MA4J8 Commutative Algebra II

2025 Lecture timetable

Mon 16:00 in MA B1.01 from Mon 6th Jan 2025
Thu 14:00 in MA B1.01
Fri 10:00 in MA B3.01
Support class: First session _Week 1_ Thu 9th Jan 13:00-14:00 in A1.01
with TA Marc Truter <Marc.Truter@warwick.ac.uk>

2024 Worksheets

Worksheet_0 on prerequisites
Worksheet_1    Assignment deadline Mon 29th Jan at 12:00
Worksheet_2    Assignment deadline Mon 12th Feb at 12:00
Worksheet_3    Assignment deadline Mon 11th Mar at 12:00
Worksheet_4    Assignment deadline Mon 25th Mar at 12:00

Write-up of Autumn 2022 lectures by Alexandros Groutides

Chapter 5    Regular sequences, the Koszul complex and regular local rings
Chapter 6    Cohen-Macaulay and Gorenstein rings
Chapter 7    Appendix on homological algebra

Lecture notes from Autumn 2022

Week1
Week2
Week3
Weeks4-5
Week3
Weeks4-5
Week6 Lecture 17 Part I Lecture 17 Part II Lecture 18 Part I Lecture 18 Part II

Here is the Warwick UG Handbook entry.

Lecture capture

To catch up on missed lectures (a few hours after they happen)
click on the Echo 360 Lecture Capture block on the right
of this Moodle page

Last year's notes

The information below refers to the course given in 2021-22.

Plans My initial plans for the course

Lect 1 What is commutative algebra? [Sorry, no lecture capture.]
Lect 2 Dedekind domain, Existence of primes [Sorry, I forgot to
wear the microphone for first 45 minutes.]
More on DVRs Further discussion and exercises on Discrete valuation rings.
Lect 3 First ideas on Spec A and the Zariski topology
Lect 4-5 Varieties versus Spec A. Chain conditions
Lect 6 Finite length modules. An Artinian ring is Noetherian
Lectures 7-10. Completion and Hensel's Lemma
Lect 12-14 Dimension theory. Graded rings and modules and their Hilbert series
Lect 15-17 Hilbert-Samuel functions and the main theorem of dimension theory
Lect 18-22 Syzygies and regular sequences
Lect 25-29 Ext and depth - preliminary draft

Homework sheets

There will be 4 Assessed worksheets, with submission deadlines 12:00 noon on

   Week 3 Fri 22nd Oct
   Week 5 Fri 12th Nov
   Week 7 Fri 26th Nov
   Week 10 Fri 9th Dec

Example sheet 1 Questions 1,8 and 11 are assessed questions.
Solution to Q10
Example sheet 2
Assessed questions from Example sheet 2
Example sheet 3
Example sheet 4

Additional resources

Download

Notes from 2020

The following is a record of the lecture course as I gave it in 2020. The final
Part 5 is a somewhat novel approach to constructing the canonical class K_C
to complete the proof of RR that seems simpler and more convincing than
treatments in the current literature.

I have not yet had time to polish this up adequately, and some parts obviously
need more work to bring them up to textbook standard. I hope to return to this at
some future point.

Part 1
Part 2
Part 3
Castenuovo free pencil trick
Max Noether's theorem
Linearly general position
Part 4
Part 5

Worksheets

Example Sheet 1
Example Sheet 2
Example Sheet 3
Example Sheet 4
Example Sheet 5
Jun 2020 exam

Scrap

The first lecture tries to outline in approachable colloquial terms
the idea that the course contents is easy, but built on
sophisticated and sometimes difficult prerequisites from
several areas.

Normal characterises DVRs A brief self-contained treatment of a
key result on nonsingularity.

Here is the old directory containing the 2019 notes and
worksheets and other scrap.
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