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Unit of study_

MATH4312: Commutative Algebra

Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.

Details

Academic unit Mathematics and Statistics Academic Operations
Unit code MATH4312
Unit name Commutative Algebra
Session, year
? 
Semester 2, 2021
Attendance mode Normal day
Location Camperdown/Darlington, Sydney
Credit points 6

Enrolment rules

Prohibitions
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None
Prerequisites
? 
None
Corequisites
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None
Assumed knowledge
? 

Familiarity with abstract algebra, e.g., MATH2922 or equivalent.

Available to study abroad and exchange students

Yes

Teaching staff and contact details

Coordinator Ruibin Zhang, ruibin.zhang@sydney.edu.au
Type Description Weight Due Length
Final exam (Take-home short release) Type D final exam Final take home exam
Take home exam
60% Formal exam period 2 hours
Outcomes assessed: LO1 LO7 LO6 LO4 LO3 LO2
Assignment Assignment 1
Assignment
20% Week 06 3000-5000 words
Outcomes assessed: LO1 LO7 LO5 LO3
Assignment Assignment 2
Assignment
20% Week 11 3000-5000 words
Outcomes assessed: LO1 LO2 LO4 LO5 LO7
Type D final exam = Type D final exam ?

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

At HD level, a student demonstrates a flair for the subject as well as a detailed and comprehensive understanding of the unit material. A ‘High Distinction’ reflects exceptional achievement and is awarded to a student who demonstrates the ability to apply their subject knowledge and understanding to produce original solutions for novel or highly complex problems and/or comprehensive critical discussions of theoretical concepts.

Distinction

75 - 84

At DI level, a student demonstrates an aptitude for the subject and a well-developed understanding of the unit material. A ‘Distinction’ reflects excellent achievement and is awarded to a student who demonstrates an ability to apply their subject knowledge and understanding of the subject to produce good solutions for challenging problems and/or a reasonably well-developed critical analysis of theoretical concepts.

Credit

65 - 74

At CR level, a student demonstrates a good command and knowledge of the unit material. A ‘Credit’ reflects solid achievement and is awarded to a student who has a broad general understanding of the unit material and can solve routine problems and/or identify and superficially discuss theoretical concepts.

Pass

50 - 64

At PS level, a student demonstrates proficiency in the unit material. A ‘Pass’ reflects satisfactory achievement and is awarded to a student who has threshold knowledge.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see sydney.edu.au/students/guide-to-grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

  • Deduction of 5% of the maximum mark for each calendar day after the due date.
  • After ten calendar days late, a mark of zero will be awarded.

Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

Academic integrity

The Current Student website provides information on academic honesty, academic dishonesty, and the resources available to all students.

The University expects students and staff to act ethically and honestly and will treat all allegations of academic dishonesty or plagiarism seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic dishonesty. If such matches indicate evidence of plagiarism or other forms of dishonesty, your teacher is required to report your work for further investigation.

WK Topic Learning activity Learning outcomes
Week 01 Introduction and Basic overview Lecture (3 hr)  
Week 02 A first view of the bridge Lecture (3 hr)  
Week 03 Modules and basic category theory Lecture (3 hr)  
Week 04 Noetherian rings Lecture (3 hr)  
Week 05 Nullstellansatz Lecture (3 hr)  
Week 06 Homological functors Lecture (3 hr)  
Week 07 Modules of fractions and localisation Lecture (3 hr)  
Week 08 Primary decomposition Lecture (3 hr)  
Week 09 Dimension theory Lecture (3 hr)  
Week 10 Algebraic geometry 1 Lecture (3 hr)  
Week 11 Algebraic geometry 2 Lecture (3 hr)  
Week 12 Algebraic geometry 3 Lecture (3 hr)  
Week 13 Algebraic geometry 4 Lecture (3 hr)  
Weekly One Tutorial per week Tutorial (13 hr)  

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

  • LO1. Demonstrate an understanding of key concepts in commutative algebra and its connections to algebraic geometry.
  • LO2. Apply these concepts to solve qualitative and quantitative problems in mathematical contexts, using appropriate mathematical techniques as necessary.
  • LO3. Distinguish and compare the properties of different types of rings, analysing them into constituent parts.
  • LO4. Formulate geometric problems and algebraic terms and determine the appropriate framework to solve them.
  • LO5. Synthesise knowledge from fundamental theorems in commutative algebra and use this to prove new results.
  • LO6. Demonstrate a broad understanding of important concepts in commutative algebra and exercise critical thinking in recognising and using these concepts to draw conclusions and analyse examples.
  • LO7. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1 Depth of disciplinary expertise

Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.

GQ2 Critical thinking and problem solving

Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.

GQ3 Oral and written communication

Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.

GQ4 Information and digital literacy

Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.

GQ5 Inventiveness

Generating novel ideas and solutions.

GQ6 Cultural competence

Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.

GQ7 Interdisciplinary effectiveness

Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.

GQ8 Integrated professional, ethical, and personal identity

An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.

GQ9 Influence

Engaging others in a process, idea or vision.

Outcome map

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9
No changes have been made since this unit was last offered.

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