Algebra 611, Fall 2009
MWF 10:10-11:00 at LGRT 1322


Covered Material, Homework Assignments, Lecture Notes

Vector Spaces. 5 lectures based on Chapter II of Knapp.
Vector spaces and linear maps. Linear independence, dimension, basis. Infinite-dimensional vector spaces. Duality for vector spaces. Linear algebra in coordinates. Quotients and direct sums. Determinants.
Homework 1 due 9/21.

Inner-Product Spaces. 3 lectures based on Chapter III of Knapp.
Inner product: real symmetric and complex Hermitian. Schwartz inequality and its friends. Bessel's inequality. Gram-Schmidt orthogonalization process. Orthogonal and unitary groups. Adjoint and self-adjoint linear maps. Spectral Theorem. Polar decomposition.
Homework 2 due 9/28.

Categories and Functors. 4 lectures based on IV.11, (parts of) VI.6 of Knapp, and lecture notes.
Categories. Examples: Vect_k, Sets, Groups, Ab, Rings, TopSpaces (topological spaces and continuous maps). A group as a category with one object. Poset as a category. Covariant and contravariant functors. Examples: forgetful functors, duality in Vect_k, TopSpaces->Rings (algebra of functions/pull-back of functions), complexification as a functor Vect_R->Vect_C. Equivalence of categories (was defined in class as a fully faithful and essentially surjective functor). Example: equivalence of the category of finite-dimensional vector spaces to the category with one object for each natural number and morphisms given by matrices. Representable functors. Products and coproducts. Examples in Vect_k and in Sets. Example of a category without a product. Natural transformations. Example: canonical isomorphism from V to V^** as a natural transformation.
Homework 3 due 10/12.

Groups and Group Actions. 4 lectures based on IV.1,2,3,6,10 and VI.1 of Knapp.
Three lectures were devoted to the proof of Sylow theorems. Along the way we reviewed the vocabulary of undergraduate group theory: groups, group actions, counting with groups, cosets, Lagrange theorem, homomorphisms, normal subgroups, quotient groups, first fundamental theorem, second fundamental theorem, center, conjugacy classes, stabilizer, normalizer, centralizer, etc. In the last lecture free groups and subgroups of free groups were discussed.
Homework 4 due 10/26.

Rings and Modules. 3 lectures based on IV.4,VIII.1,2,4 of Knapp.
Category of commutative rings. Ideals. First isomorphism theorem for rings (If f:R->S is a surjective homomorphism of rings then S is isomorphic to the qiotient ring of R by the ideal Ker f. There is a canonical bijection between ideals of S and ideals of R containing Ker f.) Category of R-modules. First isomorphism theorem for R-modules. Direct products and direct sums (coproducts) of rings and modules. Integral domains. Principal ideal domains. Principal ideal domains are factorial. Characteristic of a field. Finite fields.
Homework 5 due 11/2.

Structure theorem for finitely generated (f.g.) modules over PID. 4 lectures based on VIII.6 of Knapp.
Second isomorphism theorem for R-modules. Free R-modules. Linear algebra of free R-modules. Finitely presented R-modules. Any f.g. module over PID admits a finite presentation. Smith normal form of a matrix over PID. Any f.g. module of a PID is a direct sum of cyclic modules. Maximal ideals. Existence of maximal ideals. If Rⁿ is isomorphic to R^m then n=m. Chinese remainder theorem for PID. Young diagrams. Uniqueness theorem for f.g. modules over PID.
Homework 6 due 11/9.

Linear Operators. 2 lectures based on Knapp V.
Equivalence of a category of k[x]-modules and a category of vector spaces with a linear operator. Proof of the primary decomposition theorem, the cyclic (or rational) normal form, and existence/uniqueness of the Jordan canonical form. Characteristic polynomial. Minimal polynomials. Cayley-Hamilton theorem.
Homework 7 due 11/16.

Bilinear Forms. 2 lectures based on Knapp VI.1, VI.2, VI.3
Bilinear form, its matrix. Change of matrix under the basis change. Radical of a bilinear form. Orthogonal complement. Principal Axis Theorem for symmetric forms. Signature of a real symmetric form. Sylvester theorem. Canonical form of an alternating bilinear form. If V has a nondegenerate alternating form then dim V is even.
Take-home midterm due 11/23.

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Syllabus

Algebra is the language of several central mathematical fields, such as Number Theory, Algebraic Geometry, Representation Theory, and Algebraic Topology. It has applications to practically all mathematical (and not just mathematical) disciplines. It describes an incredibly elegant and sophisticated world of pure mathematical abstraction, which nevertheless is surprisingly useful. Perhaps one reason is that Algebra is ideally suited for inventing new rules, new objects, and new properties of old objects. Algebra has a lot of flexibility and generates new approaches to mathematical theories as quickly as our immune system generates responses to new diseases.

This semester we will concentrate on commutative aspects of algebra, more specifically on commutative rings and modules. Noncommutative algebra (Galois theory and basic representation theory) will be studied in the second semester in Algebra 612. The concept of a commutative ring is a synthesis of two notions: a ring of algebraic numbers (number theory) meets a ring of functions (analysis). It is a fact of towering importance that geometry and topology of a space can be encoded in algebra of its ring of functions. This is one of the cornerstones of modern mathematics that lead to algebraization of geometry (or geometrization of algebra, depending on your point of view).

The course has two goals. The first one is to introduce you to modern algebra on the level necessary for successful graduate studies. All mathematical disciplines from Applied Mathematics to Number Theory use algebraic tools. We will cover all material required for the advanced qualifying algebra exam (and for the Linear Algebra portion of the basic qualifying exam).

The second goal is to give you sufficient background in algebra to enable you to take more advanced graduate classes in algebraic disciplines and to participate in research seminars (Five College Number Theory Seminar, Five College Valley Geometry Seminar, Reading Seminar in Algebraic Geometry, Representation Theory Seminar). The course will also start to prepare those of you who are interested in doing research in algebra. Our department has a vibrant algebraic community that spans several research groups (Algebraic Geometry, Number Theory, and Representation Theory) and includes Tom Braden, Eduardo Cattani, Peter Dalakov, Farshid Hajir, Paul Gunnels, Paul Hacking, Eyal Markman, Ivan Mirkovic, Alexei Oblomkov, Jenia Tevelev, Peter Norman, Eric Sommers, Tom Weston, Siman Wong, and Giancarlo Urzua.

The textbook for this class is Knapp's Basic Algebra. It is very well written (I think) and provides an up-to-date account of the main concepts. However, be advised that this book contains only the minimum necessary material and a few lecture topics (specifically, basic homological algebra) are not sufficiently covered there. I will expect you to take lecture notes. The first chapter of Knapp summarizes several concepts that are usually covered in undergraduate classes such as Fundamental Concepts of Mathematics (UMass Math 300), Linear Algebra (UMass Math 235), Introduction to Algebra (Umass Math 411-412). We will not cover this chapter, however I suggest that you read it carefully for two reasons:

It explains why row reduction is useful not only for solving concrete systems of linear equations but also for proving a great deal of facts in Linear Algebra. We will use this a lot.

It discusses unique factorization of numbers and polynomials and basic facts about permutations. Sometimes instructors of undergraduate algebra courses explain these results but skip the proofs. However, I will assume that you are familiar with these arguments.

We will cover a fair amount of commutative algebra this semester. However, this is a course in Algebra and not in Commutative Algebra. It is aimed at all graduate students, and the syllabus is a result of many compromises. People who are thinking about doing research in algebra later on are advised to get a supplementary textbook, ideally Introduction to Commutative Algebra by Atiyah and MacDonald. This is an amazing book that covers commutative algebra in much greater depth and detail than our course yet philosophically it is very close to Knapp's textbook. It is also cheap, short, and contains a treasure trove of exercises.

Algebra is a language, and as with any language, the key to success is practice. I will distribute homework problem sets weekly and you should expect to spend a considerable amount of time solving these problems. At the beginning, you will probably have a feeling that each problem requires its own proof strategy and that some problems are really tricky to solve. However, at some point you will start to recognize the same old tricks and methods reused over and over again. You will become an algebraist!

There are certain things you can do to systematically improve your problem-solving techniques. For example, I am a big fan of active reading of mathematical textbooks. This means that each time you see a little statement, lemma, or theorem, you should try to come with its proof, or at least with the main idea of the proof, before you read the proof in the book. In fact, sometimes it is easier to come up with your own proof than to read a polished argument, because sophisticated notation can obscure a simple underlying principle. This is especially true for arguments (abundant in algebra) that amount to checking many little straightforward steps. However, some arguments are truly tricky and if your attempt to prove something on your own ends in failure, you will appreciate someone's ingenuity better (and then you can add this argument to your own bag of tricks).

There will be one in-class midterm on October 14, one take-home midterm (date TBA), and the final. In-class midterm's and final exam's problems will be worth certain amount of points, and you will have to choose and solve a subset of problems that adds up to 100 points. However, you won't be able to choose a subset that adds up to more than 100 points. The idea is that I don't want to punish you if you are stuck with a problem that requires a trick and you just can't come up with that trick. In this case you will have a freedom to choose another problem from the set. The take-home test and homeworks will contain A LOT of problems and will be randomly graded. Please be prepared to spend a considerable amount of time on homeworks. The course grade will be determined as follows: 20% Homework, 5% Class and Review Session participation, 25% In-class midterm, 25% Take-home midterm, 25% Final Exam.

I will have office hours on Thursday at 4pm in my office and a weekly review session of homework problems on Monday at 4pm in LGRT 1334. I will discuss homework problems after you turn them in. I will try my best to set up office hours that everybody can attend, so please keep me informed about changes in your schedule that interfere with office hours.

The course webpage is www.math.umass.edu/~tevelev/611.html. Please check it at least once a week for homework handouts and other relevant information.

Finally, let me give a brief and tentative outline of topics that remain to be covered this semester and related sections in Knapp's Basic Algebra. I encourage you to start reading these sections in advance.

Chapter VI. Multilinear Algebra. 4 lectures.
Bilinear forms, tensor products, symmetric and exterior algebras.

Chapter VII. Commutative rings and their modules. 10 lectures.
This is the main section of this course.

Lecture notes: Introduction to Homological Algebra. 4 lectures.
Tensor product of modules. Exact sequences, complexes, (co)homology, Snake Lemma, long exact sequence in cohomology. Main examples of (left,right) exact functors. Tor and Ext. The homological formalism is extremely useful in all mathematical fields, from Topology to Numerical Analysis. If you want a textbook to complement lecture notes, check out Advanced Algebra by Knapp of Lang's Algebra.