授業の目的 【日本語】
Goals of the Course(JPN) | | | Many rings come with a hidden partner (dual), associated to the same data but with structure mirrors the original in some way. One of the most famous such duality is the Koszul duality. This course develops the tools needed to recognize and construct this duality from ordinary rings and module theory, and their homological algebra. We will see how the quadratic dual of a Koszul algebra coincides the Yoneda algebra - ring that encodes the extensions of simple modules. Amongst the many beautiful consequences of this duality theory, we will focus on the BGG correspondence, which roughly relates the graded modules a Koszul algebra to the graded modules over its dual. |
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授業の目的 【英語】
Goals of the Course | | | Many rings come with a hidden partner (dual), associated to the same data but with structure mirrors the original in some way. One of the most famous such duality is the Koszul duality. This course develops the tools needed to recognize and construct this duality from ordinary rings and module theory, and their homological algebra. We will see how the quadratic dual of a Koszul algebra coincides the Yoneda algebra - ring that encodes the extensions of simple modules. Amongst the many beautiful consequences of this duality theory, we will focus on the BGG correspondence, which roughly relates the graded modules a Koszul algebra to the graded modules over its dual. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | By the end of the course, students should be able to:
1. Confidently work with duality over semisimple algebras for modules over non-commutative graded rings.
2. Define quadratic and Koszul algebras, construct quadratic duals, and verify Koszulness using the Hilbert series criterion.
3. Construct minimal graded free/projective resolutions and the classical Koszul complex directly.
4. Compute the Ext-groups between simple modules and the Yoneda algebra of a graded algebra. |
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到達目標 【英語】
Objectives of the Course | | By the end of the course, students should be able to:
1. Confidently work with duality over semisimple algebras for modules over non-commutative graded rings.
2. Define quadratic and Koszul algebras, construct quadratic duals, and verify Koszulness using the Hilbert series criterion.
3. Construct minimal graded free/projective resolutions and the classical Koszul complex directly.
4. Compute the Ext-groups between simple modules and the Yoneda algebra of a graded algebra. |
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授業の内容や構成
Course Content / Plan | | Part I — Graded Algebra and Quadratic Duality (Lectures 1–3)
Graded rings and modules, Hilbert series, tensor algebras.
Quadratic algebras as tensor algebras with relations concentrated in degree 2, and construction of the quadratic dual, computed by hand in worked examples.
Part II — Resolutions, the Koszul Complex, Koszul algebra and Yoneda algebra (Lectures 4–8)
Minimal graded resolutions, Koszul complex, leading to the definition of a Koszul algebra.
Tor, Ext, and the Yoneda algebra and identification with the quadratic dual for Koszul algebras; the Hilbert series identity for Koszulness test.
Part III — Koszul Duality and the BGG Correspondence (Lectures 9–11)
Basic categorical language. Equivalence between graded modules over the quadratic dual and linear complexes over the original algebra. The BGG correspondence between the quotient category qgr over the polynomial ring and the stable module category over the exterior algebra. |
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履修条件
Course Prerequisites | | | Essential knowledge includes linear algebra, ring, field, module. |
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関連する科目
Related Courses | | | Any course on rings and modules and homological algebras |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | |
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教科書・テキスト
Textbook | | |
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参考書
Reference Book | | 1. A. Polishchuk, L. Positselski, Quadratic Algebras, AMS University Lecture Series 37 (2005).
2. A. Beilinson, V. Ginzburg, W. Soergel, "Koszul duality patterns in representation theory," J. Amer. Math. Soc. 9 (1996), 473–527.
3. R. Martínez-Villa, "Introduction to Koszul Algebras," Revista de la Unión Matemática Argentina 48 (2007), no. 2, 67–95 |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Read the reference text whenever necessary. Feel free to contact me for any consultation. |
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注意事項
Notice for Students | | |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Algebra, homological algebra, category theory, representation theory, Koszul algebra, graded ring, Koszul duality, BGG correspondence |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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