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Commutative Algebra, Ring Theory, and Representation Theory Lab
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This lab studies the structure of module categories over commutative rings through their subcategories and triangulated categories.
AboutAbout the lab
Understanding the structure of all modules over a commutative ring
Prof. Takahashi studies commutative algebra.
Commutative algebra is a field closely related to algebraic geometry, algebraic combinatorics, and the representation theory of algebras.
The professor mainly studies the "representation theory of commutative rings," which is commutative algebra approached from the direction of the representation theory of algebras.
The aim is to understand the structure of the module category (the category of all finitely generated modules) of commutative Noetherian rings, especially Cohen–Macaulay rings.
Key terms
- commutative ring
- A number system in which addition and multiplication are possible and changing the order of multiplication does not change the result. Integers and polynomials are examples
- representation theory
- A field that studies the structure of abstract algebras by expressing them as actions on concrete objects such as matrices and modules
- triangulated category
- A category, such as a derived category, equipped with a structure that captures the relationships between objects in the form of "triangles"
ResearchResearch themes
The lab aims to understand the structure of the module category (the category of all finitely generated modules) of commutative Noetherian rings, especially Cohen–Macaulay rings. In addition to studying module categories themselves, it investigates the subcategory of all Cohen–Macaulay modules and triangulated categories such as derived categories.
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MembersFaculty
- ProfessorRyo Takahashi
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Tensor product of algebras” Jakob.scholbach / CC BY-SA 4.0 (cropped)
- Structure of Module Categories and Classification of Subcategories: “Hasse Diagram of Subgroup Lattice of D4” Spamakin / CC0 (cropped)
- Related lab (Combinatorics and Representation Theory Lab): “Line in combinatorics” Tensorproduct / CC BY-SA 4.0 (cropped)
- Related lab (Ishii Lab): “Surface of degree 7” Jean Constant / CC BY-SA 4.0 (cropped)
- Related lab (Representation Theory and Infinite-Dimensional Integrable Systems Lab): “Dynkin diagram Dn” Rogilbert / Public domain (cropped)
- Related lab (Ring Theory and Representation Theory Lab): “Combined group axiom commutative diagram” Struoc / CC BY-SA 4.0 (cropped)
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