[Paper Review] Cohomology with closed support on the overconvergent site

David Zureick-Brown|arXiv (Cornell University)|Aug 8, 2014

Homotopy and Cohomology in Algebraic Topology4 references3 citations

TL;DR

This paper establishes a sheaf-theoretic framework for cohomology with closed support on the overconvergent site using open/closed subtopoi from SGA4, proving that this construction agrees with classical rigid cohomology with supports as defined by Berthelot. The key result is a functorial isomorphism between overconvergent and rigid cohomology with supports, extending finiteness and excision properties to algebraic stacks.

ABSTRACT

Using the notions of open/closed subtopoi of SGA, we define a notion of cohomology with support in a closed subscheme on the overconvergent site, and show that this agrees with the classic notion of rigid cohomology support in a closed subscheme.

Motivation & Objective

• To extend the theory of rigid cohomology with supports to the overconvergent site using modern topos-theoretic tools.
• To resolve the lack of a systematic cohomological framework for supports in the overconvergent setting, which is essential for finiteness and duality results.
• To generalize classical results—especially finiteness and excision—for rigid cohomology with supports to algebraic stacks.
• To provide a formal, functorial definition of cohomology with support in a closed subscheme on the overconvergent site that aligns with Berthelot’s concrete constructions.
• To establish foundational tools for future applications in arithmetic geometry, such as zeta functions and counting rational points.

Proposed method

• The paper employs the theory of open and closed subtopoi from SGA4, using the formalism of sites and topoi to define cohomology with support in a closed subscheme.
• It constructs the overconvergent site $\operatorname{AN}^\dagger(X)$ as a site of overconvergent varieties with a Grothendieck topology finer than the usual topology.
• The key technical step is showing that the subtopos associated to a closed immersion $Z \hookrightarrow X$ induces a well-defined cohomology with support $\mathbb{R}p_{\operatorname{AN}^\dagger_*}\underline{\Gamma}^\dagger_Z E$ on the overconvergent site.
• The main isomorphism is established via a comparison between the overconvergent and rigid cohomology functors: $\left(\mathbb{R}p_{\operatorname{rig}}\underline{\Gamma}^{\dagger,\operatorname{Ber}}_Z E_0\right)^{\operatorname{an}} \cong \left(\mathbb{R}p_{\operatorname{AN}^\dagger_*}\underline{\Gamma}^\dagger_Z E\right)_{S_k,S_K}$, proving agreement with classical definitions.
• The theory is extended to algebraic stacks using cohomological descent and the 2-Yoneda lemma, allowing the definition of $\operatorname{AN}^\dagger(\mathcal{X})$ for fibered categories $\mathcal{X}$ over $k$.
• Finiteness results are proven using spectral sequences arising from hypercovers and the known finite-dimensionality of rigid cohomology for quasi-projective schemes.

Experimental results

Research questions

• RQ1 How can cohomology with closed support be systematically defined on the overconvergent site using topos-theoretic methods?
• RQ2 Does the overconvergent cohomology with support in a closed subscheme agree with Berthelot’s classical rigid cohomology with supports?
• RQ3 Can the standard properties—functorality, excision, and finite-dimensionality—be extended from schemes to algebraic stacks in the overconvergent setting?
• RQ4 What is the role of open and closed subtopoi in defining support conditions in non-archimedean cohomology theories?
• RQ5 How can cohomological descent be applied to prove finiteness of overconvergent cohomology with supports on stacks?

Key findings

• The paper establishes a canonical isomorphism between overconvergent cohomology with supports and classical rigid cohomology with supports, proving that the new definition is consistent with Berthelot’s theory.
• The construction of cohomology with supports via closed subtopoi ensures functoriality and compatibility with the excision exact sequence, as required by classical cohomological machinery.
• Finiteness of overconvergent cohomology with supports is proven for separated algebraic stacks of finite type over a field of positive characteristic.
• The spectral sequence $H^j(\operatorname{AN}^\dagger_{\operatorname{g}}X'_i) \Rightarrow H^{i+j}(\operatorname{AN}^\dagger_{\operatorname{g}}X)$ is used to deduce finite-dimensionality from the known finite-dimensionality of rigid cohomology on quasi-projective schemes.
• The theory extends naturally to stacks via cohomological descent, and the same finiteness result holds for $H^i_Z(\operatorname{AN}^\dagger_{\operatorname{g}}X)$ when $X$ is a separated algebraic stack of finite type over $k$.
• The framework allows for generalization to $F$-isocrystals with Frobenius action, with the same finite-dimensionality result holding in that case.