# Math 200c Spring 2012

## Announcements:

### Instructors:

#### Professor:

Name Office E-mail Phone Office Hours Lecture Time Lecture Place
Prof. Daniel Rogalski AP&M 5131 drogalsk@math.ucsd.edu 534-4421 M 4-5pm, W 3-4pm MWF 2-2:50pm AP&M 5402

#### Teaching Assistants:

Name Office E-mail Office Hours
James Berglund 6331 AP&M jberglun@math.ucsd.edu Th 3-5pm or by appointment

Course description:

This is the third quarter of the three-part graduate algebra sequence. In this quarter, we will cover commutative ring and module theory as in the text Introduction to commutative algebra" of Atiyah and MacDonald. Please find the more detailed syllabus for the class, including information about homework, exams, academic honesty, etc., below:

### Math 200c course syllabus

Lecture Summaries

• Lecture 1 (04/02/12): The prime spectrum. The set of nilpotent elements of a ring is the intersection of all prime ideals. Radical ideals. Primes minimal over an ideal.
• Lecture 2 (04/04/12): The Zariski topology on Spec R. Prime avoidance. In a noetherian ring R, there are finitely many primes minimal over an ideal I, and some power of the radical of I lies in I.
• Lecture 3 (04/06/12): Affine space and its closed subsets. Correspondence between ideals of C[x_1, dots x_n] and subsets of affine space A^n. Statement of the nullstellensatz without proof: there is a 1-1 correspondence between radical ideals of C[x_1, dots, x_n] and closed subsets of A^n. Relation between the topology on A^n and the Zariski topology on C[x_1, dots x_n].
• Lecture 4 (04/09/12): The Jacobson radical. Nakayama's lemma and applications. Contractions and extensions of ideals under a ring homomorphism A \to B. Example: Z \to Z[i].
• Lecture 5 (04/11/12): Review of localization. Study of contractions and extensions of ideals under the localization map R to S^{-1}R. Prime ideals of S^{-1}R are in 1-1 correspondence with primes of R which do not meet S.
• Lecture 6 (04/13/12): Localization is an exact functor. Local properties. Geometric interpretation of localization.
• Lecture 7 (04/16/12): Primary ideals. Definition of (minimal) primary decomposition. Associated primes. The noetherian case: irreducible ideals are primary, and every ideal is a finite intersection of irreducible ideals; so primary decompositions exist for ideals in a noetherian ring.
• Lecture 8 (04/18/12): Characterization of the associated primes of I as those primes among the ideals rad(I:x). In the notherian case the radical is not needed. Associated primes of I are the same as prime annihilators of submodules of R/I.
• Lecture 9 (04/20/12): Associated primes of 0 are exactly the union of all zerodivisors of R. Primary decomposition and localization. The primary ideals whose associated primes are minimal are uniquely determined, but the others may not be.
• Lecture 10 (04/23/12): Integral extensions. Equivalent characterizations of an integral extension A contained in B: (i) Every element of B satisfies a monic polynomial in A[x]; (ii) Every b in B satisfies that A[b] is a finite A-module; (iii) Every b \in B is contained in a subring C with A subset C subset B such that C is a finite A-module. Other lemmas on integral extensions.
• Lecture 11 (04/25/12): A PID R is integrally closed. The elements in a number field K which are integral over Z are called the ring of integers O_K of K. An element is in O_K if and only if its minimal poly over Q has Z-coefficients. Example: O_K where K = Q(\sqrt(d)) is a quadratic field extension of Q. Beginning to talk about primes in integral extensions (Cohen-Seidenberg theorems)
• Lecture 12 (04/27/12): More on prime ideals in integral extensions. Incomparability, lying over, going up, statement of going down (no proof). Krull dimension. If A to B is an integral extension, then A and B have the same Krull dimension.
• Lecture 13 (04/30/12): Chain conditions. Noetherian and artinian modules. A module which is artinian and not noetherian. Finite length modules. Length of a module and the Jordan-Holder theorem (no proof). In a short exact sequence of modules, the middle term is noetherian (or artinian) if and only if the outer two terms are. A f.g. module over a noetherian (or artinian) ring is noetherian (resp. artinian). Subrings of noetherian rings need not be noetherian; even k[x, y] has a non-noetherian subalgebra, given by k[x, xy, xy^2, \dots ].
• Lecture 14 (05/2/12): Artin-Tate lemma. Proof that if F is a field which is a f.g. k-algebra, then [F:k] is finite, using Artin-Tate. Proof of the weak Nullstellensatz: if k is algebraically closed, then the max ideals of k[x_1, ... x_m] are precisely the ideals of the form (x_1 - a_1, ... x_m - a_m).
• Lecture 15 (05/4/12): Jacobson rings. Proof that k[x_1, ... x_m] is Jacobson, using weak Nullstellensatz. Proof of the Strong Nullstellensatz: if J is an ideal of k[x_1, ..., x_m], k algebraically closed, then I(V(J)) is equal to the radical of J. Noether normalization (statement, no proof). Using Noether normalization to give an alternative proof that if F is a field which is a f. g. k-algebra, then [F:k] is finite.
• Lecture 16 (05/7/12): Artinian rings. In an Artinian ring, all primes are maximal, and there are finitely many maximal ideals. The Jacbson radical is nilpotent. An Artinian ring is noetherian. A ring is artinian if and only if it is noetherian of Krull dimension 0. An Artinian ring is isomorphic to a product of local artinian rings.
• Lecture 17 (05/9/12): Valuation rings and discrete valuation rings (dvr's). A (non-Archimedean) valuation of a field and the corresponding valuation ring. Valuation rings are local and integally closed. Basic properties of discrete valuation rings: the max ideal is principal, and every nonzero ideal is a power of the max ideal. Conversely, a noetherian local integrally closed domain of dimension 1 must be a dvr.
• Lecture 18 (05/11/12): Definition of Dedekind domain: a noetherian integrally closed domain of dimension 1. Example: ring of integers in an algebraic number field. A noetherian domain A is Dedekind if and only if all of its local rings A_m for maximal ideals m are dvr's.
• Lecture 19 (05/14/12): Applications of Dedekind Domains. In algebraic geometry, an affine curve (the space spec A for a f. g. k- algebra of dimension 1) is smooth if and only if all of its local rings A_m are dvr's, i.e. if and only if A is a Dedekind domain. The example of k[t^2, t^3], the cuspidal cubic, which fails to be smooth only at one point. Every ideal in a Dedekind domain is uniquely a product of powers of maximal ideals; this kind of unique factorization is more general than factorization of elements in a UFD. The example (2)(3) = (1 + sqrt(-5))(1 -sqrt(-5)) in the non-UFD Z[sqrt(-5)], reexamined from this perspective.
• Lecture 20 (05/16/12): Fractional ideals; basic definitions. In a Dedekind domain, all fractional ideals are invertible. The fractional ideals form a group under multiplication, which is a free abelian group of rank the number of max ideals. The class group is the group of fractional ideals modulo the group of principal fractional ideals. For the ring of integers of a number field, the class group is finite and it is an important number-theoretic invariant that is difficult to calculate in general.
• Lecture 21 (05/18/12): Graded rings and modules. The Poincare series P(M; t) = sum dim_k M_n t^n of a graded module M over a graded k-algebra A = k + A_1 + A_2 + ... which is generated in degree 1. The Hilbert polynomial of $M$ exists: this is a polynomial f such that dim_k M_n is equal to f(n) for all large n.
• Lecture 22 (05/21/12):Given a finitely generated module M over a noetherian local ring (A,m), the definition of the dimension d(M) using the Hilbert-Samuel polynomial. Namely, take the degree of the polynomial g in Q[x] such that g(n) agrees for large n with the k-dimension of M/m^nM, where k is the residue field. Basic lemma showing this definition is sensible--- we can replace m^nM in fact by any m-stable filtration, and even replace m by an m-primary ideal q (now using length instead of k-dimension). Proof that d(M/xM) is less than or equal to d(M) -1, using Artin-Rees lemma (proof of Artin-Rees omitted.)
• Lecture 23 (05/23/12): Proof that for a noetherian local ring A, d(A) as defined above agrees with Krull dimension. Corollaries: k[x_1, \dots, x_m] has dimension m. Krull's principal ideal theorem: A prime minimal over a principal ideal has height one.
• (05/25/12, 05/28/12, 05/30/12, 06/01/12): No lecture
• Lectures 24, 25 (06/04/12, 06/06/12): A brief introduction to the theory of representations of finite groups.

Homework

• Homework 1: Homework 1 due Friday 4/13

• Homework 2: Homework 2 due Friday 4/27

• Homework 3: Homework 3 due Friday 5/18

• Homework 4: Homework 4 due Friday 6/8

• ## Qualifying exam

• Qualifying exam given May 31, 2012: Spring 2012 qual