Math 200c Spring 2015

Remember that there is no final exam; the grade is based on the best 3/4 homeworks. Have a great summer!

Instructors

Professor:

Teaching Assistant:

Course Description and Syllabus:

Math 200c Spring 2015 course syllabus

Lecture Summaries

Below, we will post short summaries of what was covered in each lecture.


• Lecture 1 (3/30/15): The prime and maximal spectrum of a commutative ring R. The nilradical (ideal of nilpotent elements) is equal to the prime radical (intersection of all prime ideals). The Jacobson radical J is the intersection of all maximal ideals. An element x is in J if and only if xy + 1 is a unit for all y in R.
• Lecture 2 (4/1/15): The Zariski topology on Spec R, whose closed sets are of the form V(I) for ideals I, where V(I) is the set of prime ideals containing I. Example: if C is the complex numbers then closed sets in spec C[x] are finite sets of maximal ideals (x-a), together with the whole space. This is non-Hausdorff. Spec R/I naturally embeds as a closed set in Spec R. The radical r(I) of an ideal I. r(I) is equal to the intersection of all primes containing I. Given a prime ideal Q containing I there exists a prime P minimal over I with I subseteq P subseteq Q.
• Lecture 3 (4/3/15): Modules satisfying the ascending chain condition (ACC) (called noetherian modules) or the descending chain condition (DCC) (called artinian modules). ACC is equivalent to the maximal condition: every collection of submodules has a maximal element, and DCC is equivalent to the minimal condition: every collection of submodules has a minimal element. ACC for M is also equivalent to the condition that every submodule of M is finitely generated. Noetherian and artinian rings. In an noetherian ring, some product P_1P_2 ... P_n of primes (possibly with repeats) equals 0, and the primes minimal over 0 are among the P_i (so there are finitely many minimal primes). In an noetherian ring, the nilradical N satisfies N^m = 0 for some m. In an noetherian ring, every ideal I has finitely many primes minimal over it, and r(I)^m subseteq I for some positive m (apply the previous results to R/I).
• Lecture 4 (4/6/15): Prime avoidance: If an ideal I is contained in a finite union of primes, then I is contained in one of the primes. Extensions and contractions of ideals under a homomorphism \phi: R \to S. Example: for the homomorphism Z to Z[i], the behavior of extensions and contractions of prime ideals reflects number theoretic properties. Modules. Nakayama's Lemma: If M is a finitely generated module and IM = M for some ideal I contained in the Jacobson radical, then M = 0.
• Lecture 5 (4/8/15): Applications of Nakayama's Lemma in case R is a local ring with max ideal m. If M is a finitely generated module over R then the minimal number of generators of M is the dimension of M/mM as vector space over the residue field R/m. If M and N are finitely generated R-modules where R is local, then (M \otimes N) = 0 if and only if M = 0 or N = 0. Introduction to rings of fractions and localization. Given any multiplicative system S in a ring R, there is a localization S^{-1}R consisting of fractions x/s with x in R, s in S, under the equivalence relation x/s \sim y/t if (xt-ys)u = 0 for some u in S. The natural map \phi: R \to S^{-1}R is universal for maps from R to rings T such that every element of S becomes a unit. The kernel of \phi is those x in R such that xs = 0 for some s \in S. Two important special cases are S = all powers of some element f and S = the complement of a prime ideal of R.
• Lecture 6 (4/10/15): Behavior of ideals under a localization map phi: R to S^{-1} R under extension and contraction. If J is an ideal of S^{-1}R, then J = J^{ce}. If I is an ideal of R, then I^{ec} equals the set of elements r such that rs \in I for some s in S. There is a one-to-one correspondence between contracted and extended ideals. Under this correspondence, all prime ideals of S^{-1} R correspond to those prime ideals P of R such that P does not meet S. Topological interpretation: if Z is the image of spec S^{-1} R under the map phi^*: spec S^{-1} R \to spec R, then phi^* induces a homeomorphism between spec S^{-1} R and Z.
• Lecture 7 (4/13/15): Localization and homological algebra. Localization S^{-1}M of an R-module M, which can also be described as S^{-1}R \otimes_R M. Localization is exact, i.e. S^{-1}R is a flat R-module. S^{-1}(M \otimes_R N) is the same as S^{-1}M \otimes_{S^{-1}R} S^{-1}N. A property of a module M is local if M has the property if and only if the localizations M_P (where this means the localization at the set of elements outside P) have it for all primes P of R. Being 0 is local. A homomorphism being injective or surjective is local. Being flat is local.
• Lecture 8 (4/15/15): More on chain conditions. The Hilbert basis theorem. In a short exact sequence 0 to M to N to P to 0, N is noetherian (or artinian) if and only if both M and P are. In a noetherian (or artinian) ring, any finitely generated module M has the same property. If \phi: R to S is a ring homomorphism such that S is a finite R-module, then if R is a noetherian ring then so is S. Subrings of noetherian rings need not be noetherian.
• (4/17/15) (NO CLASS):
• Lecture 9 (4/20/15): Primary ideals. The radical of a primary ideal is prime. Every ideal whose radical is a maximal ideal is primary. Primary decomposition. Irreducible ideals. In a noetherian ring, irreducible ideals are primary, and every ideal is a finite intersection of irreducible ideals; thus primary decompositions exist for every ideal in a noetherian ring.
• Lecture 10 (4/22/15): Reduced primary decomposition. If I has a primary decomposition, its associated primes of I are the primes among the ideals r(I:x) as x varies over elements in R. If R is noetherian then the associated primes are the primes among the ideals (I:x). When R is noetherian the primes associated to I are the same as the primes occurring among the annihilators of elements of the module R/I. The zerodivisors in a noetherian ring R are the union of the primes associated to 0.
• Lecture 11 (4/24/15): In a noetherian ring R, every prime minimal over an ideal I is associated to I. Localization and primary decomposition. The primary components associated to primes minimal over I are uniquely determined. Integral extensions.
• Lecture 12 (4/27/15): Basic characterizations of integral extensions A to B: b in B is integral over A if and only if A[b] is a finitely generated A-module. Other basic theorems on integral extensions. A UFD is integrally closed in its field of fractions. The integral closure of Z in a finite extension K of Q is called the ring of integers O_K. The rings O_{Q(sqrt(d))} are the ones whose factorization theory was studied in the fall.
• Lecture 13 (4/29/15): Prime ideals in integral extensions A to B. A is a field if and only if B is. Every prime P of A has a prime Q of B lying over it. If a prime Q lies over P then Q is maximal if and only if P is. Distinct primes lying over P cannot be comparable. The going up theorem.
• Lecture 14 (5/1/15): Valuation rings. Valuations of a field into an ordered Abelian group and correspondence with valuation rings. Example of the p-adic valuation on Q and the corresponding ring Z_p. Properties of valuations rings: local and integrally closed. Special properties of discrete valuation rings: the maximal ideal is principal and all other nonzero ideals are powers of it.
• Lecture 15 (5/4/15): Algebras. The Artin-Tate lemma. Weak nullstellensatz: If F/k is a field extension which is also a finitely generated k-algebra, then F/k is a finite degree extension. Every max ideal in C[x_1, ..., x_n] is of the form (x_1 -a_1, ..., x_n - a_n).
• Lecture 16 (5/6/15): Brief introduction to the correspondence between closed subsets of affine space A^n and ideals in the polynomial ring C[x_1, ..., x_n]. A polynomial ring K[x_1, ..., x_n] is Jacobson, that is, every radical ideal of the ring is an intersection of maximal ideals. The strong nullstellensatz: If J is an ideal in C[x_1, ..., x_n], the ideal of all functions vanishing on the zero set V(J) of J is the radical r(J) of J.
• Lecture 17 (5/8/15): Artinian rings. An artinian ring has finitely many primes, and they are all maximal. The radical is nilpotent in an Artinian ring. Artinian rings are noetherian. Artinian rings are precisely the noetherian rings of dimension 0. Reduced artinian rings are direct products of finitely many fields.
• Lecture 18 (5/11/15): An Artinian ring is uniquely the product of finitely many local artinian rings (proof using primary decomposition). In a noetherian domain of dimension 1, every nonzero ideal is uniquely a product of primary ideals with distinct radicals. Dedekind domains: noetherian domains of dimension 1 which are integrally closed. Thm: a local Dedekind domain is the same as a DVR (proof omitted). A ring A is a Dedekind domain if and only if the local ring A_m is a DVR for all maximal ideals m of A. Geometric example: points of a curve where the local rings are not DVRs are singular points.
• Lecture 19 (5/13/15): In a Dedekind domain, every primary ideals is a prime power, and thus every ideal is uniquely a product of prime powers. Example: Z[sqrt(-5)] is not a PID but is a Dedekind domain. Definitions of fractional and invertible ideals.
• Lecture 20 (5/15/15): More on fractional ideals and invertible ideals. An A-submodule M of the field of fractions K is invertible if and only if M is a projective module. In a Dedekind domain, all fractional ideals are invertible. The class group of a Dedekind domain: fractional ideals under multiplication, modulo the subgroup of principal fractional ideals. Results on the class group from number theory.
• NO CLASS, qual exam period (weeks 8-9):
• Lecture 21 (6/1/15): Introduction to representations of finite groups G. Examples. The group algebra CG over the complex numbers C. A representation is the same thing as a finite-dimensional CG-module. Semisimple rings and modules.
• Lecture 22 (6/3/15): Maschke's theorem: CG is semisimple for any finite group G. CG is isomorphic to a product of matrix rings over C of sizes n_1, \dots, n_k. There are k simple CG-modules up to isomophism, of dimensions n_1, \dots, n_k. k is also the number of conjugacy classes of G, and |G| = n_1^2 + ... + n_k^2. Example of S_3.
• Lecture 23 (6/5/15): Characters. Characters are class functions (constant on conjugacy classes). The character table of S_3. values of characters are algebraic integers. Orthogonality of characters. Tensor products of representations. Finding the character table of S_4.

Homework

• Homework 1: Homework 1 due Friday April 10

• Homework 2: Homework 2 due Friday April 24

• Homework 3: Homework 3 due Friday May 8

• Homework 4: Homework 4 due Friday June 5