Initial planning for MA4JB
This is an advanced course, and some parts may involve extended
proofs, sophisticated preliminaries from set theory or category
theory, or motivation from algebraic geometry or number theory
that may seem obscure. If you are learning to juggle, you must
at some point get used to having more than one ball in play.
Going back to basics and breaking an argument into its logical
nuts and bolts is important. However, an overall narrative or
slogan may enlighten a whole subject even if it takes time to
digest. The capacity of the human intellect is infinite, and
intuition and motivation are just as important as logic or
rigour. (Get on with it!)
==
Relation with MA3G6 Commutative Algebra.
LI Chun'yi (who taught MA3G6 in 2020-21) wrote:
"I only covered around 70% contents in [UCA]. I did not cover:
Zariski topology; localisation of a module; several topics on DVRs:
(Chap 8.4-8.6, 8.8-8.13).
Also as I rarely use the original proof, I may have skipped some
technical results that may be used in an advanced module, for example,
Noetherian normalisation."
==
Shopping list
This is an inevitably incomplete list of items I would like to
cover, and I will add to it as the term progresses.
Zariski topology
local rings
comparison NSS vs localisation for ideals and modules
Factorisation, discrete valuations rings and normal rings
Finiteness of normalisation
The Artin-Rees lemma for completions and power series methods
Groebner bases and algorithmic approaches to commutative algebra
Dimension theory, the Hilbert and Hilbert-Samuel function
Systems of parameters and multiplicity
Exact sequences of modules and basic notions of homological algebra
Regular sequences, the Koszul complex, depth, Cohen-Macaulay and
Gorenstein rings
Characterisation of regular local rings.
==
Little plan to fix the starting point:
0.1 Prime ideal, the existence of primes
-> the intersection of all the prime ideals containing I
equals rad I. This is an abstract algebra analog of the NSS.
0.2 localisation, the Zariski topology on Spec A, the prime
ideal of a local ring
-> basic ideas of affine schemes in simplified terms. (If X
= Spec A, the regular sections of OX equals A, and same for
Xf.)
0.3 modules over 0.2: localisation of modules, basic
definitions related to coherent sheaves. Supp M and Ass M
for finite module M. Mention the main result from Hartshorne
Chap 2 and Shafarevich Book 2 that a coherent sheaf defined
locally on a covering of Spec A is the associated sheaf of M.
0.4 abstract valuation rings, discrete valuation ring, many
characterisations of DVR following [A&M]. A Noetherian
domain is an intersection of DVRs. Applications in alg geometry
and number theory.
0.5 mention without proof: Noether normalisation lemma,
the NSS. compare with 0.1.
0.6 state finiteness of normalisation, relation with Galois
theory and with ring of integers in a number field.
==
Completion
Understand the m-adic completion of a local ring and its use for local
fields in number theory and power series rings in complex analysis.
Includes Hensel's lemma as simple corollary. Includes classification
of ordinary double points.
Example. This illustrates the point of completion or power series
methods at a glance. p in C a point of a curve in AA^2.
False: p is a node, or ordinary double point, if and only if
$O_{C,p} \iso k[x,y]/(xy)$.
If this were true then every irreducible branch of C would be iso AA^1,
either x=0 or y=0, so C = V(xy).
True: p is a node iff the completion $\widehat{O_{C,p} \iso k[[x,y]]/(xy)$.
Appreciate the meaning of Hilbert functions and Hilbert-Samuel
polynomials, including various methods of calculation, and application
to dimension theory and multiplicity
==
Course description
Overall aim:
Develop a sophisticated command of the many facets of a major branch
of algebra with important applications across the whole of mathematics.
Apply advanced notions such as localisation, completion, dimension,
extension, regular sequences, Hilbert and Hilbert-Samuel series, to
rings and their modules, both in theoretical arguments and in practical
applications.
Give an advanced overview in different settings of the theory of
division with remainder, Euclidean algorithm and unique factorisation
Understand finitely generated modules over a principal ideal domain in
terms of Smith normal form
Treat discrete valuation rings both in terms of their factorisation
theory and integral closure
Understand the m-adic completion of a local ring and its use for local
fields in number theory and power series rings in complex analysis.
Includes Hensel's lemma as simple corollary. Includes classification
or ordinary double points.
Appreciate the meaning of Hilbert functions and Hilbert-Samuel
polynomials, including various methods of calculation, and application
to dimension theory and multiplicity
Marco Schlichting's additional aims
Use Hensel's lemma to determine solubility of polynomial
equations over local fields
Have a good grasp of Groebner bases and Buchberger's algorithm
Develop good understanding of smooth and etale extensions