Module Handbook

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Module MAT-41-20-M-7

Cohen-Macaulay and Gorenstein Rings (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-41-20-M-7 Cohen-Macaulay and Gorenstein Rings 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT)
Lecturers
Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT)
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part 		Presence-Time /
Self-Study 		SL SL is
required for exa. 		PL CP Sem.
4V MAT-41-20-K-7
Cohen-Macaulay and Gorenstein Rings
P 56 h 214 h - - PL1 9.0 irreg.
  • About [MAT-41-20-K-7]: Title: "Cohen-Macaulay and Gorenstein Rings"; Presence-Time: 56 h; Self-Study: 214 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86154 ("Cohen-Macaulay and Gorenstein Rings")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

From [MAT-41-20-K-7] Cohen-Macaulay and Gorenstein Rings:
  • regular sequences, depth, projective dimension,
  • reflexive modules,
  • Fitting ideals,
  • Cohen-Macaulay rings and modules,
  • regular and normal rings,
  • complete intersections,
  • canonical module and Gorenstein rings,
  • local duality,
  • algorithmic aspects.

Competencies / intended learning achievements

Upon successful completion of this module, the students have obtained advanced knowledge in commutative ring theory. The tools and methodology acquired opens the doors to some of the most fundamental structures in Algebraic Geometry and Singularity Theory, as well as to their algorithmic computations.

The students are able to name the essential propositions of the lecture as well as to classify and to explain the connections. They understand the proofs presented in the lecture are able to comprehend and explain them.

Literature

From [MAT-41-20-K-7] Cohen-Macaulay and Gorenstein Rings:
  • D. Eisenbud: Commutative Algebra with a View towards Algebraic Geometry,
  • W. Bruns, J. Herzog: Cohen-Macaulay Rings,
  • H. Mastumura: Commutative Ring Theory.

Requirements for attendance (informal)

Modules:

Courses

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-41-20-M-7]

Module-Pool Name
[MAT-41-MPOOL-7] Specialisation Algebraic Geometry and Computer Algebra (M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)