17Sp-520

Algebraic Structures II

Spring 2017

Instructor: Kwangho Choiy

Course Website: https://sites.google.com/site/kchoiy/home/teaching/17sp-520

Class Meeting: TTh 11:30am - 12:45pm in EGRA 208.

Textbook: Abstract Algebra 3rd edition by David S. Dummit and Richard M. Foote (other references: Introduction to Commutative Algebra by Michael F. Atiyah and Ian G. Macdonald / An Introduction to Homological Algebra by Joseph Rotman).

Objectives: Our goal is to learn basic notions and theories in modules, commutative rings, and homological algebras. A selected subset of the following will be discussed: general facts on rings/modules, tensor products, complexes, Ext, Tor, group cohomology, categories, functors, direct/inverse limits, completions, noetherian, artinian, localization, Spec, Hilbert’s Nullstellensatz, integral extensions, discrete valuation rings, Dedekind domains, basic terms on algebraic geometry, etc.

Syllabus: It is required to read carefully our syllabus linked [PDF], updated on 3/9/2017.

Project: One project per student will be given from Field theory and Galois theory (Chapters 13, 14 in the textbook). More details will be discussed in due course.

Exams: There will be one take-home mid-term and one take-home final. The schedule and further details will be discussed in due course. There is no make-up exam except exceptional cases.

Homework: Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date:

*HW Policies: You should show all your work and submit it in class on the due date. No late homework will be accepted.*

Updates and Remarks - MATH520 - Spring 2017:

[May 4, 2017] Lecture 30: more details relation between Zariski topologies in A_k^n, m-Spec(k[x1,x2...xn]), and Spec(k[x1,x2...xn]), brief introduction to direct, inverse limits, their examples, brief introduction to Dedekind domain, motivation, examples.

[May 2, 2017] Lecture 29: affine space A^n_k, algebraic set in k[x1,x2...xn], V(A), I(X), and their properties, Zariski topology on Spec(k[x1,x2...xn]) and on A^n_k, three versions of Hilbert's Nullstellensatz, relation between Zariski topologies in A_k^n, m-Spec(k[x1,x2...xn]), and Spec(k[x1,x2...xn]), example with k=C. Graded project is returned.

[Apr 27, 2017] Lecture 28: V(I), D_x, their properties, Zariski topology on Spec(R), its propertes, example, comparison Spec(C[x]) with the usual complex topology on C.

[Apr 25, 2017] Lecture 27:  M to S^{-1}M as functor from Mod(M) to Mod( S^{-1}M), proof of S^{-1}R as a flat R-module, corollaries, more properties of S^{-1}M and S^{-1}R.

[Apr 20, 2017] Lecture 26: more properties of S^{-1}R, Spec(R) and Spec(S^{-1}R), examples, localization S^{-1}M of modules, universal properties of S^{-1}R and S^{-1}M, S^{-1}M isomorphic to S^{-1}R tensor with M as S^{-1}R module, S^{-1}R as a flat R-module.

[Apr 18, 2017] Lecture 25: Motivations of localization of R at S, its connection to the rational numbers  Q from Z and the quotient field Quot(D) from an integral domain D, definition of S^{-1}R, its properties, examples. Take-Home final (due on 5/11) is handed out.

[Apr 13, 2017] Lecture 24: Hilbert's basis theorem and sketch of proof, artinian module/ring, examples, useful ascending and descending chains in commutative ring with 1, artinian ring => noetherian ring.

[Apr 11, 2017] Lecture 23: noetherian module, ring, examples, properties, characterizations of noetherian modules and rings. An additional note (4th) is handed out.

[Apr 6, 2017] Lecture 22: Hom_G(M, N) = (Hom(M,N))^G, defined a functor Hom_G(Z, _), group cohomology in terms of Ext, properties, Hilbert 90 Theorem, etc. An additional note (3rd) is emailed out.

[Apr 4, 2017] Lecture 21: G-module, Z[G]-module, group homology in terms of Tor, their properties, Hom(M,N) as G-module.  Graded take-home mid-term is returned.

[Mar 30, 2017] Lecture 20: Example for Ext, properties of Tor, Ext.

[Mar 28, 2017] Lecture 19: more properties of left/right derived functors, Tor, Ext, examples.

[Mar 23, 2017] Lecture 18: uniqueness up to homotopy of injective, projective resolutions, left, right derived functors, examples, properties.

[Mar 21, 2017] Lecture 17: injective resolution, examples, free/flat resolutions, two lemmas for the existence of projective/injective resolution.   Graded HW 3 is returned with solution.

[Mar 9, 2017] Lecture 16: outline for derived functors, homotopy, acyclic, projective resolution, examples. Office hours will be changed from 3/20 on as written above.

[Mar 7, 2017] Lecture 15: connecting homomorphism, long exact sequence of homology. Take-Home mid-term (due on 3/30) and project (due on 4/25) are handed out.

[Mar 2, 2017] Lecture 14: cohomology, morphism, exact sequence of complexes, some remarks in special cases of exact sequences.

[Feb 28, 2017] Lecture 13: characterizations of flat, injective, projective modules, examples, properties, free => projective => flat and counterexamples, cycle, boundary, homology of complexes.  Graded HW 2 is returned with solution.

[Feb 23, 2017] Lecture 12: flat, injective, projective modules, examples, properties. HW 3 (due on Mar. 9, Thur) is handed out.

[Feb 21, 2017] Lecture 11: short exact sequence from complexes, further notion of functors - additive, two Hom functors, tensor functor, covariant, contravariant, left/right-exact functors.

[Feb 17, 2017] An additional note (2nd) is emailed out.

[Feb 16, 2017] Lecture 10: complexes, exact sequences, splitness, ker f, coker f, snake lemma, category, object, morphism, functor.

[Feb 14, 2017] Lecture 9: examples for tensor product, some characterizations of operations and elements in tensor products, several remarks, properties, bimodule, algerbra.

[Feb 9, 2017] Lecture 8: definition of tensor product of modules, some descriptions, universal property. Graded HW 1 is returned / HW 2 (due on Feb. 21, Tues) is handed out.

[Feb 7, 2017] Lecture 7: proofs of Nakayama Lemma, some theorems related to Nakayama Lemma, proofs.

[Feb 2, 2017] Lecture 6: finite (finitely generated) module, minimal generating sets, a theorem related to minimal generating, Nakayama Lemma, proofs.

[Jan 31, 2017] Lecture 5: def/examples of modules, new terms, colon operators, annihilator, faithful, direct product/sum, free modules. 

[Jan 26, 2017] Lecture 4: proof of a characterization of radical, examples for radical, multiplicative subset, nil(R), jrad(R), comaximal, chinese remainder them, nilpotent element. HW 1 (due on Feb. 7, Tues) is handed out / An additional note (1st) is emailed out.

[Jan 24, 2017] Lecture 3: more examples for Spec(R), m-Spec, local, semi-local ring, radical, properties and theorems related to radical. 

[Jan 19, 2017] Lecture 2: proof of the two equivalent statements related to fields, domain, prime, maximal ideals, Spec(S) to Spec(R) from a homo R to S, Spec, m-Spec, examples.

[Jan 17, 2017] Lecture 1: Intro and outline of the course / general facts on ring theory - def, homomorphism theorem, ideals, factor rings, two equivalent statements related to fields. Syllabus distributed.

[Jan 10, 2017] Syllabus is uploaded above in PDF.