MA4J8 organisation and syllabus

My notes break up into "2024 first half" followed by organised
second half of the course
  7. Homological algebra appendix
  5. Regular sequence and Koszul complex
  6. Cohen-Macaulay and Gorenstein

==== FIRST HALF ====
This seeks to fill in prerequisite material from MA3G6
Commutative Algebra or my textbook [UCA], and go through the
basics of commutative algebra based on graded rings, localisation
and completion. This is bulky material, but a lot of it consists of
reminder of known background, some of it summarised as crib-sheet
Frequently Forgotten Facts.

Syllabus (including background and reminders of prerequisites):

-> Existence of primes using Zorn's lemma

-> Characterisation radical(I) = intersection of primes containing I

-> Krull dimension of a ring and height of a prime ideal

-> Localisation S^-1A and S^-1M, its kernel and exactness properties

-> local ring, Nakayama's lemma (the easy proof)

-> the "Determinant Trick" or Cayley-Hamilton theorem, and the proof
of Nakayama's lemma via automorphisms

-> Spec of a ring, its Zariski topology and principal open sets

-> Noetherian and Artinian conditions on rings and modules

-> Finite length modules and Jordan-Hoelder sequences, length
optional: proof that length is well defined and additive in s.e.s.

optional: proof that Artinian ring is Noetherian, so finite length

-> Associated prime, devissage of a module under Noetherian assumptions

-> Integral closure inside a finite field extension
optional: proof of finiteness

-> Discrete valuation ring DVR

-> Characterisation of DVRs as 1-dimensional Noetherian local domain

-> optional: Dedekind domain, includes ring of integers of a number
   field and affine coordinate ring of nonsingular algebraic curve

-> Completion, Hensel's lemma, the Artin-Rees Lemma

-> Graded ring, Hilbert series, proof that it is a rational function

-> Krull's Main Theorem on dimension of local rings: Krull dimension
   equals order of growth of the Hilbert-Samuel function equals minimal
   length of system of parameters

==== SECOND HALF ====
The more tightly organised second half of the module works towards
Cohen-Macaulay rings and modules, Macaulay "unmixedness" and
Gorenstein rings. This makes systematic use of complexes and
ideas from Homological Algebra, especially the Hom and Ext
functors. Almost all the textbooks summarise the required
homological algebra as an appendix -- some students have already
seen the material in earlier courses, but some need to go through
it first.

My lecture notes are supplemented by the write-up from the 2022-23
course by Alex Groutides

==== 7. Homological algebra appendix ====

-> [background: chain complexes, their homology, the Snake Lemma.

-> A short exact sequence of chain complexes gives a homology long
exact sequence, a chain homotopy gives isomorphism on homology]

-> Projective modules, definition and Ext^1, 

-> characterisation of projective modules as locally free modules

-> Injective modules, the categorical definition and Ext^1

-> divisibility, Baer's criterion and examples of injective modules

-> existence of injective embedding and resolutions (proof optional)

-> definition of the derived functors Ext^i(N,M)

-> optional: proof that Ext^i(N,M) well defined as bifunctor

-> characterisation of finite projective dimension and finite injective
dimension in terms of vanishing Exts

==== 5. Regular sequence and the Koszul complex ====

-> complexes and free resolutions

-> the explicit Koszul complex on 2 elements (or 3 elements as exercise)

-> the Hyperplane Section principle

-> the Hilbert syzygies theorem and its Auslander-Buchsbaum refinement
  (for finite graded modules over polynomial rings)

-> regular sequences

-> depth of a module = max length of regular sequence

-> exterior algebra and the general Koszul complex

-> regular sequence in terms of the exactness of the Koszul complex

-> Koszul complex as tensor product of length 1 complexes

-> counter-examples from primary decomposition with embedded points

-> many examples of rings or modules with depth 0 and depth 1

-> the Serre R1 and S2 characterisation of normal Noetherian domain
   optional proof is on example sheet 3

==== 6. Cohen-Macaulay and Gorenstein ====

-> characterisation of depth in terms of vanishing Ext^i

-> Socle A and the characterisation of 0-dimensional Gorenstein
rings in terms of A injective as A-module

-> Statement of the equivalent definitions of Gorenstein in terms
of finite injective dimension