高等交换代数

课程编码：070104D05010Z 英文名称：A Second Course in Commutative Algebra 课时：40 学分：2.00 课程属性：专业普及课 主讲教师：Manolis Tsakiris

The main purpose of this course is to strengthen the algebraic background of students pursuing pure, applied or computational algebraic geometry majors. The course studies fundamental notions of modern commutative algebra such as the Cohen-Macaulay property, local cohomology and Hilbert functions. Students taking this course are expected to be familiar with the basics of commutative algebra including the dimension theory of Noetherian rings.

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第一章 Cohen-Macaulay Rings 12.0学时
第1节 Regular Sequences
第2节 Grade and Depth
第3节 Depth and Projective Dimension
第4节 Koszul Complex
第5节 The Cohen-Macaulay Property I
第6节 The Cohen-Macaulay Property II
第二章 Local Cohomology 12.0学时
第1节 Injective Dimension and Gorenstein Rings
第2节 Injective Hulls
第3节 Matlis Duality
第4节 The Dualizing/Canonical Module
第5节 Local Cohomology Functors and Cech Complex
第6节 Grothendieck’s Vanishing and Local Duality Theorems
第三章 Hilbert Functions 16.0学时
第1节 Strongly Stable Monomial Spaces and Compression
第2节 The Generic Initial Ideal
第3节 Macaulay Representations
第4节 Green’s Hyperplane Section Theorem
第5节 Macaulay’s Characterization Theorem
第6节 Serre’s Euler Characteristic Theorem
第7节 Gotzmann’s Regularity Theorem
第8节 Gotzmann’s Persistence Theorem

Manolis C. Tsakiris，男，数学与系统科学研究院 副研