Spectral Sequences Explained Through the Beilinson t-structure and Décalage

מושגי ליבה

The Beilinson t-structure provides a powerful framework for understanding and working with spectral sequences, showing how the décalage functor connects to the pages of a spectral sequence and offering a more intuitive and homotopy-coherent approach to their construction and properties.

תקציר

Bibliographic Information:

Antieau, B. (2024). Spectral sequences, décalage, and the Beilinson t-structure. arXiv preprint arXiv:2411.09115v1.

Research Objective:

This paper aims to elucidate the theory of spectral sequences through the lens of the Beilinson t-structure, demonstrating how the concept of décalage can be used to understand the relationship between different pages of a spectral sequence.

Methodology:

The author employs methods from stable homotopy theory, particularly utilizing the Beilinson t-structure on the ∞-category of filtered objects in a stable ∞-category. The paper defines and explores the properties of the décalage functor within this framework.

Key Findings:

• The décalage functor, when applied to a filtered object, shifts the associated spectral sequence by one page (Theorem 1.1).
• This connection allows for a simplified definition of the spectral sequence associated with a filtered object, directly constructing the E1-page from the graded pieces of the filtration (Definition 1.3).
• The paper establishes the equivalence between Deligne's classical décalage functor for filtered cochain complexes and the ∞-categorical décalage functor (Theorem 1.4).

Main Conclusions:

The Beilinson t-structure and the décalage functor provide a powerful and homotopy-coherent framework for understanding and working with spectral sequences. This approach offers a more intuitive understanding of the relationship between different pages of a spectral sequence and simplifies the construction of spectral sequences from filtered objects.

Significance:

This research provides a deeper understanding of spectral sequences, which are fundamental tools in algebraic topology and homological algebra. The connection to the Beilinson t-structure offers new insights and potential applications in these fields.

Limitations and Future Research:

The paper focuses on the theoretical aspects of spectral sequences and their connection to the Beilinson t-structure. Further research could explore specific applications of this framework in areas like algebraic topology, homological algebra, and algebraic geometry. Additionally, investigating the implications of this approach for the convergence of spectral sequences could be a fruitful avenue for future work.

סטטיסטיקה

ציטוטים

"There has been a recent cultural movement best portrayed as: 'there is a spectral sequence' ⇒ 'there is a filtration'."

תובנות מפתח מזוקקות מ:

Spectral sequences, d\'ecalage, and the Beilinson t-structure

by Benjamin Ant... ב- arxiv.org 11-15-2024

https://arxiv.org/pdf/2411.09115.pdf

שאלות מעמיקות

How can the Beilinson t-structure and décalage be applied to understand more complex spectral sequences arising in specific mathematical contexts, such as algebraic geometry or representation theory?

The Beilinson t-structure and décalage provide a powerful framework for understanding complex spectral sequences by relating them to more manageable filtrations and underlying abelian categories. Here's how they can be applied in algebraic geometry and representation theory:
Algebraic Geometry:

Motivic and Filtered Derived Categories: In motivic homotopy theory, one works with categories like the motivic stable homotopy category SH(k) of a field k. These categories admit various t-structures, and the Beilinson t-structure on filtered objects in SH(k) helps analyze spectral sequences arising from motivic filtrations. For example, the slice filtration and the motivic Adams spectral sequence can be studied using these tools.
Coherent Sheaves and D-Modules: The Beilinson t-structure is closely related to the study of coherent sheaves and D-modules on schemes. For instance, the Beilinson-Bernstein localization theorem relates representations of Lie algebras to D-modules on flag varieties, and spectral sequences in these contexts can be analyzed using the Beilinson t-structure.
Hodge Theory: The Beilinson t-structure provides a natural setting for Deligne's theory of mixed Hodge structures, which are filtrations on the cohomology of complex algebraic varieties. Décalage plays a crucial role in constructing and studying these mixed Hodge structures.
Representation Theory:

Derived Categories of Representations: In representation theory, one often studies derived categories of representations of algebras or groups. The Beilinson t-structure on these categories can be used to analyze spectral sequences arising from various filtrations on representations, such as weight filtrations or filtrations coming from derived functors.
Homological Algebra and Ext-Groups: Spectral sequences often compute Ext-groups in abelian categories, which are fundamental objects in representation theory. The Beilinson t-structure and décalage can provide insights into the structure of these Ext-groups and the differentials in the spectral sequences.
Geometric Representation Theory: This field connects representation theory with geometry, often using sheaves and D-modules on algebraic varieties. The Beilinson t-structure naturally appears in this context, and décalage helps relate different geometric constructions to spectral sequences in representation theory.
General Strategy:
The general strategy for applying the Beilinson t-structure and décalage involves:

Identifying a relevant t-structure: Determine a suitable t-structure on the stable ∞-category relevant to the problem, often motivated by geometric or algebraic considerations.
Constructing filtrations: Construct filtrations on objects of interest, potentially guided by the t-structure.
Applying décalage: Use décalage to relate the spectral sequence of a filtration to the E1-page of a related filtration, potentially simplifying the analysis.
Interpreting the results: Interpret the information obtained from the spectral sequence in terms of the original problem, using the connection between the Beilinson t-structure and the underlying abelian category.

Could there be alternative categorical frameworks beyond the Beilinson t-structure that offer different perspectives or advantages in understanding spectral sequences?

While the Beilinson t-structure provides a powerful and widely applicable framework, exploring alternative categorical approaches to spectral sequences could offer new insights and advantages. Here are some possibilities:

Weight Structures: Weight structures, introduced by Bondarko, provide a different axiomatic framework for "slicing" stable ∞-categories. They are particularly well-suited for studying categories with less rigid homological properties than those typically encountered with t-structures. Exploring the interplay between weight structures and spectral sequences could lead to new perspectives.
Filtered ∞-Categories: Instead of working with the ∞-category of filtered objects in a stable ∞-category, one could directly consider ∞-categories equipped with a notion of filtration. This approach, explored by Tyler Lawson, might offer a more flexible setting for studying spectral sequences and their functoriality properties.
Higher Algebra and Operads: Spectral sequences often exhibit multiplicative structures, and operads provide a natural language for studying such structures. Investigating the connections between operads and spectral sequences, potentially in the context of stable ∞-categories, could reveal deeper algebraic structures governing spectral sequences.
Homotopy Type Theory: Homotopy type theory provides a foundational framework for mathematics based on homotopy theory. Exploring spectral sequences within this framework could lead to new computational tools and a deeper understanding of their constructive content.
Advantages of Alternatives:
These alternative frameworks could offer advantages such as:

Greater generality: Extending the applicability of spectral sequence techniques to categories beyond those with well-behaved t-structures.
New computational tools: Providing different perspectives and potentially more efficient methods for computing spectral sequences.
Deeper structural insights: Revealing hidden algebraic or homotopical structures underlying spectral sequences.

How does the concept of décalage, as a shifting operation on filtrations and spectral sequences, relate to other mathematical concepts of "shifting" or "translation," potentially revealing deeper connections across different areas of mathematics?

The concept of décalage, as a shifting operation on filtrations and spectral sequences, shares intriguing connections with other mathematical notions of "shifting" or "translation," hinting at deeper unifying principles across different areas:

Suspensions and Loop Spaces: In homotopy theory, the suspension ΣX of a space X shifts the homotopy groups: π_n(ΣX) ≅ π_{n-1}(X). Décalage can be viewed as an analogous operation on filtrations, shifting the information encoded in the graded pieces. This analogy reflects a deeper connection between stable ∞-categories and homotopy theory.
Translation Actions: In representation theory and geometry, algebraic groups often act on spaces or categories, and these actions induce "translation" operations. For example, the action of a group on itself by left multiplication induces a translation action on its representations. Décalage can sometimes be interpreted as a "translation" arising from a hidden group action on the category of filtered objects.
Shifts in Derived Categories: In algebraic geometry and homological algebra, derived categories come equipped with shift functors [1], which shift complexes. Décalage interacts with these shift functors in a controlled way, reflecting the interplay between the Beilinson t-structure and the standard t-structure on the derived category.
Spectral Sequences as Dynamical Systems: One can view a spectral sequence as a "dynamical system" where the pages evolve over time (represented by the page index r). Décalage acts as a "time evolution operator," shifting the system from one state to another. This perspective connects spectral sequences to the study of dynamical systems and ergodic theory.
Unifying Themes:
These connections suggest some unifying themes:

Categorical Actions: Décalage often arises from categorical actions, such as the action of a group or a monoidal category. Understanding these actions can provide deeper insights into the structure and behavior of spectral sequences.
Homotopical Shifting: Décalage reflects a general principle of "homotopical shifting," where information is moved between different levels of a filtration or a homotopy type. This principle connects spectral sequences to broader themes in homotopy theory and algebraic topology.
Spectral Sequences as Transformations: Viewing spectral sequences as transformations or dynamical systems opens up connections with other areas of mathematics that study change and evolution.
Exploring these connections could lead to a more unified understanding of "shifting" operations across mathematics and reveal deeper insights into the nature of spectral sequences and their applications.