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