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Pro-Algebraic Site

Hyuk Jun Kweon

TL;DR

The paper introduces a new algebraic cohomology theory for varieties over a characteristic-zero field, defined via the pro-algebraic fundamental groupoid $\Pi_1^{\mathrm{alg}}(X)$ and organized on a Betti-type site. It develops a robust sheaf-theoretic framework using groupoid schemes, proves the existence of injectives and derived functors, and establishes a concrete comparison with singular cohomology over $\mathbb{C}$, thereby unifying multiple cohomology perspectives within a common pro-algebraic foundation. The results extend naturally to other fundamental groupoids (e.g., $\Pi_1$, $\Pi_1^{\mathrm{dR}}$) and hint at connections to algebraic de Rham cohomology, suggesting a pathway toward a Weil-type, universal cohomology theory. The framework promises a uniform approach that could reconcile étale, de Rham, and Betti cohomologies and proposes an algebraic setting in which base-field automorphisms act naturally on cohomology.

Abstract

For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of étale cohomology, involving neither de Rham complexes nor hypercohomology. The main idea is to delegate the role of étale morphisms to fundamental groupoids, thereby bypassing the Grothendieck topology. To validate this theory, we prove a comparison theorem between the algebraically defined cohomology using the pro-algebraic fundamental groupoid over $\mathbb{C}$ and singular cohomology. Furthermore, our construction naturally extends to other types of fundamental groupoids, providing a uniform foundation for various cohomology theories.

Pro-Algebraic Site

TL;DR

The paper introduces a new algebraic cohomology theory for varieties over a characteristic-zero field, defined via the pro-algebraic fundamental groupoid and organized on a Betti-type site. It develops a robust sheaf-theoretic framework using groupoid schemes, proves the existence of injectives and derived functors, and establishes a concrete comparison with singular cohomology over , thereby unifying multiple cohomology perspectives within a common pro-algebraic foundation. The results extend naturally to other fundamental groupoids (e.g., , ) and hint at connections to algebraic de Rham cohomology, suggesting a pathway toward a Weil-type, universal cohomology theory. The framework promises a uniform approach that could reconcile étale, de Rham, and Betti cohomologies and proposes an algebraic setting in which base-field automorphisms act naturally on cohomology.

Abstract

For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of étale cohomology, involving neither de Rham complexes nor hypercohomology. The main idea is to delegate the role of étale morphisms to fundamental groupoids, thereby bypassing the Grothendieck topology. To validate this theory, we prove a comparison theorem between the algebraically defined cohomology using the pro-algebraic fundamental groupoid over and singular cohomology. Furthermore, our construction naturally extends to other types of fundamental groupoids, providing a uniform foundation for various cohomology theories.
Paper Structure (9 sections, 32 theorems, 94 equations)

This paper contains 9 sections, 32 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids
  4. Groupoid Schemes
  5. Pro-Algebraic Fundamental Groupoid
  6. Betti Site: Analytic Prototype
  7. Sheaf Theory via Fundamental Groupoids
  8. Cohomology Theory via Fundamental Groupoid Schemes
  9. Comparison Theorem

Key Result

Theorem 1.1

Let $X$ be a smooth variety over $\mathbb{C}$. Let $\mathcal{M}$ be a locally constant sheaf on $X_\mathrm{alg} \coloneqq (X, \Pi^\mathrm{alg}_1)$ associated to a finite-dimensional $\Pi_1^\mathrm{alg}(X)$-representation. The natural map is then an isomorphism for all $i \ge 0$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2: Deligne, Deligne
  • Lemma 2.3: Deligne, Deligne
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6: Deligne Deligne1
  • Definition 3.1
  • ...and 67 more