Date
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What we discussed/How we spent our time
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Aug 24
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Syllabus. Grade
based on HW. Motivation for and definitions of: ring,
k-algebra, and modules over either.
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Aug 26
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Algebraic languages.
Equivalent languages. Meanings of "commutative" and "abelian".
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Aug 28
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Abelianness for groups
and rings, normal subgroups and ideals. Solvability=nilpotence for ideals
in associative rings.
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Aug 31
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Properties of ideal multiplication.
Nilpotent and idempotent elements, ideals.
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Sep 2
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More properties of ideal multiplication. Proof that
a commutative ring whose ideal lattice has finite
height has a largest nilpotent ideal (= the Wedderburn radical)
and the quotient modulo this ideal is a finite product of fields.
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Sep 4
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Characterizations of the nilradical and the Jacobson radical.
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Sep 9 and 11
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Commutative rings, frames, topological spaces,
spectra.
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Sep 14
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Algebraic closure operators + additional notes.
Every endomorphism of a f.g. module satisfies its characteristic
polynomial(s).
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Sep 16-18
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Strong and weak versions of Nakayama's Lemma.
Categories, functors, representable functors,
preadditive categories, additive functors.
Notes1
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Sep 21
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Notes2
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Sep 23
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Localization:
Motivation. Definition. Universal property. All ideals
in RS are extended from R.
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Sep 25
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Localization: Bijection between primes of
RS and primes of R that are disjoint from S.
Formula for Iec along
ν: R → RS.
Any ideal maximal
among ideals disjoint from multiplicatively closed S
is prime.
Complements of saturated multiplicatively closed
sets are unions of primes.
RP := R(R-P).
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Sep 28
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Localization:
Modules over localized rings. Isomorphism between
MS and M ⊗R RS
Localization is an exact functor. RS is
a flat R-module.
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Sep 30
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Discussion of sheaves of analytic functions,
including definitions of function element,
germ, sheaf of germs, stalks,
topology on the sheaf of germs.
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Oct 2
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Definition of sections over an open set.
Proof that a continuous section over an
open set is the section induced by
a function that is locally a function element.
Definition of presheaf. Axioms (normalization,
local determination, gluing) defining a
sheaf with values in a concrete category.
Colimits and the construction of stalks.
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Oct 5
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Axioms for B-sheaves. Canonical basis for Spec(R) (= basis of principal
open sets). ``Value at P'' of f ∈ R is the image
of f under the composite map R → RP →
RP/PRP = κ(P) (= the residue field at P).
Zero set of f ∈ R. Support of f ∈ R.
Z(f) and Supp(f) are closed and Z(f) ∪ Supp(f) = Spec(R).
B-sheaf for X=Spec(R) is OX(Xf)=Rf.
Proof that Z(g) ⊆ Z(f) is equivalent to g invertible in Rf,
and consequently that the B-sheaf for Spec(R) is well-defined.
Description of the morphism of schemes Spec(S) → Spec(R) induced
by a ring homomorphism φ : R → S.
Definitions.
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Oct 7
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Primary decomposition: Definition of primary ideal.
Proof that every element
is the meet of meet-irreducibles in a lattice with the ACC.
Proof that meet-irreducible ideals are primary in a Noetherian ring.
Hence, any ideal in a Noetherian ring is a (finite)
meet of primary ideals. Definition of Laskerian ring
and a list of characterizing properties.
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Oct 9
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Primary decomposition:
Identification of primary ideals:
Q is primary iff every zero
divisor in R/Q is nilpotent. The radical of
a primary ideal is prime (= the associated prime).
If the radical of I is maximal, then I
is primary.
Constructing new primary ideals:
A finite intersection of P-primary ideals is P-primary.
If Q is P-primary and x∉ Q, then (Q:x) is P-primary
(and is larger than Q if x∈ P).
First uniqueness theorem: Definition of ``minimal''
primary decomposition. Explanation of why any
ideal in a Noetherian ring has a minimal primary decomposition.
Thm. The associated primes in any minimal decomposition are unique.
Cor. Every radical ideal of a Noetherian ring is representable
as an irredundant intersection
of prime ideals in exactly one way (up to a permutation of factors).
(This Cor. can be obtained from HW2(6)(c) plus one more line
of argument.)
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Oct 12
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Primary decomposition:
We completed the proof of the first uniqueness theorem,
then used an idea from the proof to motivate
the definition of associated primes
for modules (= prime annihilators, P=(0:m)).
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Oct 14
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Associated primes for modules: definition, characterization
via embedding R/p → M, existence of for Noetherian rings,
∪ Ass(M) = zero divisors on M when R is Noetherian.
Behavior of Ass(_) on exact sequences. F.g.
modules over a Noetherian ring have a finite filtration
with factors of the form R/p. Ass(M) is finite when M is f.g.
over a Noetherian ring.
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Oct 19
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Associated primes for modules: Behavior of Ass(_) with respect to
localization. Relationship between Ass(M) and Supp(M) when
M is f.g. over a Noetherian ring. Definition and basic properties
of primary submodules.
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Oct 21
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Associated primes for modules: More basic properties
of primary submodules, including: a submodule of a f.g.
module over a Noetherian ring is primary iff it has a unique
associated prime. Proof of First and Second Uniqueness Theorems for
primary decompositions of modules.
Proof of the Krull Intersection Theorem.
Notes for last 3 lectures.
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Oct 23
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Integral dependence: basic results about
integral extensions and integral closure.
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Oct 26
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Primes in integral extensions: Lying over, Incomparability,
Going Up. Integral closure is preserved under localization.
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Oct 28
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Primes in integral extensions:
Integral closure is a local property of a domain.
Prime avoidance.
Thm. If A is integrally closed in K, L is a normal extension of K,
B is the integeral closure of A in L, and P is a prime of A,
then Gal(L/K) acts
transitively on the primes of B lying over P.
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Oct 30
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Going Down. Noether Normalization Lemma.
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Nov 2
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Artinian rings: Artin Structure Theorem (a ring is Artinian iff
it is a finite product of Artinian local rings). A local ring
is Artinian iff it is Noetherian and has a nilpotent maximal ideal.
An arbitrary ring is Artinian
iff it is Noetherian of Krull dimension 0. Every ring is a subdirect product
of subdirectly irreducible rings. A subdirectly irreducible
ring is Noetherian iff it is Artinian.
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Nov 4
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Noetherian rings: Hilbert Basis Theorem.
Artin-Tate Lemma.
Field-theoretic version of the Nullstellensatz:
If k is a field, then any field that is an affine algebra over k
is a finite field extension of k.
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Nov 6
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We derived the Weak Nullstellensatz from the Field-Theoretic Form
of the Nullstellensatz, and then derived the Strong Nullstellensatz
from the Weak Nullstellensatz. One corollary obtained was the classification
of maximal ideals in polynomial rings over algebraically closed fields.
A second corollary was the fact that the Jacobson radical of an
ideal equals its nilradical in a polynomial ring over an algebraically
closed field.
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Nov 9
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Valuation rings: Definition. Valuation rings are the
domains whose ideal lattice is a chain.
Valuation rings are integrally closed local rings.
Given a field K and an algebraically closed field L
we can order pairs (R,φ) where R is a subring of K
and φ is a homomorphism into L by extension.
Under this ordering, if (R,φ) is maximal, then
R is a valuation ring for K and ker(φ) is its
maximal ideal.
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Nov 11
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Valuation rings: If R is a domain and K is its field of fractions,
then the intersection of the valuation rings on K that contain
R is the integral closure of R in K. Definition of the domination
order on local subrings of a field. Valuation rings on K are the
maximal local subrings in the domination order. Totally ordered groups:
Definition and examples. Order is determined by
the positive cone. Archimedean totally ordered
groups are those that embed in the reals. There exist nonarchimedean
totally ordered groups.
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Nov 13
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Valuation rings: The totally ordered group associated
to a valuation ring. A valuation ring R on a field K
defines a linear order on K*/R*. R-submodules
of K are linearly ordered. Valuation rings are
Bezout domains, hence are Noetherian iff PID.
Definition of Krull valuation and of the
canonical valuation of a valuation ring.
Valuation rings correspond to valuations.
Primes in valuation rings correspond to convex subgroups
of the value group.
A valuation ring has Krull dimension 1 iff its value group
is Archimedean. Definition of discrete valuation ring (DVR).
Proof that a valuation ring is a DVR iff it is Noetherian
iff it is a PID.
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Nov 16
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Characterization theorem for DVR's:
Thm. If R is an integral domain, TFAE:
(1) R is a DVR.
(2) R is a local PID.
(3) R is a local ring that is
(a) Noetherian,
(b) integrally closed, and
(c) of Krull dimension 1.
Item (b) can be replaced by (b)' has
a principal maximal ideal, or by
(b)'' has the property every ideal is a power of the maximal ideal.
Definition of Dedekind domain.
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Nov 18
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Characterization theorem for Dedekind domains, part 1:
Integral, invertible and fractional R-ideals. Principal ideals
are invertible.
If a product is invertible, then each factor is.
Invertible ideals are finitely generated.
Invertible ideals are fractional.
Big Thm. If R is a domain, TFAE:
(1) R is a Dedekind domain.
(2) Every nonzero (fractional) ideal of R is invertible.
(3) R is Noetherian and every maximal ideal is invertible.
(4) Every nonzero ideal of R factors uniquely as a product of
maximal ideals.
When these conditions hold, then R has Krull dimension 1.
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Nov 20
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Characterization theorem for Dedekind domains, part 2:
We completed the proof of the Big Thm. Then we proved
Thm. If R is a Noetherian domain of Krull dimension 1, TFAE:
(1) R is integrally closed.
(2) RP is a DVR for all nonzero primes P.
(3) Every P-primary ideal is a P-power.
and
Thm. If R is a Noetherian domain of Krull dimension 1, then
R is a Dedekind domain iff R is integrally closed.
Together with the Big Thm we get
Thm. If R is a domain, then R is a Dedekind domain iff it is:
(a) Noetherian,
(b) integrally closed, and
(c) of Krull dimension 1.
This imples that a Noetherian domain is a Dedekind domain iff it is
locally a DVR.
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Nov 30
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Completions: Discussion indicating that a local ring
at a point on a variety has some `global' information.
Completions allow us to look at very local information.
Background information about topologizing an algebra,
including a discussion of
the congruence metric and congruence uniformities.
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Dec 2
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Completions: the I-adic topology on a ring or module,
separated topologies. The I-adic topology is separated
iff it is
T0 iff it is
T2 iff it is a metric topology.
The ring and module operations are continuous w.r.t.
the I-adic topology. Definition of Cauchy sequence
(and equivalence of C-sequences)
for a metric or uniformity. Definition of the completion
of a ring in the I-adic topology in terms of C-sequences.
The completion of R belongs to HSP(R). There is a natural map from
R into its completion.
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Dec 4
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We discussed the construction of the completion
by inverse limits. This shows that the completion
of R lies in SPH(R), a stronger result that the earlier
result that it lies in HSP(R). We showed that
the I-adic topology on R is the restriction of a metric
topology on its completion.
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Dec 7
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We proved that the completion of R with respect to the
I-adic topology is complete (i.e., all C-sequences converge).
We discussed the statement of Hensel's Lemma. We derived
the statement
that any finite local ring contains a Galois subring with
the same residue field. We described a version of the
implicit function theorem valid over any field, which
can be derived from Hensel's Lemma.
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Dec 9
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Proof of Hensel's Lemma. Some elementary dimension theory:
any prime contained in proper principal ideal
has codimension zero.
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Dec 11
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Definition of symbolic powers of primes:
The nth symbolic power of P is
P(n) = (Pn)ec
along the canonical map R → RP.
It is the least P-primary ideal containing Pn.
P(n) and Pn have the same localization at P,
it is (PP)n.
More dimension theory:
Krull's Principal Ideal Theorem. First version proved shows that
in a Noetherian ring any prime minimal over a principal ideal
has codimension ≤ 1. Second version proved shows that
in a Noetherian ring any prime minimal over an n-generated ideal
has codimension ≤ n.
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