17Sp-520

Algebraic Structures II

Spring 2017

Office: Neckers 283 (618-453-6508)
E-mail: kchoiy_at_siu_dot_edu
Office Hours: T1-5pm / W3:30-4:30pm / Th2-3pm, or by appointment. Emails are also available.

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.

Project (due on Apr. 25, Thur; handed out on Mar. 7, Tues)

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.

Mid-term (due on Mar. 30, Thur; handed out on Mar. 7, Tues)
Final (due on May 11, Thur; handed out on Apr. 18, Tues)

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.*

HW 1 (due on Feb. 7, Tues) - updated on Feb. 2 (typo #2 fixed, etc.)
HW 2 (due on Feb. 21, Tues)
HW 3 (due on Mar. 9, Thur)

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.

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