constructing eigenvarieties and deeper insights into $p$-adic Langlands programs. Mathematicians interested in the abstraction and application of algebraic geometry and number theory may find this work opening new avenues for research.

Future Directions in AI and Mathematical Modelling

Beyond pure mathematics, this approach offers speculative insights into artificial intelligence models in representing abstract algebraic systems. By simulating the behavior of cohomological frameworks in computational terms, AI might advance theoretical research through deep learning models that capture and predict complex algebraic phenomena.

Conclusion

In essence, the paper situates itself at the intersection of traditional algebraic geometry and innovative categorical frameworks, promising substantial advancements in understanding $(\varphi,\Gamma)$-cohomologies. Its implications are profound for both theoretical explorations and practical applications in various fields of mathematics.