Math 6340 - Commutative Algebra with Applications in Algebraic Geometry

The topics in the following syllabus are tentative, and might change. Check back here often for new versions! Version 0.1

Class#	Date	Topic	Read	Problem sets
1	21 Jan 2026	Introduction Algebra-Geometry dictionary	AM Ch 1
	26 Jan 2026	Cancelled (snow)!
2	28 Jan 2026	Local rings, Spec, Jacobson and nilradicals, Modules	AM Ch 1, AM Ch2.
3	2 Feb 2026	Syzygies, Free resolutions, Hom	AM Ch 2 E Ch 2.2
4	4 Feb 2025	Tensors, adjointness, def of Ext	AM Ch 2, E Ch 2.2 E App 3
5	9 Feb 2026	Ext, Tor (and projective modules)	AM Ch 2, E App 3
6	11 Feb 2026	Flatness (and exact functors)	AM Ch 2 (Flatness) E Ch 6.1,6.2,6.3
	16 Feb 2026	No class!
7	18 Feb 2026	Macaulay2, Localizations of rings, modules	AM Ch 3 E Ch 2
8	23 Feb 2026	Localization, exactness start Primary Decompositions
9	25 Feb 2026	Primary decompositions	AM Ch 4, parts of Ch 7 E Ch 3	Expected you know: AM Ch 6, AM Ch 7
10	2 Mar 2026	Primary decompositions
11	4 Mar 2026	finish Primary decompositions Chains of primes, dim
12	9 Mar 2026	Going up/down, Nakayama's lemma, Integral extensions
13	11 Mar 2026	Integral extensions
14	16 Mar 2026	LO, INC, GU for integral extensions	AM Ch 5 pp 59-62, E Ch 4.4
15	18 Mar 2026	Graded rings and Hilbert functions	AM Ch 10 pp 106, Ch 11 pp 116-118 E Ch 5
16	23 Mar 2026	Hilbert polynomials, Rees algebra Associated graded ring	AM Ch 10 (but just filtrations), AM Ch 11 E Ch 5
17	25 Mar 2026	Filtrations, Artin-Rees lemma	AM Ch 11, E Ch 8,9
	30 Mar 2026	No class!
	1 Apr 2026	No class!
18	6 Apr 2026	Length, Hilbert-Samuel polynomials
19	8 Apr 2026	Dimension theory	AM Ch 11, E Ch 8,9
20	13 Apr 2026	Dimension Theory (cont'd) Krull's principal ideal theorem
21	15 Apr 2026	Dimension theory, cont'd
22	20 Apr 2026	Noether normalization lemma Dimension of affine rings Groebner bases	E Ch 15
23	22 Apr 2026	Groebner bases and applications	E Ch 15
24	27 Apr 2026
25	29 Apr 2026
26	4 May 2026