[Paper Review] Overconvergent global analytic geometry

Frédéric Paugam|arXiv (Cornell University)|Oct 29, 2014

Algebraic Geometry and Number Theory42 references4 citations

TL;DR

This paper introduces overconvergent global analytic geometry over general Banach rings, unifying Berkovich's and Poineau's analytic spaces with Grofle-Kl"{o}nne's $p$-adic overconvergent spaces and Bambozzi's archimedean analogs. It constructs overconvergent motives, stable homotopy theory, and a derived Chern character linking analytic $K$-theory to cyclic homology, enabling integral realizations of special $L$-values without denominators.

ABSTRACT

We define a notion of global analytic space with overconvergent structure sheaf. This gives an analog on a general base Banach ring of Grosse-Kloenne's overconvergent p-adic spaces and of Bambozzi's generalized affinoid varieties over R. This also gives an affinoid version of Berkovich's and Poineau's global analytic spaces. This affinoid approach allows the introduction of a notion of strict global analytic space, that has some relations with the ideas of Arakelov geometry, since the base extension along the identity morphism on Z (from the archimedean norm to the trivial norm) sends a strict global analytic space to a usual scheme over Z, that we interpret here as a strict analytic space over Z equipped with its trivial norm. One may also interpret some particular analytification functors as mere base extensions. We use our categories to define overconvergent motives and an overconvergent stable homotopy theory of global analytic spaces. These have natural Betti, de Rham and pro-étale realizations. Motivated by problems in global Hodge theory and integrality questions in the theory of special values of arithmetic L-functions, we also define derived overconvergent global analytic spaces and their (derived) de Rham cohomology. Finally, we use Toen and Vezzosi's derived geometric methods to define a natural (integral) Chern character on analytic Waldhausen's K-theory with values in analytic cyclic homology. The compatibility of our constructions with Banach base extensions gives new perspectives both on global analytic spaces and on the various realizations of the corresponding motives.

Motivation & Objective

• To develop a unified framework for global analytic geometry over general Banach rings, including $p$-adic and archimedean cases.
• To extend Berkovich's and Poineau's analytic spaces using overconvergent power series, ensuring compatibility with both $p$-adic and archimedean geometries.
• To define overconvergent motives and stable homotopy theory for analytic spaces with Betti, de Rham, and pro-étale realizations.
• To construct a derived Chern character from analytic Waldhausen $K$-theory to cyclic homology, enabling integral regulators for special $L$-values.
• To avoid denominators in arithmetic $L$-function special value formulas by using integral Chern characters in a derived global analytic setting.

Proposed method

• Introduces dagger algebras as overconvergent power series rings $R\{\rho^{-1}T\}^\dagger$, defined as filtered colimits of convergent power series rings $R\{\nu^{-1}T\}$ for $\nu > \rho$.
• Defines dagger analytic spaces as locally ringed spaces modeled on dagger algebras, with a topology induced by rational domains and Tate's acyclicity theorem.
• Constructs global analytic motives via stable homotopy theory of sheaves on dagger analytic spaces, with Betti, de Rham, and pro-étale realizations.
• Develops derived dagger algebras and the dagger cotangent complex to define derived de Rham cohomology for global analytic spaces.
• Applies Toen-Vezzosi's derived geometry to define a natural integral Chern character $\mathbf{K}(X) \to \mathbf{HC}^{\text{neg}}(X)$ with values in negative cyclic homology spectra.
• Uses base extension along identity morphisms (e.g., from $\mathbb{Z}$ with archimedean norm to trivial norm) to interpret strict global analytic spaces as schemes, linking to Arakelov geometry.

Experimental results

Research questions

• RQ1 How can one unify overconvergent $p$-adic geometry, archimedean geometry, and Berkovich's analytic geometry in a single framework over general Banach rings?
• RQ2 Can overconvergent power series over non-archimedean and archimedean fields provide a well-behaved analog of affinoid algebras in global analytic geometry?
• RQ3 How can one define a stable homotopy theory and motives for analytic spaces that admit Betti, de Rham, and pro-étale realizations?
• RQ4 Can a derived Chern character be constructed from analytic Waldhausen $K$-theory to cyclic homology that avoids denominators in special $L$-value formulas?
• RQ5 To what extent can the derived global analytic formalism provide an integral version of Arakelov motivic cohomology?

Key findings

• The paper constructs a category of overconvergent global analytic spaces over any Banach ring, generalizing Berkovich's and Poineau's analytic spaces.
• It defines overconvergent motives and a stable homotopy theory of analytic spaces, with natural Betti, de Rham, and pro-étale realizations.
• The derived dagger cotangent complex enables the definition of derived de Rham cohomology for global analytic spaces.
• A natural integral Chern character $\mathbf{K}(X) \to \mathbf{HC}^{\text{neg}}(X)$ is constructed, compatible with base extensions and extending Toen-Vezzosi's formalism.
• The formalism allows base extension from the archimedean norm to the trivial norm, turning strict global analytic spaces into usual schemes over $\mathbb{Z}$, linking to Arakelov geometry.
• The Chern character construction avoids denominators in special $L$-value formulas, suggesting a path to integral regulators in arithmetic $L$-functions.