Cartier duality for gerbes of vector bundles

TL;DR

The paper develops a stacky, kernel-based framework for Cartier duality to handle analytic and algebraic vector bundles and their gerbes. It locates duality as an anti-equivalence of Hopf algebras inside presentable categories of kernels arising from six-functor formalisms, using descent and a 1-étale topology to compute duals. Concrete dualities for tori, various incarnations of vector bundles, and gerbes are established, with a pivotal application relating the analytic Hodge-Tate stack $X^{HT}$ to Bhatt–Zhang's Simpson gerbe via a weight-1 equivalence $\,\mathrm{D}(X^{HT}) \cong \mathrm{D}(\mathscr{S}_X)^{\mathrm{wt}=1}$. The results yield a universal duality principle for gerbes and provide a robust toolkit for comparing geometric objects in mixed-characteristic analytic and algebraic settings, with potential implications for prismatic and Hodge–Tate theories.

Abstract

We prove a Cartier duality for gerbes of algebraic and analytic vector bundles as an anti-equivalence of Hopf algebras in the category of kernels of analytic stacks. As an application, we prove that the category of solid quasi-coherent sheaves on the Hodge-Tate stack of a smooth rigid variety over an algebraically closed field $C$ of mixed characteristic $(0,p)$ is equivalent to the category of weight $1$ sheaves on Bhatt-Zhang's Simpson gerbe.

Theorems & Definitions (130)

• Remark 1.2.1
• Theorem 1.2.2: \ref{['TheoCartierDualityVectorBundles']}
• Theorem 1.2.3: \ref{['CartierDualityHodgeTateSimpson']}
• Definition 2.1.1
• Remark 2.1.2
• Theorem 2.1.3: HeyerMannSix
• Lemma 2.1.4
• proof
• Corollary 2.1.5
• Definition 2.1.6