[Paper Review] Enhanced six operations and base change theorem for sheaves on Artin stacks

Yifeng Liu, Weizhe Zheng|arXiv (Cornell University)|Nov 26, 2012

Homotopy and Cohomology in Algebraic Topology11 citations

TL;DR

This paper develops an enhanced theory of Grothendieck's six operations for derived categories in etale cohomology on Artin stacks using Lurie's stable ∞-categories, establishing a full base change theorem in the ∞-categorical setting. It generalizes prior work by Laszlo and Olsson, extending the framework to higher Artin stacks and resolving homotopy coherence via ∞-categorical homological descent.

ABSTRACT

In this article, we develop a theory of Grothendieck's six operations for derived categories in etale cohomology of Artin stacks. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends all previous theories on this subject, including the recent one developed by Laszlo and Olsson, in which the operations are subject to more assumptions and the base change isomorphism is only constructed on the level of sheaves. Moreover, our theory works for higher Artin stacks as well. Our method differs from all previous approaches, as we exploit the theory of stable $\infty$-categories developed by Lurie. We enhance derived categories, functors, and natural isomorphisms to the level of $\infty$-categories and introduce $\infty$-categorical (co)homological descent. To handle the homotopy coherence, we apply the results of our previous article arXiv:1211.5294 and develop several other $\infty$-categorical techniques.

Motivation & Objective

• To extend Grothendieck's six operations to derived categories in etale cohomology for Artin stacks, overcoming limitations in previous formulations.
• To establish a base change isomorphism in the derived ∞-categorical setting, rather than just on the level of sheaves as in prior work.
• To generalize the theory to higher Artin stacks, where previous approaches fail or require strong assumptions.
• To resolve homotopy coherence issues in derived functors using advanced ∞-categorical techniques.
• To unify and strengthen existing theories by embedding them within Lurie’s stable ∞-category framework.

Proposed method

• Employing Lurie’s theory of stable ∞-categories to enhance derived categories, functors, and natural isomorphisms to the ∞-categorical level.
• Introducing ∞-categorical (co)homological descent to handle descent data in a coherent, higher-categorical way.
• Leveraging results from the authors’ prior work (arXiv:1211.5294) to ensure homotopy coherence in the construction of operations.
• Using ∞-categorical enhancements to define the six operations (f^*, f_*, f_!, f^!, Rf_*, Rf_!) in a way that preserves desired functoriality and duality properties.
• Constructing the base change isomorphism as a natural transformation in the ∞-category of derived categories, ensuring compatibility across all levels.
• Applying ∞-categorical techniques to prove that the base change map is an equivalence in the derived ∞-category, even for higher Artin stacks.

Experimental results

Research questions

• RQ1 How can Grothendieck’s six operations be systematically enhanced to the ∞-categorical level for Artin stacks?
• RQ2 Can the base change isomorphism be established in the derived ∞-category, rather than just on the level of sheaves?
• RQ3 To what extent can the theory be extended to higher Artin stacks beyond the classical case?
• RQ4 How can homotopy coherence in derived functors be systematically managed in this context?
• RQ5 What role does ∞-categorical homological descent play in ensuring consistency and compatibility of the operations?

Key findings

• The six operations are fully enhanced to the ∞-categorical level, ensuring coherence and compatibility in derived categories over Artin stacks.
• The base change isomorphism is established as an equivalence in the derived ∞-category, resolving a key limitation of earlier approaches.
• The theory applies to higher Artin stacks, extending the scope beyond the classical case of algebraic spaces or schemes.
• The use of ∞-categorical homological descent provides a systematic solution to homotopy coherence issues in derived functors.
• The framework generalizes and strengthens prior work by Laszlo and Olsson, removing their restrictive assumptions while preserving and enhancing their results.
• The construction is compatible with duality and preserves key properties such as projection formulas and base change in the ∞-categorical setting.