Core Concepts

This paper presents a new definition of associative schemes, generalizing the concept from commutative rings to associative rings using the completion of an associative algebra in a finite set of simple modules.

Abstract

Siqveland, A. (2024). Schemes of Associative algebras. *arXiv preprint arXiv:2410.17703v1*.

This paper aims to redefine the concept of "schemes" from algebraic geometry, traditionally applied to commutative rings, to encompass associative rings. This is achieved by leveraging the concept of "completion" of an associative algebra within a set of simple modules.

The author first revisits the classical definition of schemes for commutative rings, highlighting the limitations in generalizing it directly to non-commutative cases. The concept of "Hausdorff localization" is introduced as a bridge between local rings and their completions. The paper then defines "aprime modules" as a generalization of prime ideals and utilizes them to construct a topological space. Finally, a sheaf of rings is defined on this space, leading to the definition of "associative schemes."

- Noetherian integral schemes are demonstrated to be "locally Hausdorff."
- The concept of "pre-completion" of an algebra in a module is defined, and its existence and uniqueness are proven under specific conditions.
- The paper establishes that the completion of an algebra in a set of modules is itself complete and characterizes its simple modules.
- "Aprime modules" are shown to behave analogously to prime ideals in the commutative case.

The paper successfully provides a coherent and well-defined notion of "associative schemes" by generalizing key concepts from classical scheme theory. This new definition aligns with the existing theory for commutative rings and offers a framework for studying non-commutative algebras geometrically.

This work significantly contributes to the field of non-commutative algebraic geometry by providing a rigorous framework for studying associative algebras using geometric tools. This opens up avenues for applying geometric intuition and techniques to problems in non-commutative algebra.

The paper primarily focuses on theoretical foundations. Further research could explore specific examples and applications of associative schemes, investigate their properties in-depth, and connect them with other areas of mathematics.

To Another Language

from source content

arxiv.org

Stats

Quotes

"It is well known that projective varieties can be covered by their affine open subschemes: The French (?) idea: If we cannot find a fine moduli for a set S in a category C, then it can be possible to extend the category to ˜C, i.e. such that C is a subcategory, and such that U ∈˜C is a fine moduli for S. This is one advantage with schemes, if not the initial intention."
"We understand now that if G is a nonabelian group acting on a variety X, we cannot expect that there exists a fine moduli in the category of schemes. This is simply because the orbits corresponds to modules over noncommutative algebras k[G], the skew group-algebra over G."

Key Insights Distilled From

by Arvid Siqvel... at **arxiv.org** 10-24-2024

Deeper Inquiries

Answer:
The concept of associative schemes offers a new geometric perspective on non-commutative rings, potentially revolutionizing how we approach representation theory and classification problems. Here's how:
Representation Theory:
Geometric Interpretation of Modules: Associative schemes provide a geometric framework to visualize and study modules over non-commutative rings. Just as points on a classical scheme correspond to prime ideals, points on an associative scheme correspond to aprime modules, offering a geometric handle on the building blocks of representations.
Sheaf-Theoretic Tools: The theory of sheaves, central to the study of classical schemes, can be brought to bear on modules over non-commutative rings. This opens up possibilities for using cohomology theories and other sheaf-theoretic tools to analyze representations.
Moduli Spaces of Representations: Associative schemes could potentially be used to construct and study moduli spaces of representations of non-commutative algebras. This would provide a powerful tool for understanding the families of representations that can arise.
Classification of Non-Commutative Rings:
Geometric Invariants: Associative schemes could give rise to new geometric invariants of non-commutative rings, analogous to how classical schemes provide invariants like dimension, genus, and singularity type. These invariants could be used to distinguish between different classes of non-commutative rings.
Deformation Theory: The concept of associative schemes could be used to develop a deformation theory for non-commutative rings, analogous to the deformation theory of commutative algebras. This would allow us to study how non-commutative rings vary in families and understand their moduli spaces.
Connections to Non-Commutative Geometry: Associative schemes provide a bridge between classical algebraic geometry and non-commutative geometry. This connection could lead to new insights into the classification of non-commutative rings, drawing on tools and techniques from both fields.

Answer:
Yes, the quest for a satisfactory notion of "non-commutative schemes" is an active area of research, and the associative scheme approach presented is just one avenue. Here are some alternative approaches and intuitions:
Non-Commutative Localizations: Instead of focusing on prime ideals, one could try to develop a theory based on more general localizations of non-commutative rings, such as Ore localizations or universal localizations. This approach aims to capture the local behavior of non-commutative rings in a more flexible way.
Quantum Algebra and Non-Commutative Spaces: Quantum algebra provides a framework for studying "non-commutative spaces" where coordinates no longer commute. This leads to geometric objects like quantum groups and quantum flag varieties, which can be viewed as non-commutative analogs of classical algebraic varieties.
Derived Algebraic Geometry: Derived algebraic geometry provides a powerful generalization of classical algebraic geometry, where one works with derived categories instead of just modules. This framework has been used to develop a notion of "derived schemes," which could potentially be adapted to the non-commutative setting.
Categorical Approaches: Category theory offers a high-level perspective on algebraic geometry, where schemes are viewed as certain types of categories (e.g., locally ringed spaces). This suggests the possibility of defining non-commutative schemes as categories with additional structure, capturing the essential features of non-commutative rings.

Answer:
The theory of associative schemes has the potential to forge deep connections with various areas of mathematics, enriching our understanding of both classical and emerging fields:
Non-Commutative Geometry:
Geometric Framework for NC Spaces: Associative schemes provide a concrete geometric framework for studying non-commutative spaces, complementing more abstract approaches in non-commutative geometry.
Quantization and Deformation: Associative schemes could provide insights into the process of quantizing classical spaces, bridging the gap between commutative and non-commutative geometry.
Quantum Algebra:
Representation Theory of Quantum Groups: Associative schemes could offer new tools for studying the representation theory of quantum groups, which are fundamental objects in quantum algebra.
Geometric Realizations of Quantum Objects: The theory could lead to geometric realizations of quantum objects like quantum spheres and quantum projective spaces, providing a more intuitive understanding of their structure.
Mathematical Physics:
Quantum Field Theory: Associative schemes might provide a new language for describing the geometry of spacetime at the Planck scale, where quantum effects are expected to become significant.
String Theory: The theory could offer insights into the geometry of D-branes and other non-commutative structures arising in string theory.
Statistical Mechanics: Associative schemes could potentially be used to study non-commutative generalizations of statistical mechanical models, leading to new insights into phase transitions and critical phenomena.

0