# Math 210A: Graduate Algebra (fall quarter)

Fall 2010

Mondays, Wednesdays, and Fridays 11:00-12:15 in McCullough 122
(In the first week of class, we will meet on Monday and Friday. Likely
in future weeks, we will meet on Wednesday and Friday; Monday will be office hours.)

**About the course:**
According to the bulletin: basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, introduction to representation theory.
This course is intended to get across material important for graduate students embarking on a Ph.D. in mathematics. Others are welcome to attend the class, but should understand that the course will not be directed at them.
As part of the re-envisioning of the algebraic part of the graduate curriculum, this course was toughened last year, and this is the second year of the new regime.
**Prerequisites:** complete fluency with the ideas of Math 120 and 121. In particular, I won't revisit the foundations of group theory, and will only briefy review the theory of rings.

**Professor:** Ravi Vakil (vakil@math), 383-Q, office hours Mondays 11-12:15 (in those weeks there is no Monday class).

**Course Assistant:** Jeremy Miller (jkmiller@stanford), 381-K,
office hours Wednesdays 2:30-3:30, Thursdays 12:30-2:30, Friday 2:30-3:30.

**Text:** Lang's Algebra (revised third edition) and Dummit and Foote (3rd ed.). I see the second as a better text. Lang is useful as an encyclopediadic reference. Don't feel obliged to buy a text, but you should definitely have access to books to make sure you understand the material.

**Grading:** Problem sets will make up 100% of the grade. The problem sets will be graded by Dung Nguyen and Jeremy Miller.

**Problem sets:** Problem sets will be due every Friday (including the first week of the class), unless otherwise noted.
Please hand them in by 3 pm to Jeremy Miller's mailbox, or give them to me in class.
No lates will be allowed. But to give everyone a chance to get sick, or have busy periods, the lowest homework grade will be dropped.

Problem set 1, due Friday September 24.
Problem set 2, due Friday October 1.
Problem set 3, due Friday October 8.
Problem set 4, due Friday October 15.
(Watch out for the error in 1(b)!)
Problem set 5, due Monday October 25.
Problem set 6, due Monday November 1.
Problem set 7, due Monday November 8.
Problem set 8, due Monday November 15.
Problem set 9, due Monday December 6.
**Syllabus:**
The theme of the course will be modules over rings in different contexts.
We will discuss basics of modules; linear algebra in the language of module theory (including the structure theorem for finitely generated modules over a PID); tensors (which will require an introduction to category theory); and homological algebra for modules. If time remains at the end of the class, we may discuss some representation theory or some of the theory of binary forms.

Mon. Sept. 20: review of rings. Commtuative, 0-ring, unit, invertible, polynomial ring, formal power series, subring, center, category of rings, initial object and final object, integral domain, field, ideal, quotient by ideal, prime maximal principal ideals, Noetherian rings, euclidean algorithm, euclidean domain, greatest common divisor, principal ideal domain, associates.
Fri. Sept. 24: factoriality. Irreducible, prime, associated, UFD, prim eimplies irreducible, reverse is true in PID or UFD, PID implies UFD, PIDs are Noetherian, fraction field, Gauss' Lemma, R a UFD iff R[x] UFD, Eisenstein's criterion.
Wed. Sept. 29: modules over a commutative ring (finitely generated, cyclic, maps of modules also form a module, kernel, cokernel, image, sum, product, four isomorphism theorems, free module, localization), examples (incl.: abelian groups are precisely Z-modules; a linear transformation on a vector space over k gives a k[t]-module). Noetherian modules. Rank of a module over an integral domain; statement of theorem on submodules of finitely generated free modules over PID.
Fri. Oct. 1: proof of classification of finitely generated modules over a PID (elementary divisor and invariant factor forms), and consequences for abelian groups.
Wed. Oct. 6: interpreting classification in the case of k[t] in terms of linear algebra.
Fri Oct. 8: using the classification to understand things like the minimal polyomial, characteristic polynomial, Cayley-Hamilton theorem, Jordan canonical form. Toward tensor products: universal properties (e.g. understanding localization of rings and modules via universal properties).
Wed. Oct. 13: tensor products and bilinear (and multilinear) maps, and many interesting properties. Symmetric and alternating multilinear maps.
Fri. Oct. 15: symmetric and alternating maps; perfect pairing duality.
Wed. Oct. 20: determinants; motivation for homological algebra from topology (v-e+f=2-2g).
Fri. Oct. 22: complex, exact, homology, snake lemma, five lemma,
long exact sequence corresponding to a short exact sequence of complexes.
Wed. Oct. 27: exactness of functors (when additive functors are left or right exact), motivation for Tor; flat R-modules.
Fri. Oct. 30: Tor and many of its properties (including long exact sequence); some examples; left derived functors of right-exact covariant functors.
Wed. Nov. 3: derived functors in general; why modules over a ring have enough injectives.
Fri. Nov. 5: introduction to group cohomology; Z[G]-modules; resolution of Z as a Z[G]-module (partial description); the complex of cochains of G with values in A.
Wed. Nov. 10 class canceled due to illness.
Fri. Nov. 12: more group cohomology; H^1; Hilbert 90.
Wed. Nov. 17: introduction to representation theory; linear representation, matrix representation, faithful; examples; as F[G]-module; equivalent/similar, (ir)reducible, (in)cdecomposible, completely reducible.
Fri. Nov. 19: Maschke's Theorem, there are only finitely many irreducible representations, all appearing in F[G]; Schur's lemma; cyclic groups; abelian groups; the example of S_3; introduction to character theory.
Wed. Dec. 1: character theory continued.
Fri. Dec. 3: Schur's lemma. Number of irreducibles is number of conjugacy classes. Summary of course.

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